Modern Algebra - Structure and Method Book One [1]
The best high school algebra book ever written. This is an older SMSG text, School Mathematics Study Group, produced a
728 125 55MB
English Pages 610 Year 1965
Polecaj historie
![Withcraft Today: Book One - The Modern Craft Movement [1]
0875423779](https://dokumen.pub/img/200x200/withcraft-today-book-one-the-modern-craft-movement-1-0875423779.jpg)
![Modern Algebra - Structure and Method Book One [1]](https://dokumen.pub/img/200x200/modern-algebra-structure-and-method-book-one-1.jpg)
Table of contents :
1. Symbols and Sets
2. Variables and Open Sentences
3. Axioms, Equations, and Problem Solving
4. The Negative Numbers
5. Equations, Inequalities, and Problem solving
6. Working with Polynomials
7. Special Products and Factoring
8. Working with Fractions
9. Graphs
10. Sentences in Two Variables
11. The Real Numbers
12. Function and Variations
13. Quadratic Equations and Inequalities
14. Geometry and Trigonometry
15. Comprehensive Review and Tests
Citation preview
X*e ifLXX^ MiJjiJM
^'iLlLLi
(1
.
.
RAPID CITY.
PUBLIC SCHOOLS
TEXT
BOOK
S. D.
LIBRARY Year
NAME
Semester
DIRECTIONS Also year and semester.
Nothing
else.
Erase
Write your name marks before returning for refund. table; let the Hold the book with its back on a smooth or cohered the leaves In one hand while front board down, then the other, holding at the front and so on you open a few leaves at the back then a few pressing open the sectioris till alternately opening back and front, gently three times and you yow reach the center of the volume. Do this two or Open the volume violently or carelessly in will obtain the best results. and cause a start in any one place and you will likely break the back the leaves. Never force the back of a book. in
ink.
all
Form No. 240P-269
704:0^:
V
7^
Digitized by the Internet Archive in
2010
http://www.archive.org/details/modernalgebrabooOOdolc
}
o
MODERN ALGEBRA Structure and Method: Book 1
COVER These thousands of
units in a computer.
memory
wires, are
An
pendence on mathematics. magnetic
field
a
1,
+
electrical
which magnetizes the
reverses the magnetic field,
or
doughnut-shaped, ferromagnetic rings, threaded on
tiny,
or
—
,
a
>'e5
They are a symbol of modern man's decurrent passing along the wire sets up a
rings.
Since current in the opposite direction
the direction of the magnetic
or no condition.
storing information in the binary
Knowledge of algebra has
This
is
number system used
made
field
may
the electronic in
represent a
mechanism
for
most modern computers.
possible these computers which initially were
and engineering problems. Today, electronic data processthey help to ing systems are. invaluable in business, industry, and research, where work that untangle and to simplify, in a matter of seconds, calculations and paper built to
handle
scientific
formerly took days to do.
TITLE PAGE The
illustration
on the
title
pages indicates that future vocational plans are
dependent upon your high school preparation. chemistry, medicine?
Do
you
Are you
interested in business,
look forward to being an engineer, an architect, a
of your economist, a machinist, a housewife, or a psychologist? Regardless More imvocational plans, algebra is essential to the modern educated person.
home
portant that
is
the fact that algebra
do not even
exist today.
is
essential to
Algebra
your place as an educated person
is
in the
many
future vocational opportunities
equipment you
will
need
if
you are
to take
modern world of today and tomorrow.
MODERI*
MARY
DOLCIANI
P.
SIMON
L.
BERMAN
JULIUS FREILICH EDITORIAL ADVISER
ALBERT
E.
MEDER,
Jr.
^
,
GEBRA STRUCTURE
AND METHOD BOOK ONE
Houghton NEW YORK
.
Mifflin
ATLANTA
•
Company
GENEVA,
ILL.
•
DALLAS
Boston
•
•
PALO ALTO
ABOUT THE AUTHORS
Mary
P. Dolciani, Professor of Mathematics, Hunter
Dr. Dolciani has been a director
and teacher
State Education
in
member
New
College,
of the School Mathematics Study
Group and
numerous National Science Foundation and
Department
institutes
for mathematics
York.
teachers,
a
New York
and
visiting
secondary school lecturer for the Mathematical Association of America.
Simon L. Berman, School,
New
chairman. Department of Mathematics, Stuyvesant High
York, and formerly instructor
in
mathematics
at
Brooklyn Poly-
technic Institute.
Julius Freilich,
Principal,
Floyd Bennett School, formerly chairman of the
mathematics department of Brooklyn Technical High School and instructor
Brooklyn Polytechnic
at
Institute.
EDITORIAL ADVISER Albert E. Meder,
Jr.,
Dean and vice
Provost, Rutgers University. Dr.
Meder
was executive director of the Commission on Mathematics, College Entrance Examination Board, and
is
COPYRIGHT
®
an advisory committee member of the
1965, 1962
SMSG.
BY HOUGHTON MIFFLIN COMPANY
ALL RIGHTS RESERVED INCLUDING THE RIGHT TO REPRODUCE THIS BOOK
OR PARTS THEREOF
IN
ANY FORM. PRINTED
IN
THE
U.S.A.
CONTENTS
I
Symbols and NUMBERS AND
Sets
The Sign of Equality, Inequahty, 7
1-1 Representing Num1*1-2 Comparing Numbers:
THEIR RELATIONSHIPS
bers on a Line: Order Relations, 5
•
•
1-3 Comparing Numbers: The Signs of 1-4 IN SETS AND SUBSETS
GROUPING NUMBERS
•
•
Meaning of Membership in a Set, 10 1-5 Kinds of Sets, 13 1-6 The Graph of a Set, 16 1-7 How Subsets Relate to Sets, 18 USING NUMBERS IN ONE OR MORE OPERATIONS 1-8 Punc1-9 Order of Operations, 23 tuation Marks in Algebra, 19 •
•
•
•
•
•
•
THE
HUMAN EQUATION,
CHAPTER
TEST,
30
EXPERTS,
•
27
•
CHAPTER
SURVEYORS
JUST FOR FUN, 33
Variables and
•
25
CHAPTER SUMMARY, 26 28
REVIEW,
AND
•
EXTRA
MATHEMATICS,
•
FOR 32
•
•
Open Sentences
ANALYZING ALGEBRAIC STATEMENTS
•
35 2-1
Evaluating
Alge-
2-2 Identifying 2-3 Solving Open Factors, Coefficients, and Exponents, 40 Sentences, 44 PROBLEMS SOLVED WITH VARIABLES 2-4 Think2-5 Thinking ing with Variables: From Symbols to Words, 49 2-6 Solving Probwith Variables: From Words to Symbols, 51 braic
Expressions
Containing Variables,
35
•
•
•
•
•
•
lems with Open Sentences, 56
EXTRA FOR EXPERTS, 60
CHAPTER
THE
•
HUMAN EQUATION,
SUMMARY, 63 CHAPTER JUST FOR FUN, 67 •
REVIEW, 65
w
•
'
TEST,
64
62
•
CHAPTER
•
•
Axioms, Equations, and Problem Solving AND USING NUMBER AXIOMS
IDENTIFYING
3-1 Axioms of 3-2 The Closure Properties, 70 3-3 Commutative and Associative Properties of Arithmetic Numbers, 73 3-4 The Equality, 69
•
69
•
•
•
O6
3 means "five is greater than three." The symbol < stands for the words "is less than." When you write 3 < 5 you say, "three is less than five." The statements 5 > 3 and 3 < 5 both give the same information: 5 is a larger number than 3, and the graph of 3 is to the left of the graph of 5, or the graph of 5 is to the right of the graph of 3.
To avoid
confusing the symbols
< and >
,
think of them as arrow-
heads always pointing to the numeral for the smaller number. example, 39
-
are true statements,
33
66
X
U
For
8
SYMBOLS 9.
18
AND
+
S€TS
=
3
"S*.
X
6
=
number of
17.
50
18.
47
19.
8
X
6
>
40
20.
i
+
t
>
1
21.
12
+
22.
^
=
1
AND
SYMBOLS
For example,
set.
V
S^TS
algebra teacher
all
the teachers in your school
member
a
is
,,
ever, objects such as the letter
r,
are not elements in the set of
9,
form a
set,
or element belonging to that
the school custodian, and the
your teachers. Thus, a
all
collection of objects so well described that
or not an object belongs to the
and your
set.
you can always
How-
number
set is
tell
any
whether
set.
whole numbers. Use a capital You have no way of telling is or is not an element of R until the five whole whether the numbers are specified. If you specify the set by listing the objects forming the set within braces { } then you may have
Suppose a
letter,
set is
formed of any
name number 3
say R, to
five
or refer to the
set.
,
R = This says, "i?
is
{0,3,7,8, 14}
numbers
the set of
0, 3, 7, 8, 14."
member of
You
can easily see
and that the number 4 is not. We use a special symbol, e, to mean "is an element of," and ^ to mean "is not an element of." Thus, 3 ^ R and 4 ^ R. Specifying a set by listing its elements in braces gives you a roster or list of the set. The objects named in the listing, {our moon, the Constitution, the Alamo, California, Albert Einstein}, form a set. Note that the elements of a set need have no relation with one anthat the
number
3
is
a
this set
Furthermore, the order of
other other than being listed together. listing the
elements
element be named
Often a roster
is
is
important
is
that each
in the fisting. is
an inconvenient way of specifying a
ample, a roster of the all
What
unimportant.
set
set.
of states in the U.S.A. requires the This inconvenience
50 of the elements within braces.
is
For exlisting
of
overcome
by writing within braces a rule which describes the elements of a
set.
Thus, {the states of the U.S.A.}
says,
"The
EXAMPLE.
set
of the states of the U.S.A."
Specify the set of numbers
1, 2, 3, 4, 5, 6, 7, 8,
(b) rule.
Solution:
(a)
{1, 2, 3, 4, 5, 6, 7, 8,
(b)
{the whole {one-digit
9}
numbers between
and 10} or
numbers except 0} or
{whole numbers from 1 to
9, inclusive}
9 by
(a) roster,
12
CHAPTER
ONE
ORAL EXERCISES Specify each of the following sets by a
SAMPLE
1.
What you
say:
{the letters in the
[i,
m,
roster.
word
p, s}
word freshman}
1
{the letters in the
2.
{the letters in
your given name]
3.
{the numerals
on the face of a clock}
4.
{the
whole numbers
5.
{students in your
6.
{states of the
Mississippi}
less
row
than 20}
in the algebra class}
U.S.A. on the Gulf of Mexico}
Specify each of the following sets by a
SAMPLE
2
/i i i
What you
say:
{every fraction inator
rule.
J_\
is
whose numerator
an odd number
less
is
and whose denom-
1
than 8}
7.
{Alaska, Hawaii}
17.
{a, e,
8.
{Cahfornia, Washington, Oregon}
18.
{Saturday, Sunday}
4}
19.
{20, 10, 5, 15}
6}
20.
{i.
21.
{Washington (D.C.)}
22.
{London, Paris}
9.
{2, 3,
10.
{2, 4,
n.
{Eisenhower, Truman}
12.
{Dolciani, Freilich,
13.
••1
14. •
3
5
Berman}
7^-
2' 3' 4» 5)
Indiana, Idaho, Iowa}
15.
{Illinois,
16.
{Los Angeles, San Francisco}
Tell
whether or not each statement
27.
5
28.
5
29. 30.
E
[whole numbers
is
true.
o,
1,
i^'
u}
4' lOJ
23.
{-V,
24.
{Jefferson Davis}
25.
{I,
26.
{16,
Give a reason
G
ir, z,
y}
7,4, 13, 10} 1,
for
11, 6,
21}
each answer.
{multiples of ^}
than 5}
31.
I
^ {15, 20, 25} 3 ^ {whole numbers less than 5} 8 G {0, 8, 9}
32.
12
33.
i
^ {even numbers} G {.25, .5, .75}
34.
7
g
less
{1,9, 12,
7, 21,
15}
AND
SYMBOLS
1-5
V
SETS
13
Kinds of Sets In counting the
number of eggs
— you
one, two, three, four
basket in Figure 1-2
in the
each
really pair
egg with a number as shown, and conclude that there are as bers in
numbers
eggs as there are
num-
This pairing of eggs with
Two
a one-to-one correspondence.
is
one-to-one correspondence when set has one partner in the
sets are in
each
many
{1, 2, 3, 4}.
0000
member of one
other
set,
and no element
in either set is with-
out a partner. The pairing of point and ber on a
number
line
num-
another example of
is
t
t
I
t
one-to-one correspondence.
Can you
members of
all the
list
of whole numbers?
If
you
the set
start to write
Figure 1-2
{0, 1, 2, 3, 4, 5, 6,
you
will
never
come
to the
6 are the mathematician's
A
end of the
way of
.
.
.}
The
list.
three dots after the
indicating that the roster continues
which has so many elements that the process of counting them would never come to an end is called an infinite set. For example, you cannot list the members of without end.
set
{all the fractions
between
and 1}
although the rule enables you to identify them. Another
on a
the set of points is
not always an
Waikiki}
A
is
line.
A
infinite set.
not an
containing a large
Thus,
infinite set,
or has a
set is finite,
set
of sand on the beach at
[the grains
even though
finite
number of
it
has
=
In this example, the three dots
Can
between 8 and
empty the
{10, 11, 12, ...
9.
This
set or null set.
empty
set.
It
set
,
mean and so on
a set have no elements?
many members.
elements,
counting the elements comes to an end. Such a {two-digit numbers}
infinite set is
number of elements
Consider the
if
the process of
set is
99}.
through. set
of whole numbers
contains no elements and
is
Notice that this symbol {0} does not designate number 0. Empty braces { } might be
contains the
used, but a special symbol 0, written without braces, usually
designate the null
called the
set.
is
used to
14
ONE
CHAPTER
The notion of the empty set may seem strange at first. Still, how you reached into your pocket or purse to find it empty of coins? The set of coins in your pocket or purse was the empty set. By agreement there is only one null set. Thus, the set of whole numbers between 8 and 9 and the set of coins contained in your empty purse or pocket are one and the same set. often have
ORAL EXERCISES Use a roster
to specify
SAMPLE. What you 1
{living
2.
each of
now 2000
{Persons
The empty
say:
ihe following sets.
years of age}
set, 0.
dogs with wings}
5.
{unit fractions}
{even numbers}
6.
{three-digit
3.
{multiples of 7}
7.
{multiples of 12 less than 200}
4.
{letters
8.
{leap years
Tell
be
of the alphabet}
whether the members of the given in
sets
may be
numbers}
from 1900
2000}
paired so that the sets
wil
one-to-one correspondence.
and
9.
{a, b, c]
10.
{0,
1, 3,
11.
{vowels} and
12.
{A, tt}
and
a}
13.
{1, 3,
14.
{c, b,
5} and
5}
{a, e, u, o, i}
{vr,
A}
and {0} i i} and {i
15.
{2, 4, 6,
16.
{C, A,
8}
{.5, .3,
and
{f,
Give a
roster for
each
set,
and
state whether
SAMPLE,
{multiples of 4 between
Solution:
{4, 8, 12, 16}, finite
it
is
finite.
and 17}
1.
{the
whole numbers between 7 and 10}
2.
{the
whole numbers between
3.
{the vowels in the
4.
{all
5.
{U.S. cities each with populations greater than 20 million}
and
5,
.25}
J#,
^,
T) and {K, A, T]
WRITTEN EXERCISES
^1
to
inclusive}
word ^e/^}
five-headed people}
4}
AND
SYMBOLS
V
SETS
15 and 10}
odd numbers between
6.
the
7.
the even
8.
the
9.
the whole
numbers
less
than 20 and greater than 9}
months of the year which have fewer than 30 days} numbers greater than
but
the even
n.
the multiples of 3 between 3
12.
the multiples of 25 greater than 0}
13.
the leap years after 1960}
14.
the multiples of 20 between 31
15.
every
and
15, inclusive}
and 53}
whose denominator and whose numerator is
fraction
(1, 2, 3}
all
the
17.
all
the multiples of 2 greater than 0}
18.
the United States Presidents
19.
all
who have
whole numbers all
whole numbers between 7 and 8}
22.
all
numbers between
less
than 19
3
and
find a designation for
less
than 40}
which are squares}
21.
II
number chosen
more than 4 terms}
served
odd whole numbers whose squares are
20.
Column
even
an
is
1}
odd numbers}
16.
In
than 1776}
numbers between 200 and 1000}
10.
om
less
8 that divide 13 exactly}
each
set in
Column
I.
II
23.
{x, y, z]
(whole numbers between
24.
(0, 1}
(odd numbers between
25.
{1,9,25,...}
26.
{0, 4, 16, 36,
anti 2}
and
7,
inclusive} {all
.
1
1
.
multiples of 7}
.}
{numbers each of which equals 27.
its
square}
28.
0}
29.
1,3,5,7}
30.
2, 4, 6,
31.
0, 7, 14, 21,
32.
45}
{the
sum of 24 and
{z, y,
8}
12}
x}
{zero} .
..}
33.
36}
34.
the digits in the
numeral for the product of 9 and 3}
{squares of odd numbers}
{even numbers between 2 and
8,
inclusive}
{squares of even numbers} {the product of 15 {the digits in the
sum of
68 and 4}
and 3} numeral for the
16
CHAPTER
1-6
ONE
The Graph of a Set
Another way of specifying a set of numbers is by showing the numbers as points on the number Hne. The set of points corresponding to a set of numbers is called the graph of the set.
EXAMPLES. Graph
Set (1, 2,
{the
3}
numbers between
and
1
{the
3, including
3}
numbers greater
than 3}
A
Note:
darkened
A
in a set.
circle • represents a point corresponding to a number darkened line is used to show that all points on it
^
belong to the graph. Points not belonging to the graph are indicated
by open
o
circles
or appear on undarkened
lines.
A
darkened arrow
indicates that the graph continues indefinitely.
Specifying a 1
^>
2.
set: identifying its
elements by
— the method — describing graphic method — the roster
method
the elements, or
listing
rule
the elements, or
locating the elements on the
3. the
number
line.
ORAL EXERCISES
QPAGJRBMCHW 123456789
Specify the graph of each set by referring to the number
line
below.
^
I
\
\
\
\
\
\
\
\
\
10
SAMPLE What you
{the even
say:
J,
whole numbers between
and
9,
inclusive}
B, C.
1.
{8,2,5}
2.
{1, 8,
7.
{even whole numbers between 3 and 7}
10}
3
3.
{7,3,4}
4.
5.
{0}
6.
{4, 2,
7}
17
SYMBOLS AND SETS \i. 8.
{odd whole numbers between 3 and 7}
9.
{even whole numbers between 5 and 6}
{whole numbers greater than 2 but
10. 1
{whole numbers
1.
less
less
than 3}
than or equal to 2} 1
and
1
and 5}
12.
[whole numbers between
13.
{whole numbers between
14.
{multiples of 2 between
15.
{multiples of 5 between 4 and 6}
16.
{whole numbers between
and
6,
inclusive}
1}
and 9}
1
WRITTEN EXERCISES Draw
the
graph of each given
SAMPLE
set.
1.
{the
whole numbers between
2.
[the
numbers greater than 2^}
3
and
5,
inclusive}
Solution:
SAMPLE Solution:
1
1
the whole
numbers between
2.
the whole
numbers
less
than 5}
3.
the whole
numbers
less
than or equal to 5}
4.
the
numbers greater than
5.
the
numbers
6.
the
numbers greater than 7^}
7.
the
numbers greater than 3^ but
less
inclusive}
5}
8.
the
numbers between
the
numbers between 4^ and
10.
the
numbers
11.
the whole
numbers
12.
the whole
numbers between
13.
3,9,2}
14.
3^
2/
8,
than 5}
9.
less
and
2
3
and
6,
less
than 10}
including 6}
5,
including 4^}
than 6 and greater than or equal to 2} less
15. 16.
than 8 and greater than 3} 3
and 4}
[0}
17.
[2,4,6}
{f}
18.
{1,2,3,4,5,
9}
18
1-7
How
^ =
of
new
Subsets Relate to Sets
Suppose you form another set by removing one of the elements For example, remove the element 1, and form a {0, 1,2}.
set,
M= M
Notice that every element that
M
in set
is
also an element in set R.
Whenever a
a subset of R.
is
{0, 2}.
such as
set,
elements which are also elements of another is
A
said to be a subset of set R.
contain the
all
set,
the elements of the given set
We
how many
set
M
M, which does not
called a proper subset of
is
a proper subset of
is
V-
^fl
R
you can find. By removing either one you can form six proper subsets of R:
subsets of
R
or two elements from
-''0
^0^
2^
'1
I''-
•'^l^-
Notice again that every element in each subset R, and that none of these subsets contains
When you remove elements, the empty
except of
set,
all
You
see that
the elements of R.
is
set
with no
a proper subset of every
R
is
formed when you remove no elements.
full set
also a subset of R, but
that every set
it is
a subset of
is
The notion of
set
matics the notion
you
all
also an element of the
itself.
'0
that
is
'2^
you obtain the
the elements of R,
set, 0.
Another subset of Thus, the
is
1
2^
called an improper subset.
You
can see
itself.
appears everywhere in
human
thought; in mathe-
consciously developed as a basic, unifying idea
meet over and over again as you increase your mathe-
will
matical knowledge.
ORAL EXERCISES Tell
say
contains only
teachers}
See
is
M,
such as R, the
subset, such as
Thus, (junior high school teachers}
set.
(all
set
ONE
CHAPTER
which statements are true and which are false. Justify your answers.
1.
{1, 3,
2.
[0,
3.
{the
2}
6} is
is
a subset of
a subset of {1,
New
England
(7, 6, 5, 4, 3, 2, 1} 3, 2,
states}
is
4} a subset of {the states of the U.S.A.}
SYMBOLS
AND
19
V
SETS
(women)
4.
(red-haired people}
5.
(high school students}
6.
(people studying algebra}
7.
(0, 1, 2, 3,
.
.
,
8.
(1, 3, 5, 7, 9,
.
.
9.
(5}
is
a subset of (2, 3}
11.
10.
(0}
is
not a subset of (10,
12.
For each set,
13.
list
.
9}
is
is
.} is
'not a subset of is
a subset of (people studying algebra} is
a subset of (people studying mathematics}
not a subset of a subset of
6,
(all
the digits}
(1, 2, 3, 4, 5,
18}
the largest subset of (a)
.
(0}
odd numbers,
.}
.
is
is
a subset of (0}
not a subset of (0}
(b)
even numbers.
20
ONE
CHAPTER
Mary
wrote:
"Paul," said the teacher, "is very
intelligent."
Henry wrote Paul
said,
"The
teacher
is
very intelligent."
Both Mary and Henry had produced correct and meaningful sentences. The differences in punctuation, however, had produced a world of difference in meaning. Without punctuation marks the original statement could be interpreted in more than one way. In mathematics you eliminate statements which could be interpreted in more than one way by using mathematical punctuation marks, much as you use punctuation marks in Enghsh composition. For example, what number is represented by the set of symbols
3x2 + Is
it
10 or
18?
is it
An
4?
expression such as 3
X
+
2
4 could be called
ambiguous (am-big-u-us) because of the different interpretations. In mathematics, one way you avoid ambiguous statements using parentheses.
When you
punctuate 3
(3X2) + you mean
10.
you mean
18.
When you
pair of parentheses
2
+
by
is
4 as follows
4,
write
3
A
X
is
X
+
(2
called a
4),
symbol of inclusion because
it is
used to enclose, or include, an expression for a particular number.
The parentheses
+ 4) serve to symbol + and to in-
in the numerical expression 3
X
(2
group the numerals 2 and 4 together with the dicate that the sum of 2 and 4 is to be multiplied by 3. In writing 3 X (2 + 4), it is customary to omit the symbol
X
and
to write simply 3(2
+
4).
3X6 may be expressed in any of the forms
Similarly, the product 3(6),
(3)6,
or
(3)(6).
Brackets, braces, and a bar are used for the same purpose:
Parentheses 3(2
+
4)
Brackets 3[2
+
4]
Braces 3:2
+
4}
Bar 3 2
+
4
SYMBOLS AND SETS
21
-V
In working with fractions you have seen that the bar acts as a division sign, as well as a symbol of inclusion. For example, in the expression below, the bar groups the 16 and 4;
The bar tells you number (4 — 1).
that the
number
16-4
4-1 Each part of before
it,
this
12
~
J
it
—
(16
also groups the 4
4) is to be divided
=
and 1. by the
4
statement designates the same number as the one
but as you carry out the operations in the indicated order
becomes simpler.
the numeral
In simplifying an expression,
you use
the signs of grouping to determine the order of operation.
When you
+
X
see
a grouping inside of another grouping, such as
you always simplify the numeral in the innermost symbol of inclusion and proceed to work toward the outermost group5[3
(7
ing until
all
2)],
symbols of inclusion are removed, thus,
5[3
+
(7
X
2)]
=
5[3
=
5[17]
=
85.
+
(14)]
ORAL EXERCISES Simplify each of the following expressions.
SAMPLE.
1.
(8
^
2)
+
7
What you say:
(8
--2)
+
7
=
4
+
7
=
11
22
CHAPTER
WRITTEN EXERCISES Simplify each of the following expressions.
Q
SAMPLE.
[(14
X
2)
+
5]
-
11
Solution:
[(14
X
2)
+
5]
-
11
1.
2.
3.
5
X
[5
+
(2
X
3)]
X
[5
-
(0
X
3)]
= = =
[28 [33]
3
+ H-
5]
11
-
11
ONE
5
+ 3x2
24
CHAPTER
WRITTEN EXERCISES Simplify each of the following expressions.
SAMPLE.
II
+
2(6
+
4)
-
3(1
+
3)
Solution:
11
+
2(6
+
4)
-
3(1
+
3)
Step 2
+ 2(10) - 3(4) 11+20-12
Step 3
31
-
19,
Answer.
Step 1
Q
11
1.
(7
+
3
+
2) H- 3
2.
(7
+
3
+
2)
3.
5(7
4.
21
5.
+ +
9)
-
5(7
+
4
3)
+
(3
+ -
12
+
3
20
1
1)
ONE
THE
HUMAN
-V
EQUATION
A
A
casual visitor to the Tower of London
an unexpected for their use, a
sight.
In
the midst of
group of men,
would congregate
all
this
friends
in
year
the
1
Reserved Table
606 might have witnessed
infamous prison, at a table reserved
and guests of one of The host of
to discuss mathematics.
the prison's inmates,
this
was
unusual party
no lesser personage than the Earl of Northumberland. The leading figure
the
in
was an accomplished astronomer and mathematician, Thomas Harriot. Harriot had come to his place at the Earl's table in the Tower by way of an eventful life. Born in 1560, he was caught up in the spirit of vigor and creativity His career began which pervaded England during the reign of Elizabeth discussions
I.
with studies at Oxford, in
mathematics.
with the second
and soon
after,
he served as
was Raleigh who appointed
It
expedition to Virginia.
Walter Raleigh's
Sir
After returning to England
mathematical studies, Harriot was awarded a
it was that Earlcame into disfavor with the Crown and was imprisoned in was among the honored guests at his table. Although Harriot's last years were beset by
=
in
1
606, when the
The Jrte )
to
otflmtc it onclr fnto
tlDOO partes. OTbetcoftbcfiiftf
l«,
tthcnontnomberis
tquille yntt »ni ither. 3nO tl)C fccoiiDc Is ,>j>r» tnt htr is omptrtd ts e^OAlle intt.Ttithcrntml/ers,
The use of the sign
for equality, though introduced
his
the Tower, Harriot
80 tlietc foo;kC0 ooe ertmoc
cancer, he continued to demonstrate remarktalents.
and
pension by the Earl of North-
life
umberland, himself an amateur mathematician. So
able mathematical
tutor
Harriot to the office of surveyor
by another
ntm>
aiiuaica iDiUpng pou to rcmr bcc, that pou reOnce
partly due to Harriot,
^outnombtta , to ttjcir Icadc Ocnomtnations , ano (maUdte fo;me0>befo;te pou p:oreDc anp fartber.
who helped persuade other mathematicians of the day to adopt this notation. To Harriot alone, moreover, we owe two of the most useful mathematical notations, the symbols > and , ?> -Z-: 2.35
?
2.7
>
10
.245
1-4
1
-5
Make
a roster for each of the following
15.
{the multiples of 6 between 5
16.
(the three-digit numerals}
sets:
and 26}
Identify the following sets as being finite, infinite, or the
17.
{0, 1, 2,
18.
{4,6,8}
.
.
.}
19.
empty
{whole numbers between
set:
and
1}
23 1
-6
1-7
graph of each of the following
Draw
the
20.
{the
whole numbers between
21.
[the
numbers between
22.
{the
numbers greater than 1^}
23.
List that subset of A, all
1-8
24.
Simplify: [60
1-9
25.
Simplify:
Befor€%ou Go on
10
4, inclusive}
and 4}
A =
where
{2, 4, 6, 8}, that consists
of
that are muhiples of 4.
A
the elements of
1
sets:
and
1
ONE
CHAPTER
"^r,
-
(3
X
2)]
+
15
^
5
-
10
^ X
Simplify: 25
26.
2
-
7
X
7
Chapter 2
to
Did you miss any of
the test items?
If so,
note the section
number
Restudy that section in the that corresponds to each item you missed. Review, and do chapter. Then find the section number in the Chapter the exercises under
it.
Did you get all the items correct? and enjoy the Extra for Experts.
If so,
you may turn
to
page 30
Chapter Review Pages 1-5
Representing Numbers on a Line: Order Relations
1-1
1
A
2.
The
3.
numeral
On
a
is
name
for a
starting point of the
a
number
line,
_!
number
line is labeled
arithmetic numbers appear
Exercises 4-7 refer to the following number
',
5
3
7
4
2
4
,
1
3
4
2
4
4.
To
5.
The
6.
The coordinate of
7.
label
^
2 ^
?
5
^
4
2
4
its
3
-^
^
distance between the points labeled f and | and _! the distance between the points labeled
B
^4
5 2
4
_1_ from
the point halfway between
E
F
D
any point, you must know
of _J
line:
C
B
A
_J
in order
4
zero. is
and
the
same
F is
as
_J_.
between The points that are the graphs of the whole numbers ^_. and D, C, are 4, 1 and 4, including
1
SYMBOLS
1-2
AND
Comparing Numbers: The Sign 8.
Any name or
9.
10.
When two
for equals or
The symbol
equal to
is
>
"7
7
"Five
xy,
and
.
The expression 3«
In Ixy
+
3(;c
—
y)
is
-\
a term, and there
1
y),
and
y
-
X
ORAL EXERCISES If
a, b,
and x have the values
of the following; also, 1.
tell
15, 3,
how many
and
2, respectively, tell the
terms each contains.
value of each
38
^^
CHAPTER
10.
(vv
—
1 1
(2h'
-
12.
(3r
+
4h'
+ ~
v)
—
V
14.
13.
0(2h' t)(3r
—
3r
+
+
r)
/)
TWO
m
AND
VARIABLES
39
OPEN SENTENCES
2C 9.
Mechanical advantage of a
=
c
10.
differential pulley:
Y - y
Y =
-Jet
-;,;
X-
X
y =
1,
n.
Interior angle of a regular polygon
12.
Linear expansion of a heated rod: al(T
13.
80°,
=
20",
=
_
X
=
10", c
8", 5
=
t(R
i'Cj'
:
—
—
a)(s
+
wh
+
=
r)(R
+
110 volts,
^ =
T =
12.250",
.10
B =
ohm,
cutting
for
—
2.
let
„
=
12.
.000023,
=
/
10',
r);
let i?
=
24",
=
12",
^
—
b){s
Ih); let I
through three resistances
Tailstock. offset
L =
a
/); let
=
c);\Qta
15".
Electric current
let
—
;
x
Ex. 17
16.
£ =
6,
h
\
Surface area of a box: 2(/w
17.
15",
= ^.
15.
let
C =
20°.
Square of the area of a triangle b
X=
3,
(^)
180'
:
circular cross section:
TT
Ex. 13
14.
=
/
Area of a r
let
;
c
12".
Slope of a line:
T =
C -
a
.25
C =
ohm,
length
1.875",
d =
5'.
—
1~7;)»
ohm.
.20
taper:
=
6',h
£(— +
in parallel:
partial
D =
6.125",
= T,w =
L /D - d -I T\ 2 )^
1.125".
nE 18.
Electric
dehvered
current
by
battery
in
cells
series:
R-\\et
n
=
3,
+
E =
+
1.5 volt,
+
R =
+
12 ohms, r
n (n
„
19.
Sum,
20.
Approximate length of an open belt
1
over
diameters:
letL
4
=
9
pulleys
2L
14',
+
D =
•
•
•
of
•
)
d
=
+
\) {2 n
0.1
+
1.2'.
;
m
ohm.
1) ;
unequal
3.25 (
2.5',
1^ w^
=
let
Ex.
«
=
7.
20
TD
40
CHAPTER TWO
2-2
Identifying Factors, Coefficients,
When two
more numbers
or
and Exponents
are multiplied, each of the
numbers
Thus, 3 and 7 are factors of 21 two Note that in factoring whole numbers
called a factor of the product.
is
other factors are
and
1
21.
;
you usually consider only whole number factors. Thus, the product 6x has 1, 2, 3, 6, x, 2x, 3.v, and, of course, 6.y itself as factors. Each factor of a product is the coefficient (ko'-e-fish-ent) of the product of the other factors.
In the product ^xy, ^
the coefficient
is
and ^y is the coefficient of x. Frequently, the numerical part of a term is called the coefficient of the term. For example, the coefficient of 343x- is 343. Also, the coefficient of
^x
.TV,
of a
is 1,
is
the coefficient of
^
since a
v,
\a.
Sometimes a number appears more than once as a factor in a product. The product ^ 5 is commonly written s'~. The term s- may be read: s squared or 5-square. The small raised number is an exponent (ek-spo-nent). It shows that s, which is called the base, is to be used twice as a factor. The base is the expression used as a factor one or more times (as indicated by the exponent). •
Exponent
1
Base-
To compare an exponent with a you replace s by 15. s5^ s'
An
= = =
exponent
s
•
15
coefficient,
compare
s
2s
•
25
15
225 tells
2*
= = =
A coefficient
how many
2
s-
•
2
and
25'
when
s
15
30
is
a factor,
times another number, called the base,
is
to be used as a factor.
A
number which can be expressed by means of a base and exponent power. The exponent 1, which is seldom written, means the first power of x that the base is used only once; therefore, x^ is the same as x. Here are some other powers of x: is
called a
—
—
Third power:
x^
= x x x
Fourth power:
a-^
= x x x x
Fifth power:
a'
= x x x x x
•
•
•
•
(read x cubed or x-cube) (read •
x fourth or x exponent
(read a
fifth
4)
or x exponent 5)
VARIABLES
AND
OPEN
41
SC(JTENCES
In an expression such as 3a^, the 2 is the exponent of the base a. In an expression such as (3a) ^, the 2 is the exponent of the base 3a, because you enclosed the expression in a symbol of inclusion. Compare the examples that follow: rs^
=
r
(i'5)3
=
rs
53
=
4
5)3
=
20
4 (4
•
•
'
s
rs
•
•
s
•
5 •
•
'
rs
5
20
5
s
•
•
•
(5
5
=
500
20
=
8000
15 (15
-
if /i)2
- inn) = (5 - n)(5 -
=
- 32 = - 3)2 =
5
-
9
=
(12)2
=
144
15
n)
6
ORAL EXERCISES Read each of
the following expressions as a product.
SAMPLE
1.
SAMPLE
2.
l{y
-\-
3)
What you
say:
7 times the
sum y
plus
3.
42 In
CHAPTER TWO
Exercises
35-44, name the numerical
SAMPLE.
9(w
+
What you
6)-
say:
The (u
35.
37.
2z2
36.
38.
4>'3
Tell the
39.
A-'
40.
u''
3(K
+
4(m
-
41.
2)-^
42.
3)2
coefficient
+
y2;
What you say:
y"^
k
=
45.
k'^;
46.
n^-n =
10
47.
a^;a=
176
48.
u^;u - 2
y
=
9; the base
(a
-\-
bf
43.
5/
(a
- by
44.
17:?
its
is
is 2.
value.
9.
means y times
when y =
y;
9,
y~
a-
=
50.
(2>')-;;^
=
3
54.
51.
5h'-;h=\0 2J~; J = 5
55.
49.
5
is
exponent
6); the
meaning of each of the following terms; then give
SAMPLE.
and the exponent.
coefficient, the base,
52.
(3.y)'^
2
= 9-9 =
81.
+
a
53.
56.
(a
2)-;
=
3
= 9 (^ - 7)^; (m-9)2;m=13 (5 + Q)-; Q = 4 Z;
WRITTEN EXERCISES Rewrite each of the following expressions 1.
Find the value of
=
25.
w-'.,„
26.
n^;n = i
i
a shorter form.
in
13.
Five times the cube of
14.
Eight times the square of z
15.
One-half the second power of ^
16.
One-founh the
17.
The square of
2f*
18.
The square of
8/
19.
The cube of
20.
The cube of ab
21.
The cube of
(a
22.
The cube of
(1
23.
The cube of
the
24.
The
fifth
>»
power of /z
a.v
— 1) —a) sum
square of the
r
sum
plus 2 t
plus 7
each of the following expressions. 27.
4p-^;p
28.
8^2; r
= =
3 5
29.
(9xy; x
30.
{^y)~;
=
i
y = I
AND OPEN SENTENCES
VARIABLES
)
31.
2x2
+
4x
+
32.
5y^
-
3y
-{-
33.
7z3
+
z2
Let X
=
5,
37.
^2
+
38.
X
39.
;c2
40.
X
41.
a:2
y
-
—
+
=
>;
y
X
=
4;
y
=
3
34.
1
35.
z; z
=
and
2=3,
^2
2
36.
-
J2 _^ ^2
44.
2:^2
_
45.
2z2
-
X2
z2
46.
+
a
-\-
4; a
and evaluate the following expressions.
J
+
2a^
+
x>'
=
+ 3v4 - v3 + v; V = w^o + w5 + w + 9; w =
+
43.
-
v^
x2
^2
z^
a^
42.
_|_
-\-
>'2
2,
^
>;2
J2
+
-
5;
43
72
5
44
CHAPTER
8.
Kinetic energy:
—-
9.
Volume of
circular
=
h
a
m =
let
;
25
=
g., v
cylinder:
TWO
100 cm. /sec.
irr-h;
let
7r
=
3.14,
=
r
1.25",
12.0".
Okl —C
10.
Illumination:
n.
Centripetal force:
^
;
C =
let
;
300 candle power,
let
m =
=
15 lb., v
D =
20
ft.
A:
=
500
/sec,
feet.
r
=
10
ft.
r
12.
Resistance of an electrical conductor:
d ^ 25
13.
Law
M
14.
t
1
5.
2—3
900,000
/
=
m =
300
=
r
g.,
Heat energy from let
GmM -— —
gravitation:
.0000000667,
=
;
let
/
=
200
ft. /sec/sec, t
=
.50 sec.
10.37,
ft.,
mil.
of
G =
—
100,000
=
electricity
20 amp.,
/?
Ex. 13
let
;
:
g.,
1000 cm. 0.238/2/??;
=
10 ohms,
sec.
/ t Length of a pendulum S \:r~ '
Solving
Y )
;
let
^
=
32.2
Open Sentences
Consider
this sentence:
w
is
a city in Texas.
Suppose the replacement set of w is the cities of the U.S.A. Replacing w by Dallas produces a true statement. Putting New York in place of vv leads to a false statement. This sentence becomes true or false as the variable is replaced by one of the values from its replacement set. In general, a sentence containing a variable may be neither true nor false, as the value This sentence, as written,
of the variable variable
is
is
called
left
is
open.
neither true nor false.
Consequently, a sentence containing a
an open sentence
pattern for the various sentences,
obtain by substituting in
it
The open sentence serves as a some true, some false, which you .
the different values of the variable.
AND
VARIABLES
OPEN
45
?E|4TENCES
An algebraic
sentence is a statement composed of algebraic expresby one of the symbols =, ?^, >, 16 + l>16 + 1 > 16 19
>
16,
True
set is {6}.
solution set of an equation or of an inequality
an open sentence
{5, 6}.
you may proceed as follows:
16, False
rule form, roster form, or
x g
uses one of the symbols ^, >, ,
+ 1 > 16 + l>16 + 1 > 16 16
=
1
+
expression and the right
solve the inequality,
3x
it
+
3.x
may
be shown in
by graph. The graph of the solution
called the
graph of the sentence.
set
of
46
TWO
CHAPTER
EXAMPLE
1.
Solution:
3r
+
2
=
14;r
3r
+
2
= = =
14
3-3+2 11
12 EXAMPLE
2.
Solution:
< X
6; ^ 5;
11.
E
{5,
12.
4w >
13.
^M
14.
+ -
1
1
I
15.
2(a
16.
2(Z)
6;
+ +
w E
E
3) 3)
8.
9.
m ^
10; 5
= =
2 2
E
X X
2
-
18
Z>
{2, 4, 6, ...
{1, 2, 3, 4, ...
> K« + <
+
25;>' (>'
G
+
SAMPLE Solution:
2.
in
roster form.
Graph
3/
+
1
=
10
3-3
+
1
=
10,
2)
{7,
=
open
.YG {2},>'G
.Y
-
(y .Y
G
{10},
+
{10},
>
Answer.
(.Y
>•)
+
2
->•)
+
!
+ {7}
{0} (.Y
y G
A'G {5],y
the solution set of
{3},
I)
8,9, 10}
y G
.Y
b.
.'.
G
b.
Determine the solution set from {numbers of arithmetic}. of each equation
the
solution set.
set is {2, 4, 6}.
a.
a.
{8,9, 10, 11}
7,
The solution 3>'
= 5;>;g {0,2,4,6}
1
in
and give each
<
12 Find the solution tion set
girls
sentence.
The number of boys
^
x G [whole numbers]
girls;
set.
is {0, 1, 2,
3,
Since x must represent a whole number, the solu-
3}.
number
_
Four possible steps 1.
and since 12
is
greater
Why
is
4 not an element of the solution set?
On
line:
_
_
girls is 3,
each element of the solution set does satisfy the require-
ments of the problem. the
girls
3x
Since the largest value for the number of
than 3 times
number of
_
Answer.
I
,
solving a problem:
in
Choose a variable with an appropriate replacement in representing each described number.
set,
and use the
variable 2.
Form an open sentence by using facts given
3.
Find the solution set of the
4.
Check your answer with the words of the problem.
in
^>
the problem,
open sentence.
ORAL EXERCISES Give an open sentence
to
fit
each of the following exercises.
SAMPLE. Multiply a number by
What you
say:
2>n
3,
+8
then add 8 to the product and you get 23.
1.
Multiply a number by
2,
and you get
2.
Multiply a number by
9,
and you
6.
get 45.
=23
58 3.
Double a number, and you
4.
Add
5.
Subtract 4 from a number, and you get 15.
6.
Add
7.
Subtract 2 from a number, and you get
5.
8.
Multiply a number by one, and you get
8.
5 to a
CHAPTER
TWO
get 30.
get 52.
number, and you get
11.
20 to a number, and you get 39.
9.
Multiply a number by
3,
then multiply this product by
2,
and you
10.
Multiply a number by
5,
then multiply the product by
3^,
and you get
1 1.
Add
4 to a number, subtract 4 from the sum, and you get 39.
12.
Add
7 to a
13.
Multiply half a number by
14.
Double a number, add 2 to the product, and you get
number, then subtract 4 from the sum, and you 2,
and you
13.
get 5.
get 9. 15.
PROBLEMS Solve the following problems by the four-step method on
page 57.
Jim is 3 years more than twice as old as June is 6 years old, how old is Jim?
SAMPLE.
his sister June.
If
Solution:
Mm
Let
^ E¥
represent Jim's age;
,Y
=
2(6)
.Y
=
15
+
15 equals 3
.*.
1
.y
3
more than
Jim's age
is
3.
2 times
6.
v/
15 years, Answer.
A baseball team won 3 times as many games as lost. It won 84 games. How many games did lose? (Let represent the number lost.) A class assessed each member 5 cents to buy flowers for an entertainment. The total was 170 cents. How many members were there? it
it
2.
x e {whole numbers}
Mr. Jonas got a
roll
.y
of 50 pennies to use only for parking meters.
he used 5 pennies daily,
how many days
did the roll last?
If
AND
VARIABLES 4.
5.
59
SENTENCES
The number of boys in a certain class is seven times The number of boys is 28. How many girls are in
the class?
A
as the lot
it
6.
OPEN
house cost $18,200.
was
A man
It
What was
built.
age
y that of
What
his
is
At the end of nine months monthly salary?
his aunt.
If Si is 8 years old,
7.
Si's
8.
In a certain city ^ of the girls are blonde.
is
The perimeter of The area of a
how
old
is
his
aunt?
Find the number of
girls
of them are blonde.
in the city if 10,195
10.
on which
the cost of the lot?
takes a position at a monthly salary.
he has earned $4050.
9.
much
cost seven times as
number of girls.
the
a square
50 inches.
is
rectangle that
is
4
feet
How
wide
long
is
one side?
68 square
is
feet.
Find the
length of the rectangle.
n.
After Jack deposited $55.25, his total bank balance was $1342.70.
How much 12.
did Jack have in his account before that deposit?
After a $15.75 bank withdrawal Phil's balance was $672.39. did he have on deposit before
13.
Jane's weight
11
is
If Jane weighs 109
14.
A
is
pounds more than normal for her height and is the normal weight?
age.
What
the temperature,
is
if this
3.7 degrees less than the true reading?
Fred earns $7.50 per week more than $115, what does Bill earn per
16.
How much
the withdrawal?
pounds, what
thermometer reads 56 degrees.
reading 15.
making
Dave's golf score was 3
Bill.
If Fred's
weekly salary
is
week?
than Mark's. If Dave scored 89, what was
less
Mark's score?
17.
A man
traveled a certain
far
18.
number of miles by automobile, and then nine trip was 500 miles in length. How
by airplane. His total did he travel by automobile?
times as
fa-r
The number of Central High School freshmen studying French is onefourth the number studying Spanish. The total number of students enrolled in these languages
19.
A certain
How many
number was doubled. Then
If the result
20.
150.
is
was
84, find the
After delivering his
than 75 bottles
left.
first
freshmen
the product
elect
Spanish?
was multiplied by
3.
number.
dozen bottles of milk, a milkman had fewer bottles had he originally?
At most, how many
1 more than twice as many books how many books does John own?
Sue owns 59
21.
Sue owns
22.
The number of bolts produced daily by machine A is 600 less than four times the number produced by machine B. If machine A's output
books,
is
4800 bolts per day, what
is
as John.
B's daily output?
If
y
^"*^
60
CHAPTER
23.
Mary's bowling score was 10 more than half Jay's bowled 100, find Jay's score.
24.
If one-third of a certain
number
diminished by
is
score.
If
TWO Mary
16, the result is 21.
Find the number. 25.
The length of
a picture
is
4 feet
76 feet of framing are needed. 26.
less
than twice
width.
its
To frame
it,
Find the dimensions of the picture.
Mr. Tripp completed a journey of 640 miles. The average speed of the plane taken by Mr. Tripp was 15 times that of the automobile he used to get to the airport. If he traveled an hour by auto and an hour by jet, how far did he travel by automobile?
jet
27.
Linda said: "I sold 3 more than twice the number of tickets Jo sold." Maria replied: "I sold 32 tickets, and that is more than you sold."
What 28.
29.
the largest possible
is
tickets
Helen weighs twice as much as her sister. is less than or equal to 165 pounds, what weigh?
is
1
than three times the
less
35, find the largest
first.
Jo sold?
sum of
If the is
the
There are three numbers such that the second third
30.
number of
is
If the
their weights
most that Helen may
twice the
sum of
and the numbers is
first
the
number.
One side of a triangular lot is 13 feet less than The third side is 18 feet more than the second. To
3 times the second.
fence the lot 130 feet
of fencing are required. Find the length of each side of the
lot.
Extra for Experts
The Arithmetic of
Sets:
Union
The union of two sets consists of all the elements of both sets, but no ment is listed more than once. For example, if the union of the A = [2, 3, 4, 5} and B = {3, 4, 5, 6, 7} is called set D, then:
A
\j
{2, 3, 4, 5}
u
This example
Universe:
=
B {3, 4, 5, 6,
may
U=
7}
=
D {2, 3, 4, 5, 6,
be illustrated pictorially:
{whole numbers}
(U 7}
is
read "cup,")
ele-
sets
AND
VARIABLES
A =
OPEN
{2, 3, 4, 5}
B =
{3, 4, 5, 6,
D
Note
61
S&NTENCES
7}
A U B
two sets has each of the original symbol for union is a stylized U.
that the union of
also, that the
sets as a subset;
Questions 1.
2.
If /I
=
show
the union pictorially.
B =
and
{0, 2, 4, 3}, give the roster
A U
of
B,
and
A u B and Al and Bl Why? Give a rule to determine when A u B = B will hold. Illustrate. b. Under what conditions would A u B = A n Bl
a.
What
a.
Define 4
b.
3.
{1, 2, 3}
is
the relationship between
set
For these
^
sets,
and a
set
B
which would
In the accompanying diagram, the
of
Newville
winners}, and
High
5 =
diagram, and shade
School},
H
universal
= {honor
{scholarship recipients}. it
to
show
B and A set G =
r\
Honor graduates
b.
Award-winning honor graduates
{all
graduates
A =
Award-winning honor graduates
who
B.
{award For each item, copy the
graduates},
the indicated subset.
a.
c.
satisfy this condition.
draw Venn diagrams of A u
received scholarships
d.
Award-winning gra(juates and honor graduates
e.
Graduates receiving no honors, awards, or scholarships
HUMAN EQUATION
THE
The Amateur Father of Algebra
The time was the sixteenth century.
every war, both sides sent
their
France and Spain were at war.
messages
code
in
As
in
from the
to hide their plans
enemy. Obviously, secrecy was important. But the Spanish secret could not
When
the French captured
be
Not that the Spaniards
kept.
accurately as any Spaniard could have read
Spaniards knew
their
codes were baffling.
how
didn't try.
a Spanish messenger, they read the message as
How
it.
How
could
this
be? The
thing
could a mere Frenchman de-
could any man, unless he had the key? The conclusion
cipher them?
In
was
Something more than man must be at work. The French must be
in
obvious.
fact,
league with the
Devil.
They must be using black magic!
Pope was too wise to interit was a French lawyer named Vieta. Nor was it by magic that he did his work, but by mathematics. For Vieta was a lawyer with a hobby, and his hobby was algebra. Codebreaking was nothing to him but solving equations. The Spaniards complained to the Pope.
fere, for
it
was
not the Devil
But the
who was breaking
the codes;
The French king owed Vieta a debt of gratitude. So do generations of algebra students. For Vieta not only broke the Spanish code; he simplified the whole Before
subject of algebra.
there
was
his
time,
practically no use of signs
and symbols; everything was done the
hard
way
—
in
words.
Vieta
introduced the use of letters as variables (he used vowels for
unknown
numbers and consonants for known numbers). tion
—
to
He used
signs of
show whether
to
tract, multiply, or divide.
were these and other
opera-
add, sub-
So great
contributions
an amateur, known today as "the father of
that Vieta, though only is
algebra."
A portrait
of
Vieta, the
French lawyer
who used algebra for code-breaking.
1
VARIABLES
AND
OPEN
63
SEJ^TENCES
Chapter
Summary
Inventory of Structure and Method 1.
may
In algebraic expressions, multiplication
be indicated by no sign, ab,
parentheses, 8(7), or a raised dot, 9-5. in a term having no other numerical factor,
An
lab.
1
is
listed as the
exponent
tells
factor, but a coefficient 2.
An
numerical coefficient, as
how many is, itself,
in
ab which
number
times another
The two expressions
equal to
are called the
An open
left
member and
a
two expres-
member
the right
becomes true or the variables are replaced by numerals. Solving an open sentence of the equation or inequality.
is
a factor.
algebraic sentence represents a condition which relates
sions.
is
(the base)
sentence,
false as
in
one
variable consists in determining the elements of the replacement set of
the variable for which the sentence 3.
The 1
is
true.
steps in solving problems algebraically are as follows:
— Choose a variable with an appropriate replacement
set
and use the
variable in representing each described number.
— Form an open sentence by using given the problem. 3 — Find the solution of the open sentence. 4 — Check your answer with the words of the problem. 2
facts
in
set
Vocabulary and Spelling base {p. 40)
variable (p. 36)
replacement
set
(domain) (p. 36)
power
{p. 40)
value of a variable (p. 36)
open sentence
constant (p. 36)
algebraic sentence {p. 45)
variable (open) expression (p. 36)
equation {p. 45)
algebraic expression (p. 36)
left
value of an expression (p. 37)
inequality {p. 45)
evaluate an expression (p. 37)
solution set {p. 45)
term (p. 37)
solve {p. 45)
factor (p. 40)
root {p. 45)
coefficient {p. 40)
graph of an open sentence
exponent {p. 40)
&
right
(p. 44)
members
{p. 45)
{p. 45)
64
CHAPTER
TWO
Chapter Test 2-1
Evaluate the following expressions 8r
1.
6s
2.
3.
r
-\-
6(r
Evaluate the following expressions
ab^
5.
2-2
=
3 and
-\-
s)
-
=
2 and b
if
a
3rs
-g-.
4.
=
(r
+
s)(r
-
s)
3.
each of these expressions.
set of factors of
a
=
s
(aby
6.
Give the 7.
r
if
Pd
limn
8.
—
9.
3
Give the
missing coefficients, as indicated.
=
%qrs
12.
Identify the numerical coefficient, the base,
Write
13.
11.
)rs
?
(
-
in 3(x
2-3
^ = (^_>r
10.
and the exponent
1)2.
mathematical symbols, h used as a factor three times.
in
The replacement set for x
is
Which of
{3, 6, 9, 12}.
the elements
make
each of the following open sentences true? v
14. Let
=
=
9 and n
1
in
m -
From
{1, 2, 3,
18.
3f
the 1
+
> .
=
6
number
'
= (w -
«)
+
8
graph each of the following
4
21.
z
>
9
inequalities.
3
For each expression, give two interpretations.
-
22.
X
24.
Write an algebraic expression for the amount of
-\-
23.
5
2a:
5
in one year by Mr. Jones if his weekly income is average monthly expenses, including taxes, are
2-6
For
false.
0} determine the solution set of each sentence.
9 line,
>
5
each of the following open sentences.
16.
5n
-
X
15.
1
resulting sentence, state
20.
2-5
=
m
5
each
On
2-4
-
25.
The length of a rectangular swimming pool twice
its
width. If the pool
is
is
money saved
s dollars,
and
his
e dollars.
5 feet less than
35 feet in length, find its width.
VARIABLES
AND
OPEN
65
SENTENCES
Chapter 2-1
Review
Evaluating Algebraic Expressions Containing Variables
Pages 35-39 1.
A
variable
is
!_ which
a
represents each of the elements of a
specified set. 2.
The
whose elements may be used to replace a variable !_ set or ! of the variable.
set
is
called the
2-2
3.
Evaluate (5r
4.
An
+
s)(5r
—
Identifying Factors, Coefficients, 5.
Each factor of a term
6.
The usual way of
7.
The meaning of
8.
An
!_^ tells
is
or
how many
5=1.
called a term.
is
Pages 40-44
and Exponents
is
of the other factors.
I
•
.
'
times
is
2 and
i
called the
writing \n
s"^
=
letting r
s),
expression written as a __!
-
times another
number
is
to be used as
a factor. 9.
10.
In mathematical symbols, c used as a factor 5 times In the expression is
2-3
1
,
1 1
The exponent of
12.
Find the value of
Open
Solving
the numerical coefficient
2>a'^,
and the exponent
is
The
14.
The
,
the base
-
and the value of 3e when
e
is 15.
Pages 44-49
Sentences
Exercises 13-15 refer to the general statement: 13.
is
I
I
r in Ir is
e^
is
variable in the given statement
?_ of the variable
is
1
=
n
•
n.
iJl\
numbers
the elements of (all the
is
of arithmetic}. 15.
The expression
16.
An
1
•
«
member of
the
the equation.
open sentence may become a true statement or a
ment depending on 17.
is
the replacement for the
I
^
and
Each of the open sentences 3x called
an
6,2p
>
8,
r
—
false state-
4
;
-
1
=
3
^9.
y
^
?^ 9
21.
>^
>
4
22.
The
set
of numbers which belong to the replacement
variable and which
make
the sentence true
of the
set
?
called the
is
set
of the sentence. 23.
A number which
satisfies
an open sentence
is
L_ of the
called a
open sentence. 24.
Since 4 satisfies the condition expressed in
25.
Of the
I
=
1
19,
it is
following graphs, which represents the solution set of the
inequality
(a)
—
5^:
of the equation.
^
a
12
1
J
I
3
x
(c)
the one
2,
represented by the adjoining graph
1
3
Pages 49-51
Thinking with Variables: From Symbols to Words
Find an interpretation for each of
is
the following algebraic expressions.
In each case, identify the replacement set of the variable.
27.
3x
28.
-
+
29.
3
3(jc
-
1)
3
2-5
Pages 51-55
Thinking with Variables: From Words to Symbols In Exercises 30-35, translate from words to symbols.
than a
32.
The
33.
One-third of the
difference between 3c
30.
5 less
31.
2^ increased by 3
34.
A line is divided into three equal parts.
Using
/
sum
as the
inches in one part, represent the length of the entire
35.
2-6
When you
Open
have a problem to solve,
number of line.
first select
a
1
and use
number.
After forming an open sentence and finding
each root with the words of the problem.
66
c
Pages 56-60
Sentences
in representing each described
37.
and
3
Find an expression for the number of cents Julie received in change from a one-dollar bill after buying n five-cent articles.
Solving Problems with 36.
(/
and
its
solution
set,
I
it
Write an open sentence expressing the conditions described
38-40; then find' the solution
in
Exercises
set.
38.
Eighteen times a certain number
39.
In a school cafeteria one week, 1440 bottles of milk were sold.
is
198.
Three times as many bottles of milk as 40.
A
class of 25
If
each boy
cream bars were
sold.
boys wishes to donate from $3 to $5 to a charity.
is
inequality the
ice
to contribute the same amount, amount each boy may give.
a,
express as an
Just for Fun
Be a Magician with Numbers If
you practice a
bit;
you
will
be able to mystify your family and friends
with your seemingly magical knowledge of numbers. Tell a friend that
might choose) lowing sult,
if
tell
You say
his age (or
any other whole number he
Ask him to do the folyou give him directions. If he will then give you the rehim his age (or the number he chose).
silently, as
you can
you can give
he will follow a few instructions.
CHAPTER
3
Axioms, Equations, and
Problem Solving
diamond you recognize a symbol of value. You also know that the same size may have different values. Why is this so? The beautiful pattern you see in the upper photograph is the outward exIn
the
diamonds of
pression of a regular internal structure.
This regular pattern gives the
diamond (being examined in lower photo) decorative and practical values.
beauty and hardness,
its
Mathematics, too, has a regular structure which makes pleasure to those
many
applications. Just as a
beauty of a gem
make
who understand
full
until
its
diamond
he understands
use of mathematics
until
diamond
is
its
internal structure,
you understand
used throughout
a source of its
cutter cannot bring out the hidden
Because of the importance of structure matics, a
it
beauty and discover
internal
its
this
to both
book
to
its
you cannot
structure.
diamonds and mathemark those ideas which
form the basic structure of algebra.
IDENTIFYING 3-1
Axioms
You
AND USING NUMBER AXIOMS
of Equality
learned to perform
you abided by
many
operations with numbers because
rules, which are statements accepted axioms, or postulates- Though some of these assumptions may seem simple, you must be able to understand and use thesS rules in solving complicated problems. The first fundamental assumptions that you will meet are the axioms of equality which govern your work with equations. Perhaps the simplest of all axioms is the reflexive property of equality^ which says that any number is equal to itself.
certain rules.
These
as true, are called assumptions,
For any number a, a
The symmetric property of equality For
any numbers, a and
—
a.
states that
A,
if
a
=
b,
an equality
then b
=
is
reversible.
a.
69
The
of equality makes it possible for you to two numbers as equal if each of them equals a third number.
transitive property
identify
For any numbers,
a, b,
and
c, if
a
=
b,
=
and b
c,
=
then a
c.
ORAL EXERCISES In
each exercise, name the property of equality which
+
=
-
SAMPLE
1.
If 5
What ypu
say:
The symmetric property.
SAMPLE
2.
Given
1.
The
say:
Given that 17
2.
Given that 4
3.
r
4.
Given
-\-
s
=
+
-j-
= ~
in
3,
=
11
illustrated.
3
=
5
14
—
3,
-
2.
-
then 14
+ 6 = 11, and + 6 = 14-3
+
6
transitive property.
=
2
6
15; therefore, 15
=
10 and 10
+
= —
=
5
+
= 5
17
therefore, 4
;
+
6
=
+
5
5.
s.
-T3 8| 3 n^ 4^
aiiu and
1^3 12f
that iiiai
1^3 12f
=
13;
therefore,
13
the property of equality which
conclusions 5.
+
that
4i ^^ 3 ~r '*3
Name
r
-
14
5
then 5
What you
6
is
is
illustrated
in
each of the successive
the following examples.
+ 4) = 5(7), that 5(7) = 35, and that 35 = + 4) = 35 and 5(3 + 4) = 15 + 20. Given that 5(1 + 0) = 5, and that 5 + = 5; therefore, 5 and 5(1 + 0) = 5 + 0. Given that
5(3
15
+
=
5
20;
therefore, 5(3 6.
7.
- 6f = 16| - 6f and that 10^ = 16| - 6f 6f = lO^and 17 - 6f = 10^ Given that x ^ | = ^ x f that | X f = M> and that ff therefore, ^ X | = 3ii and | - f = 3^Given that 17
fore,
8.
3-2
16|
there-
-
,
= 3^;
The Closure Properties
When you add two whole
numbers,
is
number? To try every example would be an ing a large number of varied examples: 138
+
51
=
189;
174
+
the result always a whole
endless task. After check-
236
=
410;
and so on, you would probably assume that the answer
70
;
+
is yes.
AXIOMS,
A^D PROBLEM SOLVING
EQUATIONS,
Is this true also for
5
X
Again you
multiplication ?
=
37
23
185;
71
X
48
=
try
many examples
1104;
and so on. Again you will, no doubt, assume that the product of two whole numbers is always a whole number. Any set S is said to be closed under an operation performed on its elements, provided that each result of the operation is an element of S. This is known as the closure property of a set under an operation. Calculations in arithmetic are based on the often unstated assumption that the set of numbers is closed under addition and multiplication.
The
closure property for addition
is
stated
For every number a and every number number (one and only one number).
The
b, the
closure property for multiplication
For every
number a and every number
is
sum a
-\-
b
a unique
\s
stated:
b, the
product ab
is
a unique
number.
Closure under any operation depends on both the particular operaand the domain of numbers used. For example, the set of odd
tion
numbers is closed under multiplication but not under addition (3-5 = 15, 3 + 5 = 8). On the other hand, under division the set of whole numbers is not closed, but the set of arithmetic numbers other than
An
is.
may
operation on elements of a specified set
not be possible
For example, if you try to subtract any number of arithmetic from a smaller number, you know of no arithmetic number which could be the result. The set of numbers of arithmetic is not closed under subtraction. Important also is the assumption that an indicated sum or product of numbers does not depend on the particular names designating the unless that set
is
closed under the operation.
numbers. 3(2
+
2
+
5)
=
3
5
=
7
•
and
7
4
•
99
=
4(100
99
=
100
-
1)
because
These examples
each
-
1.
illustrate the substitution principle:
For any numbers a for
and
other.
and
b,
if
a
=
b,
then a and b
may be
substituted
You
used the substitution principle often in arithmetic.
Add:
13
+
7
+
+
4
12
Note: Each red numeral was substituted
an expression which
for
+ +
13
24
= 20; 12 = 36.
+
20
7
equaled:
it
=
4
and
24;
ORAL EXERCISES Which
of the following sets are closed under the specified operations?
SAMPLE
The even numbers,
1.
{0, 2, 4, 6,
.
.
muhiplication
.},
What you say:
Closed, because the product of two even numbers
SAMPLE
{0, 1},
2.
What you say: 1.
{0, 1, 2, 3},
2.
{0, 1),
Not
Why?
is
even.
addition
closed, because
addition
multiplication
+ 1=2
1
^
and 2
8.
{0, 1, 2, 3, 4,
9.
{0, 2, 4, 6, 8,
.
{0, 1}.
.
multiplication
3.
{1}, multiplication
10.
{1, 3, 5, 7,
{2}, subtraction
n.
{1, 3, 5, 7, 9,
.
.
5.
{0, 2},
12.
{3, 6, 9, 12,
,
.},
6.
{^,
13.
{
7.
{0, 1], division
14.
{1, 3, 5,
subtraction
1,2}, division
1
,
.},
.
.
.
.},
addition
.},
^, 2, 1, 4, 1, 8, .
addition
.},
.
4.
.
subtraction
.},
.
addition .
.
.}
,
division
multiplying by 5
WRITTEN EXERCISES Which of
each of the operations of addi-
the following sets are closed under
tion, multiplication, subtraction,
an example which shows
SAMPLE,
{fractions
Solution:
Addition
and
division?
When
from
is
not closed, give
to 1}
— not closed, as | + ^ Subtraction — not closed, as 5 — Division — not closed, as | 5 Multiplication — closed -^
72
the set
this.
is
^ is
not in the is
set
not in the
not in the
set
set
EQUATIONS,
AXIOMS,
PROBLEM
Ab^D
1.
{0}
5.
(0,1, 2; 3}
••
l^i
**•
1^5
3. 4,
73
SOLVING
(numbers between
9.
and 2}
ij
'*'•
2}
n.
(nonzero numbers of arithmetic}
12.
(all
{2}
7.
(1,
(3}
8.
(multiples of 5}
2' 4' 8' 16'
(.^»
•
•
•/
fractions of arithmetic
which
are not whole numbers}
3-3 Commutative and
Associative
Properties
of
Arithmetic
Numbers
You know 6
+
=
3
3
that
+
+
7
6;
1
=
+
1
9
7;
+
=
2
2
+
9.
you assume that when you add two numbers, you get same sum no matter what order you use in adding them. This
Jn arithmetic the
cammutative (ka-mu-ta-tiv) property
For every
Likewise,
number
6X3
=
a,
of,
addition
and every number
3X6
and 6
•
«
=
a
b,
«
may
•
-\-
b
be stated
=
b
-\-
a.
When you
6.
:
multiply
numbers, you obtain the same product, regardless of the order of the factors.
The commutative property of multiplication
For every
number
and every number
a,
b,
written:
is
ab
=
ba.
and division do not have the commutative 3 ?^ 3 — 6 and 6 -^ 3 ?^ 3 -f- 6. To find the sum of 252 + 60 + 40, you probably first add 60 and 40, obtaining 100, and then add 252 to the result, getting 352. But if you add 252 and 60, and to that sum add 40, you obtain the same total. That is, Notice
that^ subtraction
property.
For example, 6
252
+
—
(60
+
40) =
(252
+
60)
+
40.
Thus, you are free to choose any adjacent pair in addition, for the answer
is
the same.
This associative (a-so-she-ay'tiv) property of
addition states that
For every number a, every number b, and every number
a
+
{b
+
c)
=
{a
+
b)
+
c.
c,
74
CHAPTER
The
associative property
number
For every
of multiplication
every number
a,
b,
THREE
is:
and every number
c,
=
a{bc)
{ab]c.
Are subtraction and division associative? No, because
-
24
-
(6
^
2)
-
(24
-
6)
2
and 24
--
^
(6
^
2) 5^ (24
6) -- 2.
commutative (order) and associative (grouping) properties permit you to omit parentheses in a sum because the numbers may be added in any groups of two and in any order. Tile
101
+
+
33
+
46
+
67
+
14
=
99
101
+99 + 200 +
+
33
67
100
+ +
+
46
14
60
360
ORAL EXERCISES Name
the property illustrated
each of the following
in
variable has the set of the numbers of arithmetic as
SAMPLE. What you
1.6 + 2.
(12
3.
I
4.
8
•
=
2
4)
+
6
=
6
X
(0
X
•
(7
X
=
9)
(7
X
X
9)
Commutative property of
say:
2
+
X
3
+ 5
6
=
+
+
(4
1 4)
=
X
(8
X
0)
4
3
multiplication.
8.
9
+
t
12.
1
5.
X
+
(9
7
+
+
9
x
7)
=
For each u and
X
(I u',
+
5u
9)
+
X
+ \%r +
(3m
\v)
=
19.
20.
For each a and each b,l
17 18.
For each
r, {r
3)19
+
3 9)
X m X i
9
15
(17z
+
33j)
3m)
+
7
= 3) (17i = 11 X (11 X X 2 X 2) 25 X (4 X 93) = (25 x 4) X 93 f X (I X 16) = (f X J) X 16
16.
30a
=
=
15
X
+
=
14.
n.
17.99
.y,
=
7.
13.
10.
For each
a, 5{6a)
For each w, m X i X (2 + 9) = (2
16 + x + 1.01 = 17.99 + 1.01 + (58 + 11) + 139 = 58 + (11 + 139) For each z, (32 + 17z) + 332 = 32 + For each r, 11 + (4 + z) = 15 + z
9.
set.
3
6.
5)
Every
true sentences.
replacement
For each
5.
12
its
(5m
+
1
17i)
X
(4a)
X
b
w
=
2Sab
AXIOMS,
EQUATIONS,
AI^D
PROBLEM
SOLVING
Name the property that justifies each lettered A check (y/) shows that the step is justified by 21.
75
step of these chains of equality, the substitution principle.
:
76 way you
Note that distributed in this
2iS
same
get the
+
95(3
result; that
is,
+
95
=
7)
X
95
X
+
example
a(b
sum
(3
+
7),
and
number
=
c)
a,
ab
every number ac
-'r
The following show how
and every number
+
ab
or
of
stated
is
b,
is
The property shown
7).
called the distributive (dis-trib-u-tiv) property
is
-\-
THREE
7.
95, the coefficient (multipher) of the
For every
c.
3
a multipher of each term of (3
multiplication with respect to addition
b.
CHAPTER
ag^
Either
a.
:
ac
^
+
a(b
the distributive property
is
c,
c).
used
+ i) = 28 X i + 28 X i = 14 + 4 = 18 9X4| =9 (4 + I) = 9X4 + 9X1 = 36 + 15 ^ 9 15 + 9 = 6, or = 28(1
—
7
=
43
,
1
4
1(15) 4
You traction
4
4
+
= 1(15
1(9)
+
=
9)
J"
4
4
=
(24)
6
4
can readily show that the following sentences involving sub-
and multiplication are 14(f
-
20 8 •
These two sentences
1)
-
true.
= 20
•
-
14
X
f
5
=
20(8
X
14
-
5)
illustrate the distributive
1
=
=
7
60
property of multiplication
with respect to subtraction:
For each a
and each b and each c
—
a{b
Often you
For example,
=
c)
ab
—
ac, or
for which b
ab
—
ac
—
— a[b
c
is
—
a number, c).
will use these properties to simplify variable expressions.
to
5jc
show
+
3x
that 5x
+
= Ar5 + x3 = x(5 + 3) = a:8 = 8x
3x
=
8;c
for each
number
x\
Commutative property of multiplication Distributive property
Substitution principle
Commutative property of multiplication
AXIOMS,
ANO PROBLEM
EQUATIOI^S,
77
SOLVING
Similarly,
-
lab
The
=
Aab
(7
distributive property enables
or the difference (lab
Terms such
—
=
3ah
-
A)ah
you
=
3ab.
to write the
sum
{5x
+ 3.v =
8a-)
4ab) of similar terms as a single term.
5x and 3x, or lab and Aab are called similar terms. Similar terms are numerical terms or variable
and
as 7
terms or like
9,
terms whose variable factors are the same.
Sx and 3ab are unlike terms, because their variable and 7b are unlike terms. Hence, — 9 cannot be written in simpler expressions such as 8a: -f- 3ab and lb
Terms such
as
factors are different. Also, the terms 9
form.
1X3
=
3X1=3
the
multiplicative
property of 1
one times any given number equals the given
(mul-ti-pli-kay'tiv):
number
illustrates
itself.
number
For each
Since the given
you use the
number and
a,
a
•
1
3
=
•
1
=
a.
do you
I,
see
why
I
when is
the
element?
multiplicative identity
+
a
the product are always identical
multiplicative property of
Likewise,
=
+
3
=
3 illustrates that the
additive identity
is added element is 0. The additive property of states that when any given number, the sum is the given number itself, or simply:
For each
number
+
a,
a
=
a
+
=
to
a.
The multiplicative property ofO, shown in 0X3 = 3X0 = 0, states when one of the factors of a product is 0, the product itself is 0.
that
For each
number
a,
This multiplicative property of
The statement
mean
that a
3
=
=
2
X
statement true, since
means \f
b.
X
that 6
a
9^
=
/?
0,
•
a
=
a
•
=
0.
affects the use
=
3X2.
multiplicative property of
is
=
b should
no value of b can make the for each b.
indefinite in value.
is
as a divisor.
Likewise,
If
of b makes the statement true for the same reason. - either has no value or
of
that you
may
A
a
=
0,
latter
every value
Thus, the fraction
consequence of the
not divide by 0.
^
78
CHAPTER
THREE
ORAL EXERCISES Name 1.
the property of numbers which
justifies
each step
in
the following exercises.
AXIOMS,
AND
EQUATIONS,
PROBLEM
79
SOLVING
WRITTEN EXERCISES Simplify each of the following expressions
9
SAMPLE.
Ly
Solution:
(8.v
+ +
3>'
+
2 a)
2a
+
-
(3>'
3>'
by combining
similar terms.
'^
80
CHAPTER
THREE
TRANSFORMING EQUATIONS WITH EQUALITY PROPERTIES 3-5
Addition and Subtraction Properties of Equality Certain properties of equality can be proved from the properties
of equalities given in Section 3-1. will help
you
A
knowledge of these properties more readily. Consider
to solve complicated equations
the following illustration.
Man B
Man A Two men Each
$6000
receive equal salaries.
same $500
gets the
$6000
raise.
+
Their salaries change, but the
new the
man A is equal new salary of man B. salary of
= $6000
$500
=
$6000
$6500
=
$6500
+
$500
to
This example of the addition property of equality shows that same number is added to equal numbers, the sums are equal:
For each a, each b, and each
c, if
a
=
b,
then a
-\-
c
=
b
-\-
if
the
c.
This new property of equality follows from facts already learned. The reasoning leading from the assumption ^ = ^ to the conclusion a -\- c = b -\- c is shown in the following sequence of statements,
each justified by the indicated reason: a
-\-
a
a
c -\-
-\-
is
a
number
c
=
a -^ c
Reflexive property of equality
a
=
b
Given
c
=
b
-\-
c
Closure property of addition
Substitution principle
This form of logical reasoning, from conclusions, is called a proof
Can you prove ber
is
known
facts
and assumptions
the subtraction property of equality:
if
the
to
same num-
subtracted from equal numbers, the differences are equal, pro-
vided the indicated subtraction
is
possible?
For each a, each b, and each c for which a then a
—
c
=
b
—
c.
—
c
is
a number,
if
a
=
b,
AND
EQUATIONS,
AXIOMS,
PROBLEM
The addition and subtraction
To
solve equations.
100
—
In general, a
produce
a.
-
see
70 -\-
c
To undo
+ c
how
=
81
properties of equality are used to
to use them, first notice:
=
70
SOLVING
and
100
a and c
the
is
-
«
6
+
6
=
number you add
„ [o a
—
c to
a subtraction, you add.
+ 5—5 =8
y -\- 3 — ^ = y. In general, = a, and c is the number you subtract from a -\- c io a -\- c obtain a. To undo an addition you subtract. Because the operations of adding and subtracting the same number are opposite in effect, they are called inverse operations Can you Similarly,
—
8
and
r
+
X
For each 5(r
7.
each lettered step.
(«
(n
+
+ (7 + = (4 + 7) + = 11 + =
7)
X
For each
4
= = =
1)
/7)
(a)
«
(b)
n,
«
X (1 X n) (4 X 7) X n 4
(a)
(b)
2Sn
r,
+
+
1)
+
3(r
+ 5 X + 3 X r+ 3 = 5Xr + 5 + 3X/= 5Xr + 3Xr + 5 = (5 + 3)r +5 = + 8 = 8(r + 1) =
1)
X
5
r
1
X
+ + +
1
(a)
3
(b)
3
(c)
3
(d)
8/-
8.
Simplify and combine like terms:
9.
Find the value of f (39)
-
3-5
Solve:
lO.
5
3-6
Solve:
12.
7y
3-7
14.
=
73
as
$46.28. 16.
Solve:
17.
A
+
11.
42
=
/
13.
|v
=
48
Show work, fm — 4 =
Amy. At
3c/
+
2(6^
5x
—
8.
18
3
Ben
—
.3m;
is
a root of
m =
10
invests three times as
the end of the year, the total net profit
Find Ben's proportionate share of the
-
5)
=
is
profit.
5
Bonrite pen and pencil set costs $2.78, the pen costing $.80
more than 18.
28
—
2)
9f (39).
In a Junior Achievement enterprise,
much
3-8
91
+
4(a-
State whether the indicated value of the variable
the equation. 15.
=
+
+
3.y
(e)
Solve:
the pencil.
3(2/z
+
7)
-
Find the cost of each. 5
=
2(5/i
-
4)
+
4h
^
100 19.
The length of
a rectangle
The
smaller rectangle.
smaller
THREE
than the length of a
3 inches less
is
larger rectangle
is
9 inches wide; the
If the area of the larger rectangle
4 inches wide.
is
CHAPTER
is
48 square inches more than the area of the smaller, find the length of the larger rectangle.
Chapter Review
3-1
Axioms 1.
a
2.
If b
3.
Pages 69-70
of Equality
b
^
a
=
d,
then d
-{-
-
23
-\-
=
15
8,
property of equaUty.
b illustrates the
2
=
=
4
•
property.
b illustrates the 23
8,
-
15
=
2
4 illustrates the
•
_^
property.
3-2
Pages 70-73
The Closure Properties 4.
The sum of two numbers of arithmetic
5.
A
set
R
of
its
elements
6.
The
is
closed under muhiplication is
an
State whether each given set 7.
3-3
{3, 6, 9, 12,
.
.
.},
is
is
10.
You know for_^.
+1 m
For any numbers
any two
and _!
closed under the indicated operation.
division
that 4«
1
closed under
8.
{6, 4, 2, 0},
Commutative and Associative Properties 9.
be a
the product of
of R.
of numbers of arithmetic
set
will
if
1+
=
and
n,
4/7
subtraction
of Arithmetic
because of the
mn =
Numbers Pages 73-75
^_ property
because multiplication
is_^. 11.
You know the
3-4
property.
5)
+
+
12.
(8
13.
(157
14.
16
that 8 h- 4 ?^ 4
1
X
-
4)
(8
4
=
X
25
-
8
=
+
^
8
because division does not have
9 because addition
is
_?_.
157
X
100 because multiplication
-
8)
-
2) p^ (16
2
because
The Distributive Property; Special Properties of
subtraction
1
is
_! is
not
and Pages 75-79
15.
5(;c
—
y)
=
5x
—
5y because of the
tiplication with respect to
!_.
'
property for mul-
AXIOMS, 16.
EQUATIONS,
AND^ PROBLEM
When you combine
+
4u'
101
SOLVING
w
you are using the
to get Sw,
property. 17.
The expression 4
18.
0-5 =
19.
When
•
=4 because
1
because of the
must be
!_ property of is
0,
!
at least
is
L_,
21.
6«
6/7
never be
1
each lettered step. For each
-
one of the
may
In the following chain of equality, give a property of justifies
1.
1
Zero, or any expression whose value
used as a
L_ property of
of the
the product of several factors
factors
20.
1
•
+
3
X
(4
X
i)
numbers that
n,
= /J(6 - 6) + 3 X (4 X = «-0 + 3X(iX4)
i)
(a)
102
CHAPTER
3-7
=
Solve I2k
35.
=
Solve 18
36.
3.
Combining Terms and Using Transformation
THREE
f m.
Principles
Pages 86-91
-
-
+
37.
Solve 9?
39.
The sum of a number n and sented by «
3r
9
+
38.
3.
Solve
6 times that
\
+
y =
%.
number may be
repre-
or
Solve.
40.
Three times a number decreased by half the number gives Find the original number.
41.
Undo tions
=
and
indicated
and
42.
27
44.
A man
I
10.
before considering multiplica-
divisions.
-
7.2a
5
+
43.
.So
38
+
7^
-
26
+
5Z)
=
16±
2 years younger than three times his daughter's age.
is
Their ages total 50 years. Find the age of each.
3-8
Equations Having the Variable 45.
When an
in
Both
Members
Pages 91-95
equation has the variable in each member, transform
!_ member.
into an equation containing the variable in only
it
Solve.
46.
In
48.
A
+
1
=
rectangle
3(2« is
-
1)
+
47.
6
§(9/-
-
4)
=
+
r
3(r
+
f)
6 inches wide. If a rectangular strip 4 inches long
were cut from the end, the area of the remaining rectangle would be f of the original area. Find the dimensions of the original rectangle.
49.
At a County Fair, 200 ice cream cones were sold in one day, some at 15 cents each, the rest at 10 cents. If the proceeds from the sale of cones were $23,75, how many of each were sold?
Cum[\\at\oe ReufetU: Chapters \-^
State whether each of the following sets 1.
{points
2.
{all
on a circumference}
the trees in the world}
is
(a) finite, or (b) infinite.
3.
{odd numbers between
4.
{1970, 1980, 1990,
.
.
.}
3
and 5}
below
the set
If
is
PROBLEM
AND-
EQUATIONS,
AXIOMS,
103
SOLVING
given by a roster, specify
by a
it
rule;
given by a
if
rule,
specify the set by a roster. 5.
{the states in the U.S.A.
6.
{5, 11, 17, 23,
Draw 8.
the
whose names begin with the
...47}
graph of each
set
described below.
numbers between 2 and
4.5, inclusive}
and 2^}
9.
{the
numbers between
10.
{the
numbers greater than or equal
11.
Simplify sets:
1
8^8x2. A =
K]
{1,3,9,27,81}
7.
{the
Given the
letter
to 3}
Simplify 3[9
12.
{0, 2, 6, 10,
20} and B
=
-
2(1
+
1.2)].
{1, 2, 3, 9, 12}, find the set
specified as follows:
13.
{the elements in
14.
The subset of
15.
The
In
or B, or in both
containing
all
A and B)
multiples of 3
of those elements which are in both
A and B
each case, give the property which makes the conclusion
16.
17. In
set
B
A
= m; m = Conclusion: b = 6 5(5 - 2) = 55 - 10
Given:
b
18.
6
Given:
3
+
5(/
+
2)
=
x = 5x 5x
Conclusion: 19.
true.
(/
+
—
step.
20.
{2(5«
+
n,
1)}
-
lOn
= = = = =
- 10« {2 + 10/7} - 10/7 2 + {10/7 - 10/7} 2 + /7(10 - 10) {10«
2
+
+
2}
/7-0
(a)
(b) (c)
(d)
=
1
3
-\-
x
2)5
the following chain of equalities, give the property that justifies
For each
1
—
each lettered
104 If
a
27.
28.
CHAPTER
=
=
b
1,
+
4(6
f)
2Z>
+
c
2b
—
c
—
Express
in
=
4,
and
3^(a
+
6)
32, c
-
=
c/
0, find the value of
each expression. ^
29.
4^2 30.
-
16(f
by^
algebraic symbols.
31.
Twice a number
32.
One-fifth of the
33.
The
larger of
two numbers when
it is
34.
The
larger of
two numbers when
their
35.
The square of
36.
Twice the sum of the squares of two numbers (R and
If
decreased by 3
n,
sum of a and b
37. 38. 39.
make each
2(« 3« > 2/t
1
= 1)
sum
is
\s
U =
the smaller, s
50 and the smaller r) r)
3}, find the subset of
[0, 1, 2,
x
is
U whose
of the following sentences true.
5
=
more than
5
sum of two numbers (R and
the
the replacement set for n
elements
THREE
5
+
40.
4«
41.
n{5n)
A2.
n{n
-
> 6 = n-5-n-n 1
=
\)
X
n
-
n
n
Write an algebraic expression for each of the following. 43.
44.
a.
A
b.
Give the replacement
whole number that
is
third side of a triangle
side
is
first,
more than
set for
The
twice the
5
a given number,
/
t.
whose perimeter
is
27,
and whose second
a
Solve each equation.
45.
8w -
46.
7.y
47.
-
l{5y
13
=
67
+
3w
=
31
+
4
-
^
-
4)
=
li
+
48. 25
-
5
49.
\{6y
+
8)
50.
+ + fu' = ^-w 3(2/ + 5) - 7 = 9/ - 4 12 + 3(2^ - 1) = 2(3A: +
fu-
1
1
1)
Solve each problem.
51.
Find a number such that the difference of twice the number and twothirds of the
52.
number
68.
The sum of two numbers times the smaller.
53.
is
is
20.
Four times the larger
For Mother's Day, Tony bought three he bought two
Tony
is
1
less
than
five
Find the numbers.
ties.
If a tie cost
spent a total of $5.95,
kerchiefs.
For Father's Day,
50 cents more than a kerchief and
how much
did a
tie
cost?
AND -PROBLEM
EQUATIONS,
AXIOMS,
105
SOLVING
Extra for Experts
The Algebra
and
of Logic
Sets
Consider the following compound sentences: 3
Each of these may be broken 3
< X
jc
^
+
(-a)
=
a
number and
Several facts about a
—a
a unique number
is
such that
0.
opposite follow from this
its
assumption 1.
2.
a
If if
a
is
—a
positive,
—a
is 0,
negative;
is
if
a
—a
negative,
is
positive;
is
is 0.
—a
The opposite of
that
is a,
—( — a) =
is,
a.
All these relationships help in simplifying expressions.
EXAMPLE
Show on
1.
-
b. c.
Solution:
number
+
-(^3
lines that a.
=
+4)
7 and
+ (- ^4) = -7; therefore, (-3 + +4) = - -3 + (- -4). ^3
Add ^3 and
a.
H
the
~^4,
and then
find the opposite of this result, ~7.
—
1 \
\
-6
-7
b.
Add
\
\
1
-4
-5
-3
\
-2
-6
e.
-5
-2
\
1
\
\
\
-1
'4
'3
S
\
-6
"7
the opposite of "3 and the opposite of '4; that
"3 and ~4. This
-7
\
-1
-4
-3
is,
add
is ~7.
-2
-1
-1
~2
+3
'4
-5
+6
'7
"By the transitive property of equality, as both expressions equal ~7, they equal each other.
Example
1
illustrates
a very important property of opposites
The opposite (additive inverse) of the sum of two numbers
is
the
sum of
their opposites:
-(a
+
ft)
= -a
+
i-b).
Try problems similar to Example 1 until you see clearly that this statement is true for any numbers a and b, positive, negative, or zero. Assuming that every directed number has an opposite is a device that would have enabled us to invent the negative numbers without having to think of them as partners of points on the number line. This way, ~3 is simply the number whose sum with "^3 is 0, 3 = —^3. By
,
I 122
FOUR
CHAPTER
agreement (page 112), +3 = 3. As a result, hereafter we will simplify notation by dropping the small + and " signs. Thus, write 3 rather than ^3 and
—3
for
3 or —^3.
You now
read
—3
either as negative 3 or as the
opposite of 3.
Using addition on the number
EXAMPLE
2.
EXAMPLE
3.
line,
you may
sums:
find the following
+ (-5) = 2 + (-5) = -3 [-(-2) + 3] + (-4) = [2 + 3] + (-4) ^ -(-2)
1
ORAL EXERCISES Name
the additive inverse of each of the following directed numbers.
SAMPLE. 1.
—12
What you
say:
Twelve or positive twelve.
+
7.
2
8.
-(3
9.
9
+
7)
10.
-0
11.
7
12.
(-9)
+
(-8)
+
4
.,
.
)
13.
^%.
124
FOUR
CHAPTER
ORAL EXERCISES Tell
which of these statements are true and which are false.
=
9
1.
|9|
2.
|-345|
3.
4.
= -345 |-56| = 56 |i2^i =
-m
>
10.
-3 < |17|
12.
118.91
9.
|-8|
5.
|8|
6.
|-234|
7.
|32|
8.
1-4.91
>
>
|234|
1-3.41
|17
<
^|
d -h d\
Directed
21.
22.
6, c
a
= \y\ -y = \y\
y
13. 14.
= — Sg,
+
\a\
\b\
(-b)
\b\
\c\
and d
+
\c\
=
+
d
15.
\y\
16.
|>'|
> >
3^.
23.
a -h d
\c\
24.
\b
\d\
25.
a
+ -{-
c\
\b
+ -\-
+
\c\
d c\
Numbers
To treat addition of directed numbers without the number scale, you must Hst the addition properties assumed for the directed numbers.
NUMBERS
NEGATIVE
THE
For
all
members
125
126
^>
THE
NEGATIVE
NUMBERS
WRITTEN EXERCISES Add
the following.
127
128
CHAPTER
3^^
FOUR
PROBLEMS 1.
A
merchant's transactions had the following results: a gain of $35,
a gain of $14.75, a loss of $26.10, a gain of $18.15
a loss of $7.50.
Represent his net gain or loss by means of a signed number. 2.
A
girls'
club took in $13.00 for the semester's dues and paid out $7.50
for refreshments, $1.25 for programs,
and $2.00 for a charity project. Use a signed
Their share in the proceeds of the class play was $6.50.
number 3.
A
to represent the financial condition of the club.
housewife
made
the following entries in her household account one
day: groceries $13.68, bakery $1.09, meat $4.17, return for bottles $.37, Joan's allowance $.75.
and 4.
find the
To buy graduation more than
Represent each item by a directed number
sum. prizes,
the $73 in
its
the Parent-Teacher Association needed
treasury.
The members presented a
play, for
which they paid a royalty of $25. Scenery and costumes cost $18, and the programs, $15. The sale of tickets amounted to $185. Program advertisements brought $64. Find the amount they then had. 5.
A
rises
6.
7.
below sea level fires a rocket which far above sea level does the rocket go ?
submarine submerged 375 650
feet.
How
feet
A football player made the following yardage on five plays: 15, —3, 8, — 9, — 12. What was his total net gain in yards? If G = { — 4, —1,0, 1, 3}, find the set of all sums of pairs of elements of G.
Is
G
closed under addition?
8.
Bob lost 3 pounds the first week on his 900-calorie diet, gained 1^ pounds the second week, gained f pound the third week, and lost 4 pounds the fourth week. What was his total gain or loss?
9.
The temperature at noon was 49°F and was the net change in temperature?
4-7
Subtracting Directed
at 5 p.m.
it
was 21.5°F. What
Numbers
One evening the thermometer read 11° above zero. The next morning it read 5° below zero. How much had the temperature changed? Do you get a 16° drop? Do you realize that you just subtracted 11 from -5?
THE
NUMBERS
NEGATIVE
129
..
Another situation illustrates something else you know about subIf you buy 85 cents' worth of goods and give the clerk a dollar, he may count your change saying, 85, 95 (handing you a dime), one dollar (handing you a nickel). The clerk did a subtraction problem (100-85) by adding. Another way of saying this is that x has the same value in both of these equations: traction.
100
Similarly,
x has
-
the
(-5)
85
=
and
a:
same value
-\\ =
85
+
.v
=
100.
both of these equations:
in
and
X
11
+
= -5.
x
Guided by these results, we make this definition For all directed numbers a and b, any directed number satisfying the equation b -\- x = a is called the difference of a and b, that is, a — b. Using only this definition, you do a subtraction problem by asking yourself, "What number added to b gives a?" You can find a simple expression for a — b by transforming the equation b -{- x = a: :
b
-\-
X
X
-\-
b
X
+ +
jc
The
last
A
= = (-b) = = .. X =
+
a a a a
a
+ + ^
(-b) (-b) i-b)
equation evidently has just one root, a
b A b
+ +
a
+
(-b)
-
b
-\-
a
^
a
+
a
^ ^
a a
>/
-\-
x = a\s,a
=
a
^
+
(-Z)),
it
follows that:
(-b).
To perform a subtraction, replace the subtrahend by
add.
Checking
X = a
a
a
(
i-b) = a
+
Since the one and only root of ^
+ — b).
you have:
this root in the original equation,
its
opposite,
and
130
CHAPTER
FOUR
Does this rule give a meaningful expression for a — /?? As every number has an opposite, if you know b then you know —b. Also,
+ ( — Z?) is a sum, it represents a definite number. Hence, the shows that the set of directed numbers is closed under subtraction. Using this rule, you always can replace a subtraction by an addition:
since a rule
Check
Subtraction
2=6
+ 8 + (-2) -8 + 2 -4 + (-2) -7 + + 7) 4
(fl
=
6
= -6 = -6
=
fl
THE
NUMBERS
NEGATIVE
131
^
WRITTEN EXERCISES Rewrite these subtraction exercises as additions, and then find the sums.
SAMPLE
1.
89
Solution:
99
Check:
89
-10
+ +
(-99)
= -10
99
I =
89
SAMPLE
2.
Solution:
Check.
I
1. 1
9
k
+
5]
-
[/•
-
4]
[r
+
5]
-
[r
-
4]
+ [r-4] =
= = =
r r
+ +
5
-
[r
5
+
i-r)
+
9
9-4 +
r
=
r
+5
(-4)]
+
4
89
89
v/
.
132
j^
V31.
Prove: -(a
-
b)
32.
Prove: -(a
-\-
b
Which of these
= -a -\- b. -\- c) = (-a)
+
(-b)
+
are closed under subtraction?
sets
(-c). Explain.
33.
{odd integers}
35.
(odd
34.
(even integers}
36.
^
{0,
FOUR
CHAPTER
integers, 0}
-^ f,
-|,
-f}
f,
PROBLEMS Solve these problenns by using directed numbers. 1
Find the difference in altitude between Salton Sea, California, 244 feet below sea level, and a spot in Death Valley, 276 feet below sea level.
2.
Find the change in temperature on a winter day when the temperature dropped from 3° below zero to a low of 1 1° below zero.
3.
At 6 5°
4.
thermometer read 8° above zero. At midnight below zero. Find the change in temperature. P.M. the
above sea change in
City Rapid Transit System a high point is 161 feet and a low point is 113 feet below sea level. Find the altitude in going from the high point to the low. level,
5.
The Peloponnesian War began in 431 404 B.C. How long did the war last?
6.
The Greek mathematician Archimedes was born
7.
212
read
New York
In the
in
it
B.C.
John owes
How
B.C.
Peace was
in
finally
287
B.C.
made
in
and died
long did he Hve?
his father $4.38.
How much
must John pay to acquire a
credit of $1.25 with his father? 8.
In playing a
game Ellen was make to have
points must she 9.
10.
175 points "in the hole."
Carthage was destroyed
in 146 B.C.
event? (Assume no year
0.)
If the sea is 37,800 feet
How many
a score of 250 points?
How many
years ago
deep and the highest mountain
is
was
that
29,012 feet
high, find the difference in elevation between them.
n.
Mr. Lescaire had a bank balance of $317.25 on Monday. On Friday the bank said he was overdrawn by $9.47. How much had Mr. Lescaire spent during that period
12.
If
New
is 4rN and Rio de Janeiro's between the two cities.
York's latitude
difference in latitude
? is
23°S, find the
4-8
V
NUMBERS
NEGATIVE
THE
Multiplying Directed
133
Numbers
Probably you were
first introduced to mulby some explanation such
tiplication in arithmetic
"When we
as this:
you have
3X2
X
write 3
you have
three times. If
You
apples.
we mean take
2,
2
boxes each with 2 apples,
3
+
also have 2
2
+
2
apples." If
you
try to
meaning
give
cannot talk about apples
in
X — 2),
to 3
(
you
boxes, but you can
— 2) + — 2) + — 2), so you can say — 2) means. But, when you that that is what 3 X — try to talk about 3) X 2, you have trouble. about
talk
(
(
(
(
(
You
cannot take 2 "minus three" times conven-
iently.
To
help solve this dilemma, consider a dif-
ferent kind of example.
Suppose water
is
flowing into a tank at the rate
of 3 gallons per minute
(i).
You
can make the
following statements.
1
Two minutes hence (2), there will be 6 gallons more (6) in the tank. (2)(3)
positive
2.
Two
number
X
positive
=
6
number
gives positive
minutes ago (—2), there were 6 gallons
(-2)0) = negative
number
X
positive
less
(
Two minutes hence (2),
number
positive
4.
Two
number
X
negative
in the tank,
,
gives negative
there will be 6 gallons less
(2)(-3)
— 6)
-6
Suppose that water is flowing out of the tank at the per minute ( — i). You can make these statements. 3.
number
number
rate of 3 gallons
(— 6)
in the tank.
= -6
number
gives negative
minutes ago (—2), there were 6 gallons more
number
(6) in the tank.
(-2)(-3) = 6 negative
number
X
negative
number
gives positive
number
.
:
:
134
CHAPTER
-.
The
rules suggested
FOUR
by these examples can be developed from the
following assumptions for multiplication.
and
For a, b,
c,
members
The closure property: for every a and
1
I
of the set of directed numbers:
product ab
the
b,
is
a
unique directed number.
= ba. associative property: a{bc) — (ab)c. distributive property: a(b -\- c) = ab = multiplicative property of 0: a = multiplicative property of 1: a
2.
The commutative property: ab
3.
The
4.
The
5.
The
6.
The
1
•
ac.
-{-
•
1
•
a
•
a
= =
0. a.
property might make you curious about the product Would you guess that it would be —a? To verify this guess, show that the sum ofa{—\) and a is zero:
This
a
•
last
(—1).
+
a
— 1) +
a(l)
a(-l) a(
a[(—
1)
+
1]
a(0)
I ^
Multiplicative property of 1
^0
Distributive property
^ =
Property of opposites Multiplicative property of zero
v/
—1
is
gives
its
Therefore, the multiplicative property of Multiplying any number For any a,
A
special case
(— 1)( — 1) =
1.
of
this
by
—
a( — 1) =
1
(—
= —a.
l)a
when a = —\;
property occurs
You now
opposite.
this
gives
can justify the products you obtained in the
four cases of multiplication illustrated by the water tank problem by writing the following chains of equalities
=
6
1.
(2)(3)
2.
(-2)(3)
=
[(-1)(2)](3)
=
(-1)[2(3)]
3.
(2)(_3)
=
(2)[(-l)(3)]
=
[(-1)(2)](3)
4.
(-2)(-3) =
Similarly, for
all
[(-1)(2)][(-1)(3)]
numbers a and
=
=
(-1)(6)
= =
= -6
(-1)[2(3)]
= -6 [(-1)(-1)][(2)(3)] = 1(6) = (-1)(6)
b:
b(-a) = (-a)(b) = [-Ha)]b
=
{-l){ab)
(-a)(-b) = [-Ha)][-l(b)] = [(-1)(-I)](fl6)
= -ab = Hab) =
ab
6
NEGATIVE
THE
Do »
you
1.
NUMBERS
135
see that the following statements are true ?
The absolute value of the product of two directed numbers
the
is
product of the absolute values of the numbers. 2.
The product of a positive and negative number
3.
The product of two positive numbers or of two negative numbers
a negative number.
is
is
"{'y
a
positive number.
By
pairing
(—1)(—1) =
1
,
you can extend these
any number
rules to
of factors. 1.
The absolute value of an indicated product of numbers
is
the product
of the absolute values of the numbers. 2.
An is
3.
indicated product containing an
An indicated product is
odd number
of negative factors
^>
a negative number. containing an even
number of negative factors
a positive number.
Since the distributive property
numbers, variables
assumed to hold for directed an expression are treated as they
is
in the terms of
have been previously.
EXAMPLE. Solution:
Simplify:
6x
-
4y
6x
-
-
5x
4y
+
-
Sy
5x
-^
^
Ix
Sy -h Ix
-
9y
-
9y
-
=
{6
+
(-4
5
^
+
8
l)x
-
9)y
ORAL EXERCISES Find
each of the indicated products.
1.
(4)(5)
8.
-4(7)(-l)
15.
A{-^){-2b)b^
2.
(-6X-2)
9.
-1(1)(-1)
16.
7(-i)(-3^y3
3.
(7)(-3)
10.
K-2)
17.
(-4)(-2)(0)(-l)
4.
(2)(9)
11.
(-3)(-i)
18.
{-x){-xy^){-y){Qi)
5.
(-5)(-4)
12.
4a(-5a)(10)
19.
(-2)3
6.
(-15)(-2)
13.
(-3)(2fl)(a)
20.
(-3)3
(7«)(0)(-6^)
21.
2(-3)4
7.
(-3)(5)
14.
136
THE
V
NUMBERS
NEGATIVE
137
WRITTEN EXERCISES Evaluate each of the following numerical expressions (a) by combining terms,
and
(b)
by
using the distributive property.
SAMPLE.
(-3)[7
Solution:
a.
-
(-2)]
-
(-3)[7
(-2)]
= (-3)[7 + = (-3)(9) = -27
+
1.
(-4)(10
2.
(-3)[(-2)
3.
(20
4.
(5)(-6
-
5^ 0(-8
+
-
[-3
7.
4(3)
8.
(-2)9
9.
(-99)(12)
10.
21.
22. 23. 24.
25. 26.
27. 28. 29. 30.
2a 4r
3)
(-2)
13.
14(-5
14.
(-48)(-3^)
15.
-(-7 + 12) -(-34 - 20)
+
19.
(40)(-99)
20.
3
7
-\-
+
5a -3^
in
-
\0a
+
-
9
-{-
-}-
S
-{-
+ 5(-r - s) - 3b) + 9{-b + a) 35)
31. 32.
8r
-{-
+
-
-
(-3)(-2)
6
(-7)(13)
\)
- (2.5 - .32) 6 - (-1.9 + .27) -.3 - .2(-.7 - .3) -.7 - .2(-.l - .4) .8
each of the following expressions.
\
-l(a
i)
(-12)
-\-
42.
+
18.
>'
-2(r
+
(-12)(4
17.
-6x + 5 - 2x + X - 12 -8;^ + 8 - 6>' + 14 -4nt + 8 - 6 - «/ + 3a7/ -9hk 5hk - S -^ hk 5 4r r 6r 5s s \6a - 9b - a + b - lb 1.5« - 8 - 3.5/7 + 5 y - i-h + + y
41.
(-13)(-7)
4(13)
the similar terms
-
11.
16.
+
(_3)(7)
12.
(-7)]0
+
40(99)
Combine
(-5)]
5)
6.
(-2)]
= -21 = -27
21)(4)
-
-
(-3)[7
=
2]
11)
+
b.
33. 34. 35. 36.
37. 38.
39. 40.
-3fl
9m
—
+ M
b-\-4a
+
5
—
+ a-b 6w — 6 +
2m
33A:3 - k^ + Ak^ - AOk^ xyz — %xyz + 5xyz — 2xyz -d^ - Ad^ + .5^2 ^ 1,8^2 5(r + 5) - 6(r + s) - 8(r + 4x^ — 5x — 6x^ + 7x -13;; + 2y^ - 4y^ + 6y 4m w^ — 2m — 3 — m^ «3 - 2w2 + 4w + 3m2 4m -\-
-
43.
3{p
44.
-4(-7v
2q)
-
+
(5q
-
-
2p)
(-f
-
3v)
s)
138
a^
45.
2[3(a
-
1)
46.
3[-6
+
2(fl
+
5]
-
-
4)]
4
+
26
CHAPTER
47.
-In -
48.
-lO/z
-
5[2(1
4[-l
Evaluate each of the following algebraic expressions, using a c
= —2,
49.
X
=
.3,
and y
= —3.
+
-
3]
3(3/z
-
2n)
-
FOUR
= — 1,
b
=
2)]
.2,
NEGATIVE
THE
A
NUMBERS
139
"^^
statement about reciprocals that corresponds to the property of
opposites on page 121
is:
The reciprocal of a product of two numbers, each
diflFerent
from 0,
is
the
product of the reciprocals of the numbers.
_
J_ ab
I a
I b
Reciprocals enable you to express a quotient as a product. If/j the root o{
xb
=
^
is
a
•
-
,
?^
0,
you can show.
as
xb 1
.
xb
-
X
•
b
1
=
a
=
a
=
a
1
'
b
X = a
This
last
equation has just one root, a
-
-.
Checking
in the original
equation
xb = a
(-1) b ^
a
\
=
a
a
=
a
a
•
Since the one and only root of xb
a
^
b
=
=
a-,
a
\/
is
a
b
•
b
,
it
follows that:
9^
b To perform a
division,
replace the divisor by
its
reciprocal,
and
multiply.
m
140
j^
Does
this rule give a
number except
CHAPTER
meaningful expression for a
has a reciprocal, and b
9^
0,
there
FOUR
-^
bl As every
is
a
number
7
•
b
As a
•
b
directed
is
a product,
numbers
is
^
it
represents a definite number. Thus, the set of
closed under division, not including division by 0.
= -111-
=
EXAMPLE
1.
EXAMPLE
2.
EXAMPLE
3.
^ = if-
EXAMPLE
4.
?.('-^U?(_5)=_('?.?U_?
^)
-2 = -=-2y /1\
i^
4
--2
= -
-
3X5/3X4/
\3 4/
ORAL EXERCISES Give
6
THE
NEGATIVE
Give the 33.
1
NUMBERS
multiplicative inverse of
141 each number.
142
Q
CHAPTER (r/)2
—
u'
31
»
(r
33.
-
f3
34.
FOUR
THE
NEGATIVE
NUMBERS
143
The better your guess, the smaller the average deviation, but what you guess really doesn't matter. Suppose you assumed 210 pounds as the average.
Weight
144 5.
CHAPTER
'S,
Fahrenheit temperature readings taken at
one week were:
8 a.m.
-6°, -9°, -14°, -4°,
13°, 5°,
FOUR
each morning during 2°.
Find the average
8 A.M. temperature for that week. 6.
The following
daily net changes in the price of one stock were observed
two weeks: — 1|-, — ^, — |, Find the average daily net change in price.
in the course of
7.
At
2^, \^,
— ^,
3^, 0,
—5,
4^.
determined experimentally the value
different times, eight scientists
of a constant, as follows: 4.177, 4.188, 4.196, 4.196, 4.186, 4.188, 4.184, 4.181. 8.
A
Find the average value.
high school physics class was trying to verify a rule.
the following values for a constant:
1645.1,1646.0,1649.6,1645.8.
1654.8,
1655.6,
They obtained 1654.8,
1643.5,
Find the average value.
Chapter
Summary
Inventory of Structure and Method 1.
The number
scale can be extended to the left as well as to the right of the
zero-point.
Points to the
numerals include minus
left
correspond to negative numbers, whose
Points to the right correspond to positive numbers, whose numerals may include plus signs. For example, ~3
marks the point
signs.
3 units to the left of 0; +3, or 3, the point 3 units to the
right of 0.
A
number a is greater than every number number to its right on the number line.
to
its left
and
less
than every
2.
The sum a -\- b, for any numbers a and b on the number hne is found by moving from a a distance \b\ units in the direction associated with b. Every number a on the number line has an opposite or additive inverse such that a + (-a) = 0. Thus, -(+3) = "3 and -(-3) = +3. -0 = 0. The addition of directed numbers can be developed without reference to the number line if you assume closure, commutative, and associative properties, and properties of zero and of opposites.
3.
The
difference a
gives a.
—
of directed numbers 4.
b
Subtracting b is
is is
number which added to b The set
defined as that the
same
as adding the opposite of b.
closed under subtraction.
You can deduce other properties of multiplication of directed numbers from the closure, commutative, associative, and distributive properties and properties of 1 and 0. You have (— 1) a = —a. For a and b, any •
THE
145
NUMBERS
NEGATIVE
directed numbers,
=
ab\
or both negative, and ab
^
if
0,
the quotient a
^
Z?,
=
a
tipHed by b gives a; -
is
inverse,
-
A
to find
relatively easy
deviation.
its
way
A = G
-\-
other,
number which mula
a,
9^ 0,
has
a
a and - are both positive or a
;
= ai-j
b 9^
,
0.
^ G from
of finding an average
to use a guessed average G.
is
and the
positive
Every number
both negative. Zero has no reciprocal. -
numbers
is
defined as that
is
,
a
6. (Optional)
a and b are both positive
0.
not defined.
or multiplicative
reciprocal
or b
if
one factor
=
ab
>
ab
\b\,
b
line that if point
of point
left
FIVE
c,
a
lies
to
then point a
c.
illustrates the transitive
property of inequality in the set of directed
numbers: For any directed numbers
On
points in
if
a
2.
if
a
number
the
—4
from
1
< >
< >
b and b
A and b
line,
—4
lies
3
1
it
then a
c.
—4 < -4
follows that
-4 +
and
4- (5)
(_5)
b
b
— —
C; similarly, c.
EQUATIONS,
Notice what happens
-4 < it
3
AND
INEQUALITIES,
by
if
PROBLEM
161
SOLVING
you multiply each member of the inequahty = -8and(3)(2) = 6, and also -8 < 6,
Since (-4)(2)
2.
follows that
—6,
member of —4
member of
(3)(-2).
the inequality by
—2
reverses the
sense of the inequality. -4
When you
2v
-
3/-
and
2/-
7 or 4v
< 5/7 - 1)(.Y +
16.
11/7
12 or
17.
(a-
2)
13
>0
-1 > 3
+ -
3) 3) 1)
> <
185
roll.
EQUATIONS,
^
INEQUALITIES,
AND PROBLEM SOLVING
167
«^
168
CHAPTER
FIVE
PROBLEMS Take the four steps given on page 57 sketch 1.
when
Michael and Robert are going If the total
catch
The Red Cross
is
19 fish,
Michael owns the boat; there-
fishing.
fore the boys have agreed that he
2.
each problem, making a
solving
in
possible.
to get 5
is
how many
more
fish
than Robert.
each receive?
will
The Junior Red
knitted 50 sweaters within ten days.
Cross assisted, contributing 2 dozen fewer than the senior organization.
How many 3.
did the Junior
Red Cross knit?
The Jowett family budgets part of its weekly income of SI 50 for food. Half the remainder of the income exceeds the amount spent on food by from $ 5 to $30. How much do they spend on food per week ? 1
4.
Tom
5.
berries. Tom picked How many did each pick ?
and Otto picked 36 quarts of
half the
number Otto
picked.
Day
Mrs. Abbott decided on Christmas
mas. She started saving $5 a week, but could reduce that amount. By saving and 6.
The length of feet
7.
8.
still
at the
a playground exceeds twice
of fencing are needed to enclose
it.
more than
to save $150 for next Christ-
at the
how much
have $150 or more
3
end of 12 weeks saw she
could she reduce her weekly
end of the year? its
width by 25
Find
its
feet,
and 650
dimensions.
The length of
a rectangle exceeds three times the width by 6 feet, and
the perimeter
is
188
feet.
Find the dimensions of the rectangle.
In a certain puzzle, the larger of two numbers must exceed three times the smaller by
5,
and
their difference
must be
at least 31.
Find the
least
possible value of the smaller number. 9.
To be
called
more than at 63 miles
10.
A
child's
thirds as
"Limited" a
train's average speed
must be
5 miles
an hour
twice the average speed of a "Local." If the Limited travels
an hour, what
is
the speed of the Local?
bank contained twice
many dimes
as
many
nickels as pennies
and two-
as nickels, the total value being at least $3.65.
Find the smallest possible number of coins in the bank. 11.
Bill
Jones wanted Sally Smith's telephone number.
Sally said that
ninety. added to her age equaled six times her telephone number,
minus 6060. Bill knew that Sally was eighteen years old, but he didn't know enough algebra to call her. Find Sally's telephone number. 12.
The area of wide.
a twelve-foot square equals the area of a rectangle 9 feet
Find the length of the rectangle.
EQUATIONS,
A
rectangle
is
9 feet by 8 feet.
rectangle 12 feet long.
^
In a
PROBLEM
INEQUALITIES, 'A.ND
Its
169
SOLVING
area
is
three times the area of a
Find the width of the second rectangle.
new school building, 270 cubic feet of air are to be allowed for To meet this requirement, what should be the height of
each pupil.
the ceiling of a classroom, 30 feet by 24 feet, seating 36 pupils? 15.
A
coal bin
is
and 9
15 feet long, 10 feet wide,
feet high.
If 10| tons of
egg coal which runs 28 pounds to the cubic foot are put
in, to
what
height will the coal reach? 16.
A
bicycle wheel has a diameter of 2 feet.
it
make
in
ference of a circle
17.
^
David
is
given by the equation c
making a model of a rectangular
52 inches long. The length
be 1
is
8.
1
An
inch
more than
isosceles triangle
is
a whole
revolutions will
the width. is
number and
of the two equal
sides.
number between
What
4
solid
use
tt
=
-y^-.)
from a piece of wire and the height is to
feet less
feet.
two
A
sides
j^gng^e
isosceles triangle
than the
The perimeter
and 75
rrd;
are the dimensions of the solid?
a triangle having
is
=
to be twice the width,
equal in length. The base of an is
How many
going 5500 feet? {Hint: The rule for finding the circum-
is
sum
a whole
Find the possible
lengths of each side. 19.
Mrs. Fry weighs 50 pounds
less
Their combined
than her husband.
220 pounds more than that of their daughter, who weighs half as much as Mr. Fry. What is Mrs. Fry's minimum weight? weight
is
at least
20.
Mr. Martin earns three times as much in his regular job as he does as a writer.' His total income is at least $14,000 more than that of his sister, who earns only half as much as Mr. Martin does in his regular job. What is the least amount he earns in his regular job?
21.
Farmer Brown needs .03 acre of land to grow 1 bushel of corn and .06 acre to grow 1 bushel of wheat. He has at most 480 acres of land for planting and wants to use at least half of that acreage. If he decides to grow twice as much corn as wheat, find (a) the maximum and (b) the minimum number of bushels of corn he can grow.
22.
In a factory the time required to assemble a table a chair, 30 minutes.
The
able each day, but can provide as as
many
much
20 minutes and
as 140 hours daily. If four times
chairs as tables are produced,
maximum number
is
factory has at least 126 hours of labor avail-
find the
minimum and
of chairs the factory can produce.
the
^^
170
5-5
CHAPTER
FIVE
Problems about Consecutive Integers
is another name for any whole number, positive, negaThe integers have many interesting properties, and to talk about them you need a few descriptive terms. An integer which is twice some integer is called even; all others are called odd. For example,
Integer
tive,
or zero.
2, 126, 0,
—10
are even integers;
The word consecutive exactly as in
is
—15, 77 are odd.
3,
used here to
ordinary language when you
vatdin following in order, just
say, "I got
A
in algebra for
three consecutive weeks." Counting by ones gives consecutive integers: ...
2, 3, 4,
1,
.
The
three largest consecutive two-digit integers are
Likewise, —6, —5, —4, —3,
97, 98, 99.
is
a set of consecutive integers.
Counting by twos from an even integer gives consecutive even integers: 2, 4, 6, 8, or —4, —2, 0. Counting by twos from an odd integer gives consecutive odd integers: 15, 17, 19, or —5, —3, —1, 1. Some consecutive multiples
Two
of
five are 5, 10, 15, 20.
two consecutive even integers differ by 2. Two consecutive odd integers also differ by 2. If x represents any integer, then .t + 1 is the next larger integer and .t — 1 is the next smaller integer. If x represents an even integer, then .t + 2 consecutive integers differ by
What
the next larger even integer.
is
an odd
If
X
is
the next smaller
is
EXAMPLE
integer, then
.v
1
+
2
;
is
the next smaller even integer?
the next larger
is
odd
integer.
What
odd integer?
Find three consecutive integers whose sum
is
48.
I
Solution:
» Let n Then
(/i
+
The sum of
1)
=
the
=
the second integer and (w
+
2)
=
the third integer.
+
1)
+
(«
+
first integer.
the integers
Solve the equation:
is
n
48:
3/i
+
3
3/1 /I
.-.
and
/I
/I
+ +
1
2
+ = = = = =
(/i
48 45 15
16 17
2)
=
48
j
EQUATIONS,
A
Is the
INEQUALIIIES,
sum of
IkUD
these integers 48?
PROBLEM
SOLVING
171
172 9.
CHAPTER George Dean plans
to use
FIVE
60 inches of lumber for four shelves whose
lengths are to be a series of consecutive even numbers.
How
long shall
he make each shelf? 10.
The
1
1.
240
feet, find
the length of each side.
The smaller of two consecutive even larger.
12.
The
numbers. If the perimeter of
sides of a triangle are consecutive
this triangle is
integers
is
2
more than twice
the
Find the numbers.
larger of
Jl
two consecutive odd integers
is
4 less than ^ the smaller.
Find the numbers.
1
3.
Find four consecutive integers such that by twice the second is 7.
five
times the fourth diminished
14.
Find four consecutive even integers such that four times the fourth decreased by one-half the second is 9.
15.
Three times the smaller of two consecutive odd integers twice the larger.
16.
What
is
less
than
are the largest possible values for the integers?
Three consecutive even integers are such that their sum is more than 24 decreased by twice the third integer. What are the smallest possible values for the integers?
17.
The
larger of
the smaller. 18.
two consecutive integers
What
is
greater than 4
more than half
are the smallest possible values for the integers?
Three consecutive integers are such that the sum of the first arid thiid What are the largest is less than 18 increased by half the second. possible values for the integers?
5-6
Problems about Angles Think of the
figure
composed of two
rays/?
and q drawn from a
Then think of the ray q as having turned or rotated about O, p and going to its indicated position. As shown, the rota-
point O.
starting at
tion
may
be clockwise or counterclockwise.
Terminal side Vertex
Terminal side
q Initial side
Counterclockwise Rotation
Clockwise Rotation
I
EQUATIONS,
The
figure
AND
SOLVING
173
composed of two rays drawn from a
point, together with
INEQUALITIES,
PROBLEM
the rotation that sends one ray into the other
is called a directed angle- Counterclockwise rotation yields a positive directed angle;
clockwise rotation yields a negative directed angle. Ray p is the and ray q is the terminal side. The point O
side of the angle
initial is
the
vertex of the angle.
A common A degree
is
unit of measure of an angle
is
a degree, written as
1°.
3^0 of a complete rotation of a ray about a point. The whose measures are 1°, 30° (read "30 degrees"), 90®,
directed angles 180°,
-45°, -180°, and -360° are shown:
-180
zlZA o 180°
Two
complementary angles if the sum of their measures Each is the complement of the other. If an angle contains n degrees, its complement contains (90 — n) degrees.
is
Two is
angles are
90°.
180°.
angles are supplementary angles
Each
is
if
the
sum of
their
measures
an angle contains n dedegrees. The diagrams on
the supplement of the other.
If
supplement contains (180 — n) the next page show complementary and supplementary angles. grees, its
174
CHAPTER
FIVE
*S.
Complementary Angles
Supplementary Angles
EXAMPLE. How
large
is
an angle whose supplement contains 21°
less
than
complement?
four times
its
Let n
=
the
number of degrees
in the angle.
— —
= =
the
number of degrees
in its
the
number of degrees
in its supplement.
Solution:
Then (90 and
(180
/i)
n)
The supplement
complement,
four times the complement
less i
(180
-
4(90
n)
Steps 3 and 4 are
left to
-
n)
21 i
21
you.
The three line segments that compose a triangle intersect by pairs and so form three angles. If you tear off the corners from any paper triangle and fit them together as shown in Figure 5-1, you will notice
EQUATIONS,
INEQUALITIES,
that the three angles
fit
AND
PROBLEM
175
SOLVING
together to form a straight angle.
gests a property of all triangles
which
is
proved
in
The sum of the measures of the angles of any triangle
figure 5-1
ORAL EXERCISES
1.
This sug-
geometry. is
180°.
176
CHAPTER
FIVE.
I
'^S. In
Exercises 13-18, find the
number of degrees
a
in
+
b,
if
the measures of
a and b are as indicated. a
=
30°, b
=
^ of a complete rotation clockwise
b
=
15°,
a
=
5 of a complete rotation clockwise
15.
a
16.
a
= =
f of a complete rotation clockwise
b
=
^ of a complete rotation counterclockwise
a
= = = —
^ of a complete rotation counterclockwise
13.
14.
17.
b 18.
a b
Exercises
a.
positive straight angle, b
a.
negative straight angle
I of a complete rotation clockwise
5 of a complete rotation counterclockwise 5 of a complete rotation clockwise 1
9 and 20 refer
to the
Law
of Reflection:
/
=
r.
20.
19.
/
r
= =
Find
Exercises is
=
(In
+
30)'
a
(4/7
-
10)'
b
10)'
Find m.
n.
21-26
refer to the science of navigation
expressed as a bearing. The bearing of a
with the north
= 2m° = {m +
line,
the observations are
line
measured clockwise from
made.
Find
in
which a compass direction
of motion
is
north, through
each bearing.
the angle
it
makes
a point at which
EQUATIONS, 24.
AND
INEQUALITIES,
PROBLEM
25.
^,
E
177
SOLVING 26.
.,
W
E
W
PROBLEMS 1.
An
angle
in the
is
12°
more than
its
complement. Find the number of degrees
complement. 28° less than the other.
2.
Find two complementary angles
3.
An
4.
Find two complementary angles
if
one
is
18° less than 3 times the other.
four times the other.
angle
15° less than twice
is
if
its
one
is
complement. Find the angle.
5.
Find two supplementary angles
if
one
is
6.
Find two supplementary angles
if
one
is five
7.
One angle of a triangle is twice as large as another. The third angle contains 5° more than the larger of these. Find each angle.
8.
One
angle of a triangle
angle
is
20° less than the
is
times the other.
three times as large as another.
sum of the
first
The
third
two angles. Find the number
of degrees in each angle. 9.
In any isosceles triangle two angles are equal to each other.
angle of one isosceles triangle
Find each angle of the
Two
less
first
two.
How many
sum of the
other two. Find
A triangle is to be drawn in which and the
its
third
than the
one
me
degrees are in
The
each angle ?
third angle
is
6°
angles.
a;a^le is 18° larger
sum of the
23° less than
is
others.
than another,
Find the angles.
an angle whose complement contains 5° more than half supplement?
-How its
third, 12° less
The
other two.
triangle.
angle of a triangle exceeds another by ^3°.
than the
How
sum of the
angles of a triangle are equal, but the third angle
2^ times the sum of the
One
36° less than the
is
large
is
large
is
an angle whose supplement contains 12°
complement?
less
than twice
:
178
CHAPTER
5-7
in
FIVE
Uniform Motion Problems
An object which moves without changing its speed is said to be uniform motion. Often, charts can help you in organizing the given
facts in will
problems involving uniform motion. The basic principle you
need in such cases
EXAMPLE
is
(Motion
1
distance
=
rate
X
time
d
=
r
•
t
in
Opposite Directions) Mr. Rush and Mr. Slow
arrange to meet at an airport that line
is
between, and in a straight
home airports. Mr. Rush's jet travels at per hour; Mr. Slow's plane travels at 320 miles They leave their home airports, which are 1380
with, their
600 miles per hour.
miles apart, at the
a nonstop
flight, in
same time. If each plane is scheduled how many hours will they meet?
for
Solution:
^
Let n
=
Make
a sketch
the
number of hours before
the
men
meet.
illustrating
Mr. Rush's jet rate is 600 m.p.h. Mr. Slow's plane rate is 320 m.p.h.
the given facts.
Total distance
Each
600/?
is
travels the
1380 miles.
same number of hours.
320/?
Rule 1380
\
Arrange the \ facts in \
chart form.
EQUATIONS,
INEQUALITIES,
179
AND PROBLEM SOLVING
Solve the equation:
+
600/1
320/1
=
1380
920/1
= =
1380
n
To check whether
the
How
man
far did each
men met
in l|^ hours,
you have
this question:
=
900 miles
ij =
480 miles
these distances
^
1380 miles
1380
=
1380
Mr. Rush
flew 600/1 miles:
600
•
Mr. Slow
flew 320/i miles:
320
•
The men
2.
answer
fly?
The sum of
IXAMPLE
to
li
ij
will
meet
in
v/ 1^ hours, Answer.
(Motion in the Same Direction) An airplane which maintains an average speed of 350 miles per hour passed an airport at 8 A.M.
A jet
following that course, at a diff'erent
same airport at 10 A.M. and overtook noon. At what rate was the jet flying?
altitude, passed the
the airplane at
Solution.
Let X
=
Make
a chart
the rate of the jet in m.p.h.
Rate of airplane
is
350 m.p.h.
of the facts
Periods of time under consideration
given in the
Airplane :
problem.
8 A.M. to noon, or 4 hours
Jet: 10 A.M. to noon, or 2 hours
Each plane covered
Rule
the
same
distance.
\
Make \
a sketch
illustrating
the facts
\ 1
given in the
problem.
180
CHAPTER
EXAMPLE
3.
(Round Trip)
A man
leaves his
home and
FIVE
drives to a conven-
tion at an average rate of 50 miles per hour.
Upon
arrival,
he finds a telegram advising him to return at once.
He
catches a plane that takes him back at an average rate of
300 miles per hour.
how long
did
it
If the total traveling time
take him to
fly
back?
How far
was if hours, from
his
home
was the convention? Solution:
Then I
Let A
number of hours
flown,
-
number of hours
driven.
h
Home
The given
miles driven
is
the
Convention City
The total time is if hours, b. The driving rate is The flying rate is 300 miles per hour. d. The number of same as the number of miles flown.
facts are these: a.
50 miles per hour.
Rule
City
c.
Nymber System
Structure
Faced with such problems as counting the animals in his flock or comparing the band with that of his enemy, man eventually conceived the
size of his warrior
To count you need a
natural or counting numbers.
know what number comes next
properties of the set of natural numbers are:
namely,
1;
{»')
there
Hence, two important a
is
its
succession of equally
spaced
the set, such that
in
immediate successor no other natural numbers
Consequently, the set of natural numbers
can be inserted.
natural number,
first
every natural number has an immediate successor
(//')
between a natural number and
a
number, and you have to
first
any given number.
after
is
end
points extending without
usually pictured as a
in
one direction along
line.
The operations of addition and multiplication arise when you seek to count the members in the set formed by combining the elements in two or more sets having no members in common. For example, if you have 3 pennies in one row and 2 in another row, you have, in ail, 5 pennies, a fact expressed by the symbols: 3
+
=
2
On
5.
have 6 pennies
and
tion
the other hand,
in all:
3X2 =
6.
if
you have 3 pennies
multiplication:
For any natural numbers a and b the sum a ab are both definite natural numbers. This
=
c and d such that a
in
enlarged
>
For example, 2
are also called integers. than 0. 3
—
2
every number
If
=