Basic Algebra [Teacher's Edition] 0395278643, 9780395278642
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English Pages 464 [648] Year 1980
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Teacher's Edition
THIS BOOK OrDER OF-
c4^
TUj^f^ -
A^fX. cu-~J
National "Bank of Detroit
II
Be
careful!
for
somebody if
Now why 1.
2.
It
you
2.
19
yfof.
/-^
1
?P
DOLLARS
^
your
'0
$J5^
ii:90Bi.o;se
check
No
-r-
JoinimA^
Pay TO THE
to
Write
—00, ^ when loo' 100
there are
no
cents.
ifi
might be possible else to cash the
lose
it.
not try writing
Write a check for $ Write a check for broke.
3.
Write a check for bicycle.
4.
Write a check for $ a car.
6.
Write a check for $ rock concert.
©
1974 United Feature Syndicate.
1
190
T9
Supplementary Materials This Teacher's Edition contains lesson-by-lesson commentary on the student text, with teaching suggestions, chalkboard examples, and ex-
Chapter tests and answers, an individualized assignment and a table of related references are also provided. Answers to classroom and written exercises appear on full-sized facsimiles of the tensions.
guide,
student pages. Progress Tests are a convenient
way
to
keep track of each
dent's performance.
Each
stutest Test 9
is
keyed
to the
student
text.
{Sections
appear in EdiAnnotated Teacher's the
Answers
to all tests
ions
AND NEGATIVE NUMBERS
POSITIVE
Write the answers
1
in
and
2)
the spaces provided
QUESTIONS
tion of the Progress Tests. 1
The
Solution
Key
provides
step-by-step solutions for every
Write a positive or negati\ 1.
a
liis^ i>r
S piiinls
2.
a
gam
of $12
written exercise and puzzle.
Solution
Key
r
Chapter Tests
Chapter If
AC
\.
=
1
6, find
+
X
ym/M/M/m/M/M/M/m/m/m/m/A
the value of the expression.
1
3x
2.
3.
X
3
-=-
-
5{x
4.
2)
Find the value of the expression. 5.
3
-
7
•
2
6.
(5
+
6)(7
-
7.
5)
-
(9
-
3)
-
3
2
Simplify. 8.
+
3x
7j»c
+
X
—
5a
9.
a
10. 2>x
?>a
-\-
Ay
-\-
-\-
-
X
7>y
Find the value of the expression. 11.
52
25
12.
13.
3^
power
2 to the fourth
14.
Simplify.
X X
15.
3-(5x)
16.
4-6a
4)
Find the value
if
-
26.
X X
lb)
24. 4(5
-
4m)
If not, write impossible.
possible.
8)
-
23. 6{2>a
1)
+ —
27.
X
— 28.
— X
Simplify.
+
+
4x
-
29.
3x
Tell
which of the numbers are solutions.
32.
3x
34.
X
Copyright
-
+ ©
8
5
7
=
1 1
> 20
5
30.
4, 6,
-
6b
+
6b
-
3a
or 8?
11, 12,
1977 by Houghton Mifflin
la
13
Company. Permission
to
reproduce
is
granted to users of
31.
+
33.
3
35.
3>'
2x
+
8
BASIC ALGEBRA.
\2a
+
6(5
-
=
9
1, 2,
or
'
=
X
+J
2.
4y
4.
;c
6.
2a
-2y = +
3x
-
3-??
+ +
ly 2^ 2c
=
2a
=
4x
-
3Z)
+ +
67 2c
4
•PROPERTIES OF ADDITION
10-11)
(pp.
Teaching These Pages To
reinforce the
meaning of commutative, show the students some opera-
tions that are not
commutative.
-
4
-4
2 7^ 2
4
-
2 7^ 2
-4
meaning of associative could be reinforced by showan operation that is not associative. Use numerical examples to point out that the commutative and associative prop3 8 10 erties are helpful in developing mental In a similar way, the
ing that division
is
i
1
arithmetic.
10
7
'
2
'
20
Chalkboard Examples
Some
students can check their work by substituting a value and finding the value of the expression.
for the variable
Simplify. 1.
3a
+2 +4a +6 =la +
3.
5x
+
2x
—
4v
+
X
=
^x
^
—
Ay
+6 -lb -2=
2,
4b
4.
12m
-I-
\0n
+
lOw
lb
-
+
2n
A
= 22m +
Sn
Objectives for Pages 12-21 1.
To
2.
To use exponents
3.
To use the commutative and
find the value of expressions with exponents. to simplify expressions.
associative properties to simplify expres-
sions. 4.
To use the
5.
To use
5
distributive property to simplify expressions.
the properties of
and
1
to simplify expressions.
-EXPONENTS AND FACTORS
(pp.
12-13)
Teaching These Pages is another shorthand used in matheExponents are often used to write large numbers in a concise form. For example, one bilhon is written as 10^, one trillion as 10^^^ and the distance from the earth to the sun as 150 x 10*^ km.
Point out that the use of exponents
matics.
T37
Chalkboard Examples Find the value of the expression.
=
23
1.
8
32
2.
=
9
103
3.
^
1000
4.
3^
5.
1
51
=
5
Simplify. 6.
y
'
y y = y^
1.
a
•
a
•
a
•
=
b
a^b
8.
S
X X X 'y 'y '
'
•
=
Sx^y^
=
\2k^
Suggested Extensions
Which
1.
2^; 32; 2^
2^ or 52? 32 or 23? 2^ or 42?
the larger value:
is
The students should continue the above theme by picking other numbers and recording their results.
How many squares can you find How many squares in a 3-unit
2.
in this figure?
=
42
pairs of
5
by 3-unit square? The stu-
number of squares, etc. The
dents should find a pattern in order to predict the
by
squares in 4-
6
•
by 5-unit by 2-unit square is
4-unit squares, 5-
number of squares
in a 2-
by 3-unit square
32 -f 22
is
-|-
F,
22
+
P, in a 3-
etc.
PROPERTIES OF MULTIPLICATION
(pp.
14-15)
Teaching These Pages This section extends the commutative and associative properties to multiplication.
Again, these properties can be applied to mental arithmetic. To class, illustrate how quickly one can multiply by 50.
motivate the 50
.
= (y
24
•
100)
meaning of
Point out the
•
24
=y
•
(
100
•
=y
24)
•
2400
=
1200
evaluate.
Chalkboard Examples 1.
Evaluate 2 2
•
16
•
5
16
•
= = = =
2 (2
•
5
•
•
Show how you
5.
•
16
5)
•
16
use the properties.
commutative property associative property
10-16 160
Simplify.
=
2.
3-5v
5.
(4s)i5s)
T38
I5y
=
20.v2
3.
2y3y =
6.
5
•
a
•
fl
•
6y^ 6
=
30fl2
4.
i3k)(4k)
7.
3'k-2'k =6k^
Suggested Extension Explain each step of the method in Teaching These Pages to muhiply by
Have
50.
the students use that
method
compute these products mentally.
to
1.
50
•
12
=
600
2.
24
•
50
=
1200
3.
52
•
50
=
2600
4.
50
•
76
=
3800
5.
38
•
50
=
1900
6.
50
•
98
=
4900
54)
=
Provide a sheet of questions based on the above
Have
skill.
the students try the
questions over a period of time to see the improvement in their mental tation
7
•
compu-
skills.
THE DISTRIBUTIVE PROPERTY
(pp. 16-17)
Teaching These Pages To introduce
this topic, illustrate
how
quickly certain problems can be solved
mentally by using the distributive property. 3
•
99
=
3(100
-
1)
=
-
300
^
3
297
class may be motivated by randomly picking products that involve 99 and seeing how quickly the product can be found.
The
Chalkboard Examples Evaluate each expression in two different ways. 1.
+
5(2
8)
=
50
2.
+
4(97
3)
=
400
+
3.
12(46
6.
l{a
-
2b)
9.
5(jc
-
\\y)
1200
State the expression without parentheses. 4. 1.
+ 2) = 3x + 6 8(3x - 2) = 24jc -
3(x
Use the 10.
16
distributive property.
3(2x
+
3)
-
6
=
6jc
+
-
5.
4(y
8.
6{2a
6)
+
4b)
Then combine
3
11. 4{a
+
=
b)
-
4y
=
24
+
12a
24b
=
-
la
=
5x
\4b
-
55/
like terms.
-
4a
=
4b
12.
4{m
+
n)
—
2n
3.
40
6.
99-99 =9801
= 4m +
2n
Suggested Extension Explain each step of the method in Teaching These Pages to multiply by 99.
Have
the students use that
method
to
compute these products mentally.
1.
99-
15
=
1485
2.
22-99 =2178
4.
99
65
=
6435
5.
99-75
•
=
Provide a sheet of questions based on the above try the questions over a period of time to see the
mental computation
7425
-
99
=
3960
Have the students improvement in their
skill.
skills.
T39
8
PROPERTIES OF
•
18-19)
(pp.
Teaching These Pages
A
"mind-reading trick" can be used in motivating the the students perform the following steps.
class to study zero.
Have
Results
Steps
x 2x
2.
Choose a whole number. Double the number.
3.
Add
4.
Divide by
5.
Subtract your original number.
6.
Subtract
1.
Ask the
2x
6.
x
2.
+6 +
3
3
3.
obtained
class if they all
made mechanical
an answer.
for
any students have
If
errors, retrace the steps in order to locate the errors as
well as review the
work
for the rest of the class.
Chalkboard Examples Find the value
if
possible.
6+0=6
1.
2.
(NP
indicates not possible.)
0-9=0
A^O
3.
4.
-^^^ =
8.
-?-
12
=
+
45
5.
45
6.
02
=
II 4?-
7.
NP
"
"
'
Let 9.
jc
=
4.
X
+
4
NP
Find the value of the expression.
=
8
10.
6(je
-
4)
=
^
'^^
11.
=
12.
5(3jc
-
1
1)
=
5
Simplify. 9
13.
+
16. {Ix
2fl
+
X
-
2fl
-
+
3jc)
9
=
-
18
NP
14.
8
+
17. 4fl
2x
+
9
-
2jc
-
8
=
-
4fl
+
8
=
17
15.
3(m
18.
5(3
Suggested Extension 1.
Can you
explain
always result
T40
the steps of the "mind-reading trick" given above
who can answer Question own mind-reading tricks.
Students their
why
in zero? 1
can be encouraged
to
make up
-f
+
4)
2x)
-
12 \5
= 3m =
\0x
9
PROPERTIES OF
•
1 (pp.
20-21)
Teaching These Pages
A
"mind-reading trick" can be used to introduce
this section.
Have
the
students perform the following steps. Steps 1.
Results
Choose a whole number. the next whole number
x
3.
Add Add
4.
Divide by
5.
Subtract your original number.
5
6.
Subtract
1
2.
to
2x
+ +
10
x
-\-
5
2x
it.
9. 2.
4.
All students should obtain the answer
If
1.
1
any students have made me-
chanical errors, retrace the steps.
Objectives for Pages 22-25 1.
To check whether
a given
number
is
a solution of an equation.
2.
To check whether
a given
number
is
a solution of an inequality.
10
•
INTRODUCTION TO EQUATIONS
22-23)
(pp.
Teaching These Pages To introduce this lesson, use some recent baseball standings. Ask the students which of the following statements are true, and which are false. 1.
2. 3.
Oakland
Kansas City is at the top of the standings. Texas is ahead of Minnesota in the standings. Chicago is in last place.
Chalkboard Examples
+
=
1.
Is
X
2.
Is
4 a solution of 6
3.
Is
6 a solution of 12
Which of 4.
7
-h
6
5
9 true
if
the numbers
=
9
jc
=
5?
+7 =
=
v
3 or 9?
3
6?
is
x
=
4?
no; yes
no
9?
+
shown
if
yes
a solution? 5.
a
-
.
.
.
Suggested Extension
—
Replace each * with +, need to use parentheses
=
27
1.
4
3.
16*4*2 =
*
5 * 3
X, or
,
=
(4
+
16
-4-2=2
2
to
-r-
make
a true sentence.
5)
X
3
27
3*9*3=
2.
12
4.
INTRODUCTION TO INEQUALITIES
11 •
You may
also.
=
*22
* 3
3x9 + 3=
30
12
^
3
-
22
30
=
24-25)
(pp.
Teaching These Pages Review the terms introduced in the last section and relate them to the de velopment of inequalities in this section: number sentence, true number
number
sentence, false
sentence, solution, root.
Point out that the slash
mean
Compare
no.
meanings of
=
this
and
often used to
is
example
to the
P
7^.
No
Parking
parking
Chalkboard Examples
/
1.
Is
X
2.
Is
6
3.
Is 8 7^
Tell
/:
true if
x
true if y
is
true
is
k
if
is
replaced by 3? by 6?
replaced by 4? by 9? replaced by 3?
yes;
no
yes;
no
by 8?
no
yes;
which of the numbers shown are solutions.
4.
6>k
3,4,5,6
3,4,5
5.y2
0, 1,2, 3
3
1.9
8.
jc
+
3
2f -
3
6,7,8,9
0, 10,
20
7,8,10 6,7
a true sentence.
5x2 > ? 22
Suggested Extension
Use 1.
all
the symbols to
2,3,4,
T42
14,
X,
+,
,0
vary.) 3.
=, ^, -, 24,
8, 3,
Chapter
2.
Solving Equations
In the previous chapter students learned that equations are mathematical In this chapter the students will use equations as problem-solv-
sentences.
In order to
ing tools.
word problems,
overcome the
practice
is
difficulty that students
have
in solving
regularly provided in translating words into
symbols.
Objectives for Pages 34-41 1.
To
solve equations by addition.
2.
To
solve equations by subtraction.
3.
To
solve equations
4.
To
solve equations by division.
1 •
by multiplication.
SOLVING EQUATIONS BY ADDITION
(pp.
34-35)
Teaching These Pages
Ask
the students to find a solution to each of these equations.
X
The
+
last
5
=
25
2
+m =
10
+
3Z)
=
5
2Z>
+
10
equation illustrates that the "guess and evaluate" method of find-
ing solutions will sometimes take quite a while.
Point out the need to find
organized ways of solving equations.
Use the balanced
show how an equation might be formed.
scale to
3
=A =4 -
3
=
X
X X
-
Point out that to solve the equation x
3
1
—
3
=
1
we
reverse or
"undo" the
steps shown.
Encourage students to check and 4.
their solutions mentally in this section
and
in Sections 2, 3,
Chalkboard Examples Solve. will
Encourage students
be a useful
1.
X
-
3
=
8
3.
a
—
4
=
11
to write a final statement of the solution.
The The
solution
This
problem solving.
skill in later
is
solution
is
11.
15.
=7 -
2.
8
4.
12
The
4
=y —
1
solution
The
is
solution
is
12. 19.
T43
2 •
SOLVING EQUATIONS BY SUBTRACTION
(pp.
36-37)
Teaching These Pages
The balanced
scale technique of building
and unbuilding equations can be
applied to this section also.
Chalkboard Examples Solve. I,
a
+
9
=
\5
The
solution
is 6.
3.
y +
1
=
20
The
solution
is
2.
13.
4.
19
=
y —
1
jc
+
=
10
20
The
The
solution
solution
Suggested Extension
A
path
tion
is
hidden from Start
and study the solutions
vertically.
X
Start
-
4
=
to Finish.
To
find the path, solve each equa-
to find a pattern.
Move
only horizontally or
is
is 9.
27.
Think of an input-output machine. 5.
IN]
Tell 6.
what
is
missing in the table. 7.
8.
Write each sentence as an equation. Then solve.
more than twice
a
8.
8
9.
6 less than 3 times a
10.
When
9
is
added
number
is
number
to twice a
2x
18.
3x
is 9.
+ —
number, the
8
6
= =
x
18; 9;
result
x
is
=
21.
=
5
5
2x
+ 9=21;x=6
Suggested Extension Choose a number between 10 and
sum of
For example, choose the number four.
25.
the solutions of the equations 25.
Create four equations so that the
number you chose. The equations may vary, but here is
equal to the
are
Chalkboard Examples 1.
Copy and complete. Steps
Write an equation and solve.
Twice a number plus the number
3.
What 2n
+
is
4/7
n
= —
is
36.
number?
the
36 6
Point out that the result should be checked in the original problem, not in the equation.
Suggested Extension
Use
the terms and symbols to write an equation.
may
vary.)
1.
6«, 2, 30, 2n,
2.
2x,
3.
ly, 14, 6, 2y,
3, X, 12,
-, +,
=
=
+, -,
-, -,
-
6n
2
+
2n
=
30; n
Solve.
Check. (Answers
=4
+ X - 3 = 12; x = 5 Iv - 6 - 2y = \4; y = 4
2x
=
Objectives for Pages 50-57 1.
To
2.
To use equations
7
•
translate sentences into algebraic equations.
word problems.
to solve
WRITING ALGEBRAIC EQUATIONS
(pp.
50-53)
Teaching These Pages Use a newspaper clipping
NATIONAL LEAGUE to introduce a real-life
problem. Omit some of the
entries.
Have
East
the stu-
Won
dents write variable expressions based on the standings.
Philadelphia has
The number
won
10
lost is n, the
more games than they have
number won
is
«
-|-
10.
lost.
82
?
Philadelph
? ?
68 70
New York
74
72
Chicago
70
77
68
81
St.
Louis
[Montreal
Chalkboard Examples 1.
2.
Chicago lost 5 more games than they won. Let X = number of games won. Then ? = number of games lost, x +
George
is
y
years old today.
His age in 5 years will be His age
T48
last
year was
?
V
? .
>'
—
1
+
5
5
Lost
Pittsburg
.
Georgina
3.
Let g
=
Then
is
Sam.
3 years older than
Georgina's age.
=
?
Sam's age.
g
—
3
In 3 years, Georgina's age will be In 3 years, Sam's age will be
+
"L
?
3
^
.
Suggested Extension Part of the standings shown have missing entries. Use the information that follows to complete the standings. 1.
Chicago won 6 more games than they
2.
Vancouver
3.
Chicago won
4.
Minnesota
more games than
lost 3
3
more games than times as
lost 5
lost.
St.
NHL
Louis won.
many games
Chicago
Kansas City
as
Vancouver tied. .
many games
5.
Kansas City
Minnesota
lost.
6.
Minnesota won 6 more games than Vancouver
tied.
8
•
lost as
as
Standings
Louis won.
St.
APPLIED PROBLEMS AND PUZZLES
St.
Louis
Minnesota Kansas City
(pp.
54-57)
Teaching These Pages Refer the students to the Guide for Problem Solving on student's page 54 as the following
problem
is
solved.
Chalkboard Examples
The school play ran for two nights. were 79 more people on the second
A
total
night.
of 535 people attended. There
How many
people attended
each night? 1.
2.
Read
the problem.
Let n
= number of people on the + 19 = number of people
Then n 3.
n
+
n
2n
+ 19 = +19 =
535
=
456
2« n 4.
Answer:
Check:
night.
on the second
night.
535
=228
228 people attended the (228
5.
first
228
+
+
first
night.
79) or 307 people attended the second night.
307
=
535
/ T49
Suggested Extension
On
day you are given an amount of money. Each day after that is doubled. On the sixth day you have 252 cents. much money were you given on the first day? 4 cents
the
first
your amount of money
How
Objectives for Pages 58-63 1.
To
2.
To use equations
9
solve equations having the variable
VARIABLE
•
on both
sides.
word problems.
to solve
ON BOTH
SIDES OF THE EQUATION
(pp.
58-59)
Teaching These Pages Introduce this section by posing a puzzle problem that results in an equa-
on both
tion with the variable
sides.
number decreased by number increased by 7.
Six times a
times the
6n
—
=
3
4n
+
3
is
equal to 4
1
Point out that in this section the students will learn to solve equations like this.
Chalkboard Examples Solve and check. 1.
4x 2jc
X 5.
6.
= = =
8
+
2x
25
4
Six times a
number
is
- 3m = 2m 25 = 5m 5 = m more than
15
3.
1
+
x 5 \
3 times the
= = =
6x 5x X
number.
-
4
4.
4a
+
6
2a a
6^
—
15
=
3n; n
=
= = =
2a 6 3
5
Solve the problem posed in the introduction to this lesson. Check the answer in the original problem,
10
2.
8
•
dn
—
3
—
4n
-\-
1: n
=
5
EQUATIONS WITH PARENTHESES
(pp.
60-61)
Teaching These Pages Review the various methods acquired thus far for solving equations by using examples. Encourage students to discuss how they would solve each one. .X
-f 5
=
25
+
\2
Introduce an equation with parentheses and ask the students to discuss their
first
step in solving the equation.
Chalkboard Examples Use 1.
the distributive property to simplify.
4(x
+
5)
=
+
4x
20
2.
3(x
-
5.
3(m
+
6)
=
3x
-
3.
li
-
4(2jc
1)
=
8jc
-
4
Solve and check. 4.
2(x
2x
-
5)
10
Ix
X
11 •
= = = =
8
3w +
8
18
= 2m + = 2m + m —6 1)
9
3
9
6. 4(_y ^^
+ 2) + 9 = + 8+9 = 21
=
4(27
87
"
1)
-4
ly
3=y
9
PUZZLES
(pp.
62-63)
Teaching These Pages Review the Guide
for
Problem Solving on student's page
54.
Apply those
steps to solve the following puzzles.
Chalkboard Examples 1.
Jean has $16 more than Jose.
Four times
Jose's
amount
How much money Let n 2{n
In
—
Jose's
+ 16) = + 32 ^
32= 16 =
is
the
same
as 2 times Jean's
amount.
does each have?
amount of money; n
+
16
=
Jean's
amount
An An In A2
Jose has $16; Jean has $32. 2.
One number is 6 more than another. Three times the larger number is 5 more than 4 times the smaller. What are the numbers? Let n 'in
3n
-
= 5
5
\9
The
the larger
= = =
A{n
An
number; n
—
6
=
the smaller.
- 6) - 2A
n
larger
number
is
19; the
smaller
number
is
13.
T51
Chapter
and Negative Numbers
Positive
3.
The main objective of this chapter is to teach the four operations with positive and negative numbers. The number line is used to develop the skills, as well as to help the students compare numbers. These skills are then applied to simplifying variable expressions and solving equations.
Objectives for Pages 72-75 1.
To compare
2.
To graph the solutions of an
1 •
integers.
POSITIVE
inequality.
AND NEGATIVE NUMBERS
(pp.
72-73)
Teaching These Pages
To illustrate the occurrence of positive and negative numbers in evervday life, use the weather report in the newspaper, or a portion of the stock market ings in the newspaper.
show
Discuss the need to use
a gain of 5 dollars
—4
and
to
show
CanCab
list-
+5
to
a loss of
four dollars.
Have
the students
name
other ways in which positive and negative
num-
bers might be used.
Chalkboard Examples Write a positive or negative number. 1.
a debt of S40
3.
30
m
below sea
Compare 5.
-3*^5
-40 level
—30
the numbers. Write
2?
< -2 >
2.
a deposit of SIO
4.
a profit of SIO
or
7.
0? -7
>
+10 +10
2
•
INEQUALITIES AND GRAPHS
(pp.
74-75)
Teaching These Pages
Remind tences.
the students that equations
Compare
Sentence jc
and inequalities are mathematical sen-
the following.
-
2
Graph of the Solutionis)
Solutionis)
=5
1 3
jc
'
7.
-2)= -iy y =
8
-
3(5
2)
=
-(5
9=9
5
DIVISION
•
Check:
14)
(pp.
-
14)
/
88-89)
Teaching These Pages Review the relationship between addition and 3
X
Use whole numbers
+2 =: +y =
subtraction.
5
5-2 =
3
5
5
—y =
X
to review the relationship
between multiplication and
division.
3x2=6 Have
6-2=3 show they understand how Then develop the rules for dividing
the students state further examples to
multiplication and division are related. positive
and negative numbers.
Point out the similarities of these summaries. Division
Multiplication
X positive = positive x positive = negative positive X negative = negative negative X negative = positive
= positive = negative positive -^ negative = negative negative -^ negative = positive
positive
positive H- positive
negative
negative
-=-
positive
Chalkboard Examples Divide. 1.
.
9
-(-9) = -1
-36 =
-4
T58
9
-9
-9 =
-
-18 ^ _9
3.
= -5
Suggested Extension Find the value of each expression I.
a^
5,
a
9
=4 +
b
= -5
= — 2,
a
b
=
—3, c
=6
2.
ab
6.
abc
SOLVING EQUATIONS
•
if
=
18
=
3.
a^b
7.
ab
d = —I.
3 and
= -12 +
cd
=
3
4.
-a 2 _ -4
8.
^5
= -
90-91)
(pp.
Teaching These Pages These four examples review that to solve an equation, the same number added (subtracted, multiplied, divided) to both sides of the equation. 3
=
X
+
6
= -3
Review the vocabulary related amples are completed.
is
-3v = -24
to solving equations as the
Chalkboard Ex-
Chalkboard Examples Solve and check. 1.
8;c
-
3jc
X 10
= -5 = -\
2.
3^
+
=
15
ABSOLUTE VALUE (OPTIONAL)
•
2y
3.
3x
-
{5x
-1) = -2
-4=x
j=-15 (pp.
92-93)
Teaching These Pages Introduce the symbol for absolute value. Point out that
and
\x\
= —X
if
x
|x|
=
x
if
x
>
< 0.
Chalkboard Examples Find the value of the expression. 1.
|-4|
=4
2.
-|7|
5.
|x|
= -1
-
3.
|-9|
6.
W -3=
|3|
=6
Solve. 4.
|a|
a
+2 =4 = 6 or —6
X
- 4 = 10 = 14 or —14
y =
\
or
-2 —\
T59
Chapter Skills
4.
needed
chapter.
P =
2/
-I-
will
^ =
2h' (p. 102)
(p.
/vv
D =rt
108)
students' earlier
formulas.
problem solving are developed work with the following formulas.
to apply formulas to
The students
V = Bh The
Formulas
(p. (p.
A =]-bh{^.
104)
C ^pn
112)
work with solving equations
is
(p.
related to
in this
XOA)
112)
work with
Also, throughout the chapter the students obtain practice in writ-
ing formulas.
Objectives for Pages 102-111 1.
To use
the formula
2.
To use
the formulas
f*
=
A =
2/
+ 2^
Iw and
to find perimeters.
A =
— bh
to find areas.
2 3.
1 •
To use
the formula
V = Bh
to find
PERIMETER FORMULAS
volumes.
(pp.
102-103)
Teaching These Pages Reinforce the meaning of perimeter as "distance around" by having the students find the perimeters of these figures.
6
A
cm 16
4
cm
4
cm
m/
4x
\l6m
/
\
\
/ 6
cm
2x 2x
16m
2x
Write a formula for the perimeter of each figure. ^-^
1.
3x 3x
2x 6x
P =
\lx
Suggested Extension Chain-link fencing
sells for
cost of fencing the field.
$2.10 per meter. Find the
$777
Find the area of each shaded region. 1.
.
2.
3x
3x
A =9jc2
3.
Express the volumes in Exercises
4.
A
box measures
5.
A
cube has sides that measure
12
cm by
and 2
1
cm by
15
3
24 L; 21 L
as liters.
20 cm. Find the volume.
cm. Find the volume.
3600 cm^
27 cm^
Suggested Extension
How much
NATURE
cereal
FLAKES
1.
paper is needed to cover the box shown? 1724 cm^
Find the volume of the box
2.
meters.
When
the
cereal
is
3.
in cubic centi-
25 cm
4050 cm^
8
box
is
opened, the top of the
cm from
How many
the top.
cubic centimeters of cereal are in the box?
2754 cm^
9 18
cm
cm
Objectives for Pages 112-121 1.
To
write formulas.
2.
To
solve motion problems.
3.
To
solve cost problems.
4
•
WRITING FORMULAS
(pp.
112-113)
Teaching These Pages
SOCCER STANDINGS
Sports standings from the newspaper can be
used to introduce
When
^Northampton ....
this topic.
a soccer team wins a game, they are
given 2 points. After winning n games, the
number of winning formula
w =
w
points
is
given by the
!
Won
Lost
15
4
Lincoln
Reading
Tranmere
Bournemouth
In.
Develop the formulas for distance and cost. These formulas will be dealt with in
detail in Sections 5
and
6.
Chalkboard Examples Complete. 1.
You walk
5
km/h
Your distance 2.
You rode your Your distance
3.
You buy 5 The cost C
for
D =
t
hours.
?
-
bicycle at
D =
?
.
5/
y km/h
for n hours.
vn
tennis balls at s cents each.
= _2
5s
T63
4.
In hockey, 2 points are given for a win.
Winning
6.
tie /
A
games
4
points.
1
point.
/
hockey team won
The
_A
scores
game, a iiockey team receives points. games scores _1
For a Tying
5.
2
total points
vv
games and
T = _A
tied
2w +
t
games.
/
Suggested Extension Clip the hockey standings from the newspaper.
Use
the formula developed in Exercises 4-6 above to
check the 5
total points for the various teams.
For example, games.
St.
Louis has
T = 2w +
5
•
t
=
won
+
2(10)
MOTION FORMULAS
10
5
(pp.
games and
=
tied
25
114-117)
Teaching These Pages Review the problem-solving methods of Chapter throughout
2.
They
this section.
Chalkboard Examples Jean and John
live 18
km
apart.
same time and walk toward each Jean walks at 5 km/h and John walks at 4 km/h. How long will it take them to meet?
They
leave at the
Let
= number
/
of hours until they meet.
other.
are applied
6
•COST AND MONEY PROBLEMS
118-121)
(pp.
Teaching These Pages Use several examples
to review the
formula for
cost.
Point out the mean-
ing o{ profit. 1.
A
pencil costs \5l.
2.
A
tennis ball costs $1.25.
3.
You bought
Find the cost of 10
pencils.
Find the cost of
10 pens at 35c each,
3 tennis balls.
and sold them
for
50*1;
each.
Find
the profit.
Solve the Chalkboard Example and ask the class to indicate each prob-
lem-solving step.
Chalkboard Examples George and Robin bought greeting cards at 70
2)
= -6a
^
\5ab^ a^b^
= -la
-aW
9ab
Sa^x^b^
9aY = 3a
5a
3y^a
-3a4/>3
—64a^b^x^
=
-\6a%^ = -4a Aab^ 36a6
= —a
=
6a
6a^
Finish
10
'EXPONENT RULES (OPTIONAL)
(pp.
156-157)
Teaching These Pages Encourage students
Discuss the exponent rules on page 156. rules,
but point out that they
when
necessary.
may
to use these
use the methods of the previous section
Chalkboard Examples Divide. 2.
x^ 11 •
— = n^ n
-3b
9a^b'^ 3.
POLYNOMIALS DIVIDED BY MONOMIALS
(pp.
158-159)
Teaching These Pages Use examples from arithmetic to illustrate ^-^ c example below illustrates a common error.
6+12 1 3
§_ 3
18 3
,
}2^ 3
-9a^b
-3
3a^b^
ab
4.
3d^b
—=
.
1
c
6+12
The second
c
+
12
Chalkboard Examples Divide.
20a
.
-
12
=
-
5a
3
=
2.
+
3a
ab .
6m'^n
4.
+ 4mn — — Imn
\2mn^
= —5m —
^
-j
1
on
—-;
3.
3ab r—.
5.
+
'iab'^
=
c,
do
3ab
Suggested Extension Find the errors in the examples below. 1.
^^
^^
=2a +6b
5—
2.
12
j"
— 4m«"^
=
Did not divide both terms by
3m^«2
Overlooked
m
2.
term in denominator.
•DIVIDING POLYNOMIALS (OPTIONAL)
(pp.
160-161)
Teaching These Pages
To introduce
this topic,
have the
(jc
Then have
+
4)(x
+9x + X
+
20
=
x
Chalkboard
1.
+
x2
+
Expl
x^ 5
to develop a
the students suggest a
rect.
Divide.
=
4
Use the example on page 160
Have
5)
9x
+
20
the class explain these equations.
x^
als.
class find the product.
+
method
to
+
9x
X
+
+
20
=x +
4
5
method
=
3ab
—
Mab"^
c
/:
,
-\-
6b
for dividing
polynomi-
check that the division
is
cor-
—
^
I
2a
—
b
Chapter
Factoring
6.
This chapter extends work with polynomials to factoring. learn three basic factoring techniques: finding a factoring trinomials with
term having coefficient of
first
The
students will
common monomial 1,
factor,
and factoring
the diff'erence of squares. In later chapters the skills developed in this chapter will be applied to
equation solving as well as other algebraic problems.
Objectives for Pages 170-175 1.
To
factor positive integers.
2.
To
find a
1 •
common monomial
factor in a polynomial.
FACTORING IN ARITHMETIC
170-171)
(pp.
Teaching These Pages Discuss the importance of factoring in adding fractions. Factoring can
common
help us find a
denominator.
9^
5
3-3-3
5-5
^25
27
60
60
2-2-3-52-2-3-5
20
12
Introduce the meanings oi prime number, prime factor,
common factor
common factor and
above example. Point out that factors can be found by division. If the remainder
greatest
the divisor
as they relate to the
is
zero,
a factor.
is
Chalkboard Examples Find the prime factors.
=
1.
24
5.
Find a
2
•
2
2
•
•
common
Find the greatest 6.
3
12,30
2.
=
42
factor of 12
common
6
2
3
•
and
7
•
=
3.
99
8.
17,27
3
•
3
•
1 1
=
4.
53
9.
18,54
53
2 or A
20.
factor.
7.
100,125
25
1
18
Suggested Extension
A
number has
example, 135
Which numbers 1.
45
yes
9 as a factor
is
divisible
if
the
sum
by 9 since
1
of
+
3
its
+
digits
5
is
is
divisible
divisible
by
by
9.
For
9.
are divisible by 9? 2.
147
no
3.
909
yes
4.
242
no
T75
A number
has 4 as a factor
ample, 2348
Which numbers 1323
5.
2 •
two
if its last
by 4 since 48
divisible
is
is
digits are divisible
by
divisible
by
For ex-
4.
4.
are divisible by 4?
no
4156
6.
324
7.
yes
COMMON MONOMIAL FACTORS
yes
(pp.
3929
8.
no
162-173)
Teaching These Pages Review the
distributive property.
+
2x(3x
Use the terms
and Sxy
6x'^
=
4y)
6x2
_,_
to introduce the
g^^^
meaning of
greatest
common
factor.
6x^
Sxy
= =
X
2
•
2
'I'l' X y
3
•
jc
•
The
'
greatest
common
factor
2x.
is
Encourage students to factor completely, that is, to find the greatest common factor, and to check their factorizations by applying the distributive property.
Chalkboard Examples Find the greatest 1.
50mn,
XOrrfin^
common
factor.
lOmn
2.
12x2y, 16X72
3.
4;cy
Uab^, 18^2^2
9^^2
Factor. 4.
3/ - 6/ =
6. 6fl2
3
•
_
24ab
3y^(y
+
\Sb^
-
=
5.
2)
6(a^
USING FACTORING
-
Aab
(pp.
+
3^2)
7^
IOjc^
^(^
+
30x2
+ y) -
_^ 50;^
x{x
=
10x(x2
+ y) =
{x
174-175)
Teaching These Pages
Ask
the class to suggest different ways to evaluate the following expres-
sion.
Some
students will suggest finding each product and then adding.
Encourage the students
to look for a
common
factor, using the
methods of
the previous section.
25
T76
X
32
+
25
X
18 -h 25
X
50
=
25(32
-F
18
+
50)
=
25(100)
=
2500
+
3x
+ y){y -
+
5)
x)
Chalkboard Examples 1.
Find the 12
= = =
X
of the sides of the box.
+ 12 X 25 + 12 X +25 + 15 +25)
15
12(15
+
15
X
12
25
12
cm
12(80)
15
960
The area 2.
total area
is
cm
25 cm
960 cm^.
Find the area of the shaded and r.
Leave your answer
part.
Area of rectangle: Ar^ Area of 2 semicircles: irr^ Area of shaded part: Ar^ — ur^
—
—
r^(4
in
terms of
77
tt)
Suggested Extension
The
surface area of a cylinder
Find an expression
shown by
is
the
model
for the surface area.
Area of 2 ends: lirr^ Area of curved surface:
iTirh
Total surface area: 277^2
+
iTrrh
=
2'nr{r
+
h)
Objectives for Pages 176-187 1.
To
factor trinomials with factors of the
form {x
+
a){x
2.
To
factor trinomials with factors of the
form (x
—
3.
To
factor trinomial squares.
4.
To
factor trinomials with factors of the
form {x
—
4
a){x
+ —
b).
a)(x
+
b).
•FACTORING TRINOMIALS— TWO SUMS
(pp.
b).
176-179)
Teaching These Pages
Have
the students find these products at sight. (x (x (x
+ + +
2)(x 3)(x
6)(x
+ + +
4) 5) 3)
=
x2
= =
x^ x2
+ + +
6jc
Sx 9jc
+ + +
8
\5
18
Discuss the method the students used to find the products. in this section
we
reverse that
method
Point out that
to find the factors that yield
each
trinomial.
T77
Discuss the examples on pages 176 and 177. Ask the class to suggest a
way
to
check that the factorizations are correct.
Chalkboard Examples
Name
the two numbers.
1.
Their
sum
is 9,
2.
Their
sum
is
12, their
product
is
36.
6,
3.
Their
sum
is
13, their
product
is
40.
5, 8
4.
Factor d^
their product
+ Ua +
30.
{a
+
3,6
18.
is
+
2>){a
6
10)
Suggested Extension
The work of
the previous section
used in
is
this alternative
approach
to
factoring. 1.
Tina factored x^
+
+
x2
2.
5
\2x
Use Tina's method
+
\2x
21 as shown. Explain the steps she used.
+
21
= = =
+ 3x + 9x + 21 xix + 3) + 9(x + 3) (x + 3)(x + 9)
x^
to factor Exercises
31-36 on page
179.
•FACTORING TRINOMIALS— TWO DIFFERENCES
(pp.
180-181)
Teaching These Pages
Have
the students find these products at sight. (x {x
(x
-
2){x
3)ix 6)(x
-
4) 5)
3)
= = =
x^ x2 jc2
-
6x 8x 9jc
+ + +
S 15
18
Discuss the examples on page 180. Be sure students understand
look for negative numbers in Step
3.
Chalkboard Examples
Name
the two numbers.
—2,-6
1.
Their
sum
is
—8,
2.
Their
sum
is
—12,
their product
is
32.
—4,-8
3.
Their
sum
is
—10,
their product
is
25.
—5,
4.
Factor m^
—
10m
T78
their product
+
24.
(m
—
is
12.
6)(m
—
4)
—5
why we
Suggested Extension
The
factoring
tion can
6
•
method used
be used for
in the
Suggested Extension for the previous sec-
this section also.
FACTORING TRINOMIAL SQUARES
(pp.
182-183)
Teaching These Pages
Have
how each
the students find the squares of these binomials and note
term of the trinomial square (jc 4-
3)2
+
5)2
{x
Develop the
test
= =
x2 jc2
+ +
6x
related to the original factors.
is
+9 +25
-
(jc
lOx
{x
3)2 5)2
= =
x^ jc2
-
+9 + 25
6x
IOjc
outlined on page 182 for recognizing trinomial squares.
Point out that the students can always use the methods of the previous sec-
how
tions for factoring if they forget
to recognize trinomial squares.
Chalkboard Examples Complete.
+
16x
+
64
1.
jc2
3.
m2 - 10m + 25
= (_X + =
(m
8)2
_J_f
=(a_J_f
+6
=(^^-4)2
y
X
2.
fl2
+
12a
-5
4.
/
-
87
6.
x2
+
16x
+
64
=
(x
8.
x2
-
\4x
+
49
=
(X
+
+
36
16
Factor. 5.
m2
7.
^2
- 18m + 81 = (m - 9)2 _ g^ ^ 15 ^ (^ _ 2,){a -
5)
+ -
8)2
7)2
Suggested Extension
We
can use what we know about trinomial squares to find squares of numFor example, we can write 25 as 20 + 5 and find 252 ^5 shown
bers.
below. (20
+
5)2
=
202
+
2(5
•
20)
Write the square of each number. 1.
7
21
441
2.
16
+
=
52
400
How many
256
3.
'FACTORING TRINOMIALS
(pp.
+
200
+
25
=
625
can you do mentally? 35
1225
4.
28
784
184-187)
Teaching These Pages
Have (;c
the students find these products.
+
8)(x
-
6)
=
x2
+
2x
-
48
(x
-
8)(jc
+
6)
=
jc2
-
2x
-
48
T79
Ask students page
How
the two examples.
compare
to
How
same?
are they the
Relate the students' observations to the examples on
are thev different? 184.
Chalkboard Examples
Name
the two numbers.
1.
Their
sum
is
—5.
their product
is
—14.
2.
Their
sum
is
—4.
their product
is
—5.
3.
Their
sum
is 6.
their product
is
-
5)
—7,2 —5.
1
—2.8
—16.
Factor.
A-
4.
+
2.x
-
35
=
+
(.V
7)(.v
5.
-
x^
3.x
-
=
40
{.x
-
8)(.v
+
5)
Objectives for Pages 188-193 1.
To multiply
2.
To
8
•
sum of two monomials by
the
factor the difference of
their difi'erence.
two squares.
A SPECIAL PRODUCT
(pp.
188-189)
Teaching These Pages
Have
the students tind these products. (.V
(X
Ask
the students
if
+ -
1)(A-
3)(.Y
+
1)
(A-
-2)(A
3)
(A
+
4)(A
+ -
2)
4)
they notice anything special about the products.
Chalivboard Examples Complete. 1.
(V
-
+
3)
= _J_ - _J_
-
9)
=
V-
-
5)(m-
-
5)
=
An"'
3)( V
)•-;
9
2.
(2m
-
4)(2m
4.
(4a
-
v)(4a
6.
(a
+
4)
=
_JL_
-
.JL_ 4m-;
Multiply.
+
81
3.
(V
5.
(m-
7.
Find the product (45)(55). Use the rule (a
9)(v
+
-
25
(45)(55)
T80
= = = =
-
b){a
+
2475
v)
=
16a2
=
x^
2v)(a
+
2v)
=
a-
-
b)
- 5)(50 + _ 52 2500 - 25
(50 502
+
5)
b'-.
-
-/ 4/2
16
Suggested Extension 1, Sections 6 and 7; Chapter 6, Section 6; and Example 6 above suggest methods for computing products mentally. Combine exercises for each method to have a Mental Multiplication Quiz. The students might work in teams to arrive at a class championship.
Extensions for Chapter
9
FACTORING THE DIFFERENCE OF SQUARES
•
(pp.
190-191)
Teaching These Pages Review a few examples from the previous section on finding products of the form (a + b){a — b). Discuss the examples on page 190. Emphasize that this method is for factoring the difference of two squares. Point out that x^ + 4y^ and 9^2
+
cannot be factored.
1
Introduce the need to recognize the difference of two squares. ple,
—16 + y'^.
ask the class to factor
ten as y^
—
For exam-
—16 + y^ can be
Point out that
writ-
16.
Chalkboard Examples Complete. I.
-
a^
b^
=
{a
_^){a ^I_)
+b;
-b
2.
4x2
4.
x^
-/
=
(^_ + /)(^_ - j)
2jc; 2jc
Factor. 3.
m'^
-
16
= (m -
5.
m^
-
16
=
(m2
4)(w
-
+
4){m^
4)
+
4)
=
{m
-
2)(m
+
- 9/ =
2)(m^
+
(x
-
3y)ix
+
3y)
4)
Suggested Extension 1.
— y)(x + y). Students may show that x^ — y^ = (x — y)(x + y). Then since x"^ — y"^ y£ x^ + y^ (unless y = 0), x^ + y^ Show why
The x^
x^
+ y^
:^
(x
students might also assign values to x and
+ y^
and (x
— y)(x + y)
to
show
that the
y
:p^
(x
— y){x + y).
then evaluate
two expressions are not
equal.
10*
MANY TYPES OF FACTORING
(pp.
192-193)
Teaching These Pages Review the various factoring techniques developed the class work a few examples using each method.
in this chapter.
Have
T81
Point out that in this section
more than one technique may be needed
to
factor each example.
Chalkboard Examples Factor. 1.
3.
^
2x2
i2;c
_
3x2
48
+
= =
= =
18
3(^2
3(x
2(jc2
2(x
+ 6x + + 3)2
- 16) + 4){x -
2.
9)
x3
6jc2
-x2 +
4.
+
6jc
Sx
+
= =
16
4)
x(x^
x(x
- 6x 4- 8) - 4)(x - 2)
= - 1(^2 -6x - 16) = -l(x -S)(x + 2)
Suggested Extension Factor. 1.
2.
3.
-
+ 9)(x2 - 4) (x2 + 9)(x - 2){x + 2) (^2 - 9)(x2 + 4) x^ - 5x2 _ 36 (x - 3)(x + 3)(x2 + 4) x* - 13x2 + 36 = (x2 - 4)(x2 - 9) .= (X - 2)(x + 2)(x - 3)(x + x^
11 •
+
5jc2
MORE
36
= = = =
(jc2
3)
DIFFICULT FACTORING (OPTIONAL)
(pp.
194-195)
Teaching These Pages Introduce this section by having the students find products. (3x (2x
+ -
l)(x
l)(x
+ +
2) 3)
Discuss the examples on page 194. the
first
term of the trinomial
is
= =
3x2 2x2
+ +
+ -
7x 5x
Point out that
not
1,
it
may
2 3
when
the coefficient of
be necessary to
try
many
factors before finding the correct ones.
Chalkboard Examples Factor. 1.
3x2
T82
_
i2x
+
4
=
(3x
-
l)(x
-
4)
2.
3/ +
5/
-
2
=
(3/
-
1)(/
+
2)
Chapter
7.
Graphs
Every day we are confronted with graphs occurring in newspapers, magaand so on. Thus it is important to be able to interpret information displayed by means of a graph. This chapter develops skills for working with pictographs and bar graphs and then extends the development to include broken line graphs as well as graphs of linear equations. zines, reports,
In preparation for the next chapter, the graphs of equations in two un-
knowns
are studied
and related vocabulary
is
introduced.
Objectives for Pages 204-215 1.
To read
2.
To read and draw
a bar graph.
3.
To read and draw
a broken line graph.
4.
To
a pictograph.
identify
and graph ordered
pairs of
GRAPHS YOU OFTEN SEE
1 •
numbers on
(pp.
a coordinate plane.
204-207)
Teaching These Pages Throughout
this chapter,
emphasize the usefulness of a graph
to display
information in a concise manner.
Have zines. sion.
the class collect examples of graphs from newspapers and magaTextbooks for other subjects might also provide examples for discus-
Use the example on page 205
to introduce the nature
of misleading
graphs.
Point out that the scale on a graph should always start at zero.
When
would cause the size of the graph to be too large, we delete a portion of the scale and indicate the deletion by a jagged rule. (See page 206.)
this
Chalkboard Examples
NUMBER OF OLYMPIC MEDALS WON
1896-1972
Sweden Great Britain
USSR United States = 100 medals 1.
What
information does the graph show?
The number of medals won
in
the Olympics from 1896 to 1972.
T83
2.
How many
medals does each symbol represent?
3.
What does
a portion of a symbol represent?
4.
Draw
2
•
100
Fewer than 100 medals.
shows the same information.
a bar graph that
GRAPHS WITH LINES AND CURVES
(pp.
208-211)
Teaching These Pages Record the temperature
for a certain city for 7 consecutive days.
Make
a
table to illustrate the data.
Then show
the class the use of a bar graph to
show
the data, the construc-
graph from the bar graph, and the drawing of a "smooth" line graph. Discuss how the three graphs are related and why we can draw a smooth curve in the third graph. Have the class create questions based on the information shown by the tion of a
broken
line
graphs.
Chalkboard Example Each year a group of students hold
a car
WASH-ATHON EARNINGS
wash-a-thon to raise money for charity. The
amount of money collected shown on the bar graph.
for the last 6 years
is
Refer to the graph to answer the questions. 1.
Estimate the amount of 1972 and in 1975.
2.
How much more money 1976 than in 1971?
3.
money was
collected in
$60
Estimate the average amount of lected each year since 1971.
4.
collected in
$482; $525
Predict the
might
amount of money
collect in 1977.
money
col-
about $500 the group
probably more than
1971 1972 1973 1974 1975 1976
$500
Suggested Extension Each day the newspaper publishes high and low temperatures for various Have the class, or groups of students, select a city and collect data
cities.
about the daily temperatures. 1.
Draw
2.
Use the graph
T84
a graph to
show
the data.
to predict the next day's temperature.
3
POINTS ON GRAPH
•
212-215)
(pp.
Teaching These Pages
The temperature data used
in the previous section
can be used here to
introduce graphing points on a coordinate plane.
Introduce the vocabulary related to the graph on page 212: ordered pair, origin, coordinate, horizontal axis, vertical axis.
Chalkboard Examples Give the coordinates of each point.
A (-4,2)
I.
Name
C
the point by
(1,3)
4.
2.
E
5.
(2,-2)
3.
Z)
6.
(-1,-3)
(-2,0)
its letter.
(-1,3)
G
/
Give the coordinates of the point closest to each point.
F E
1.
Name
10.
S.
G E
4 points that are
H
9.
J
at the vertices
of a
rectangle.
Objectives for Pages 216-227 1.
To
2.
To draw
3.
To
4
•
identify solution pairs for a linear equation in
two
variables.
the graph of a linear equation in two variables.
find the slope of a line.
SOLUTION PAIRS
(pp.
218-219)
Teaching These Pages
Show
the class an equation in one variable
and an equation
in
two
variables.
3x+6 = List
some
X
12
solutions for the equation in
X
=
\,y
=
5
X
=
2,y
+y =
6
two variables.
=
4
x
= -3,y =
9
Point out that the equation in one variable has only one solution, the equation in two variables has
many
2,
but
solution pairs.
Introduce the use of an ordered pair {x,y) to record a solution for an
equation in two variables.
Chalkboard Examples
=
2x
+7 =
1.
Find X
3.
Which ordered pair is not a (1,-4), (-2,-5), (6,1)
if V
3 in
x
+
4
1 1.
solution pair for
x
—
2y
ii
=
x
=
2 in
x
S'!
(6,1)
=
4.
Find a solution pair
5.
Write an equation to go with the table.
for
Find y
2.
3/
9.
x X
y
y = 3x
+y =
Suggested Extension 1.
Draw
the graph of
x
2.
Draw
the graph of
x
3.
Name
6
•
+y = —y =
the point that Hes
SLOPE OF A LINE
5.
1
on the same
on both hnes.
(pp.
set
of axes.
(3, 2)
222-223)
Teaching These Pages Introduce the meaning of slope by comparing the slopes of the roofs of various houses.
The roof of an A-frame cottage has a steeper ing rain and snow to run off more quickly. Introduce the meanings of rise and run.
slope than
Chalkboard Examples Find the slope.
l./ 3.
=
f+
How
is
2
i
same
y = -2x +
3
the slope of each graph of the equa-
tions in Exercises efficient
2.
1
and 2 related
to the co-
of the X term in each equation?
the
many
roofs, allow-
Suggested Extension
Draw
y = 3x +
I.
Do
3.
2?
Draw
the graphs in Exercises
They
Do
7.
6?
•
and 2
set of axes.
y = 3x -
\
cross each other?
No.
notice about the slopes of the graphs in Exercises
1
and
notice about the slopes of the graphs in Exercises 5
and
3
6.
the graphs in Exercises 5
They
same
are the same.
What do you
8.
1
the graph of each equation on the
y = 2x +
5.
2.
\
What do you
4.
7
the graph of each equation on the
and 6
same
set of axes.
y = X - 4
cross each other?
Yes.
are different.
FUNCTIONS
(pp.
224-227)
Teaching These Pages Discuss the examples of functions on pages 224 and 225.
Although
this section takes a
very intuitive approach to functions, some
students will learn in later courses that a function can be thought of as a set
of ordered pairs in which no two different pairs have the same
first
element.
Chalkboard Examples 1.
A machine adds 2 to the Complete the table.
input.
X
=
INPUT
^
—
vj»j
r
1
vj 1
10 2.
3.
An
equation relating x and
y
in Exercise
1
is:
Study the function in Example 2 on page 224.
y
Is
lating the age of the car to the value of the car?
Suggested Extension
=
?
.
8
x
+
-10
2
there an equation re-
No.
Chapter
8.
Equations with Two Variables
In this chapter students will learn to solve a pair of hnear equations in
two
The graphic method is first used, but for greater accuracy the methods of substitution and add-or-subtract are developed. These skills are
variables.
applied to problem solving in Sections
5, 7,
and
8.
Objectives for Pages 236-243 1.
2.
To
find the solutions of pairs of linear equations
To recognize whether a no solution pairs, or all
1 •
by graphing.
pair of linear equations shares one solution pair, solution pairs.
THE GRAPHING METHOD
(pp.
236-239)
——r~
Teaching Tliese Pages Use a
map
street
\
— CO to
review locating points on a
coordinate system and to introduce the meaning
of intersection.
Rogers Ave.
"CO \
"(5
ro
O
\
Davis Ave.
c QQ
O Ebos Ave.
Park Ave. CO
Hillcrest Ave.
Chalkboard Examples Complete the 1.
.V
+7 = X
table.
12
y 10
-2 6 5
CO
NO SOLUTION, MANY SOLUTIONS
2 •
(pp.
240-243)
Teaching These Pages Use Ask
the following graph to introduce this lesson. the students to name a pair of
equations that have intersecting graphs. Introduce the meaning o( parallel
You may want
lines.
parallel lines
Ask
lie
to point out that
same
in the
the students to
name
plane. a pair of
equations that have graphs that are Point out that the slopes
parallel lines.
are the same.
Point out that the graphs of
X
-\-
y = A and 2x
same, and so
2x
4-
2y
=
a:
-\-
2y = % are the 4 and
+/ =
8 share all solutions.
Chalkboard Examples Find the slope of the line defined by each equation. 1.
^ = 6x +
3
6
2.
jc
+
V
=
4
-
1
3.
X
+
2y
=
4
Chalkboard Examples Solve.
X
1.
+J =
8
2.
x-y=4 P =
4.
21
Find
4
+
P
2w-
I
=
y = 4x X
(6,2) 4k,
in terms of k.
3.
-y = -9
+
2b
=4 -26
(8,
-2)
w = 2k, P = \2k
THE ADD-OR-SUBTRACT METHOD
•
a
b-3a=
(3, 12)
(pp.
1A6-1A9)
Teaching These Pages Use the substitution method
3x-y = 2x+/ =
to find a solution pair for these equations.
l
(1)
8
(2)
y = 2x
+
3x-l
-7) =
(3x 5jc
-
= =
7
X Introduce the add-or-subtract method. Point out that (1)
and
(2)
we
will obtain the starred equation.
pairs of equations the add-or-subtract
method
if
8* 8 2>,y
=
2
we add equations
Suggest that for certain
more convenient than
is
the
substitution method.
Chalkboard Examples Solve.
3^
1.
2y
Encourage students to check the solution
- 2jc = + 2x=4
2.
1
(-1,3)
3y 2y
+ +
in the original equations.
2x 2x
=4 =
2
(-1,2)
Objectives for Pages 250-257 1.
To use a
2.
To use
pair of linear equations to solve problems.
multiplication
and the add-or-subtract method
to find the solu-
tions of pairs of linear equations.
To use
3.
5
•
a pair of linear equations to solve cost problems.
WORD PROBLEMS, TWO
VARIABLES
(pp.
250-253)
Teaching These Pages Review the Guide
for
Problem Solving on page
54.
Point out that
we
fol-
low similar steps when solving problems with two variables. Solve a problem with one variable and compare its solution to the solutions of the
Chalkboard Examples.
T91
Chalkboard Examples Solve. 1.
Use two
variables and two equations.
The sum of two numbers is 26. One number is 6 more than the other. Find
2.
y = X X
=
How many
score?
+ y = 26 — 6 =y
=
Let X
the smaller.
=
16;/
Write a problem to
=
X X
10
"fit" these
Lesley's score;
+y = —y =
•
2S
—y =
2
The sum of Alec's and Georgina's ages The difference in their ages is 2. Find
6
+y =
is
1
X
=
76,
28 years.
their ages.
USING MULTIPLICATION
(pp.
254-255)
Teaching These Pages
Show an example
of a pair of equations that cannot be solved by the sub-
method or the add-or-subtract method. Review that we can multiply members of an equation by the same number to obtain an equivalent equation. Thus we can transform the second
stitution
equation and solve the pair as shown.
Ax 2x
-\-
5y
+
2>y
= =
1 3>
Ax Ax 4x
Point out that tional
number
-\-
5y
+
6y
-
5
= = =
1
6
7;
X
=
3
we can multiply members of an equation by if it is more convenient.
the
same
Chalkboard Examples Solve. \.
Zx 2x
T92
-2y = + y =
10
9
2.
(4, 1)
3x
2x
-2y = -1 + 2>y = 4
y =
Mike's score
\45
Solve your problem.
equations.
X X Sample:
more
points did each
number;
the larger
Suggested Extension 1.
points alto-
Lesley scored 7 points
than Mike.
the numbers.
Let X
Mike and Lesley scored 145 gether.
(-1,2)
frac-
/ =
69
7
•
COST PROBLEMS
256-257)
(pp.
Teaching These Pages Review the cost formula from Chapter 4. Guide for Problem Solving on page 54.
C =p X
n.
Also review the
Chalkboard Examples 1.
2.
A
A
hot dog costs 50c.
a.
Write the total cost of x hot dogs and
b.
Write the total cost of
hamburger
costs
60(1;.
y hamburgers.
hot dogs and 2y hamburgers.
3jx:
50x
+
6O7 cents
150x
+
120^^ cents
Movie tickets cost $3 for adults and $2 for children. On Saturday, $66 was paid for 28 tickets. How many adult tickets were sold?
— number
Let a
a
3a
+ +
of tickets for adults; c
= number
of tickets for children.
z=2S
c
=
2c
66
There were 10 adult
tickets sold.
Suggested Extension Clip advertisements from the newspaper to obtain prices for various items.
Have
8
•
the students create
and exchange
cost
problems based on the
BOAT AND AIRCRAFT PROBLEMS (OPTIONAL)
ads.
(pp.
258-259)
Teaching These Pages Review the
skills
developed in Chapter 4 for solving motion problems with
one variable.
Show
the students the difficulty in attempting to solve the
example on
page 258 with only one variable.
Chalkboard Example 1.
Janie and Jacques traveled 12 return trip took only 3 hours. the rate of the boat (in
Let X
=
rate in
6{x
-y) = +y) =
upstream in a boat in 6 hours. The Find the rate of the flowing water and
water).
km/h of the boat km/h of the current.
rate in
y = 3(x
still
km
in
still
water;
12 12
X
= 3;y =
\
T93
Chapter
9.
Working with Fractions
In earlier chapters the students have developed skills for working with pol-
ynomials, and applied these
skills to
solving problems.
In this chapter stu-
working with fractions in algebra, essential to solving problems. Throughout the chapter skills with fractions in algebra are related to similar skills the students have learned in working with arithmetic. Once the students have learned to add, subtract, multiply and divide dents develop
skills for
algebraic expressions involving fractions, these
skills
become
useful for
This work with equations
solving equations that involve fractions.
is
then
applied to solving problems that deal with rates of work.
Objective for Pages 268-273
To use
1.
1 •
factoring to simplify fractions.
FRACTIONS IN ALGEBRA
(pp.
268-269)
Teaching These Pages Review
that a
Remind
number has
different
9
12-3
students that a fractional
4
names.
5+4
32
number
also has different names.
32
8
Point out that to simplify a fraction
means
to find a simpler
name
for the
fraction.
Use the examples on page 268
to
compare simplifying
fractions in arith-
metic to simplifying fractions in algebra. it would be helpful to review the rules for numbers (page 88), and the exponent rules
Before assigning the exercises dividing positive and negative
(page 156).
Chalkboard Examples Simplify.
-8fl
. '
\6a
_ _\_ ~ 2
2 '
30m _ _ -25m ~
6^
5
3 '
-Xla^b -24ab^
^ ~
_a_
-36ab^c^
^ '
2b
^
3b^c
-Ud^bc
a
State the value of x for which the fraction has no meaning. 5.
^
— X -2
T94
2
6.
-^^— \
-
X
1
7.
^r^—r 2x - 4
2
8.
-r-^ 6-3x
2
Suggested Extension Simplify each fraction then add or subtract.
-36a^b J
— lAm^n — Arn^
= -la
Uab
j6rV
=
— 3>r'^s
4
—Ars
SIMPLIFYING FRACTIONS
2 •
(pp.
—Arn^n^
-.Sn
2„2 Irrrn
6^W
-16A:5m4
3km
Ak^m^
—Ik'^m
270-271)
Teaching These Pages Review the various factoring techniques students learned a^
+ 6m +
m^
9
=
+ 3f
(m
in
Chapter
6.
+ 2a -S =(a + A)(a - 2) - / = {2x - y){2x + y)
4x2
Chalkboard Examples Simplify.
X
+
1.
2x
2ab 3.
3^2
3
•
2
+A
jc
2(x
-6a _ _ 9^ -
4-
3y-6 ^ 3{y-
2
+
2
2)
3
2a(b
-
2b(b
-3)~Jb
_
3)
THE -1 FACTOR
+6 +4k +A
2a
(pp.
2)
3k 4.
k^
= y-2
3
3(k (k
+
+
2){k
2)
+
2)
k
+
2
272-273)
Teaching These Pages
Ask
the students to
compare these X
-
pairs of binomials.
1
a
—
-X
b
—
I 1
Use the
b
x^
— j2
a
yi-
_
I
show
distributive property to
-
-\{x
1
that
\)
1
—
jc
= -X ^ = -JC
x^
= — l(jc —
1).
\
1
Point out the usefulness of the ;c \
-
1
-X
-
factor in simplifying fractions.
1
X
-\ -
-l(jc
1
1)
-1 T95
Chalkboard Examples Simplify.
1.
—
.V
X
= —
= —X +
2.
1
X
-\-
V
3.
y
—
— jc2
4-
2xy
— y'^
= ^
—y
Objectives for Pages 274-283 1.
To
solve problems involving ratios.
2.
To
solve problems involving proportions.
4
•
RATIO
274-275)
(pp.
Teaching These Pages Introduce the meaning of ratio with familiar examples. 1.
2.
number of students over 165 cm number of students under 165 cm tall.
Ratio of the to the
tall
Ratio of the number of pages in the math book to the
number of pages
in the dictionary.
Point out that the following two statements are equivalent.
The
ratio of the
teachers
is
20 to
number of
students to the
number of
1.
There are 20 students
for every
1
teacher.
Chalkboard Examples Give the ratio
in simplest form.
1.
20 meters to 100 meters.
3.
A
team won 24 games and
What
is
—20— = — 1
lost 18.
the ratio of losses to wins?
2.
What
•
the ratio of wins to losses?
— — 4
3
;
3
5
is
2 years to 2 months.
4
PROBLEMS INVOLVING RATIOS
(pp.
276-277)
Teaching These Pages Introduce this topic by posing a problem.
The
ratio of the
number of
tended the school play
tended the play.
T96
How
adults to the
number of
children
who
at-
Three hundred seventy-five persons many adults were there? is
2 to
3.
at-
—24- = —12
Point out that the ratio as
Ix
if
the ratio of two
numbers
2 to
is
3,
then we can write
to 3x.
2x 3x
Review the Guide
to
_ 2_ ~y
Problem Solving on page
54.
Chalkboard Examples ratio of the number of adults to the number of children who attended the school play was 2 to 3. Three hundred seventy-five persons attended the play. How many adults were there?
The
1.
2jc
There were 150 adults 2.
in the
size cars
How many
3 to 5.
3x X
=
375
=:
75
there.
There are 480 cars is
+
parking
3x
+
compact cars were there?
ratio of
cars
to full-
= 480 =60
5x jc
There were 180 compact
The
lot.
compact
cars.
Suggested Extension
A
box
is filled
with 500 beans, brown beans and white beans. Have the
brown beans number of samples, have
students take a sample from the box and record the ratio of to white beans.
them
6
•
predict the
After the students have taken a
number of brown beans
PROPORTION
(pp.
in the box.
278-279)
Teaching These Pages Review
that fractional
numbers have
In a similar way, ratios
may be
2_^
4.
3
6
~
equal.
2 to 3
equals
3
4 to 6 t
I
1
names.
different
=
1 6
Introduce the meaning o{ proportion and the property of proportions.
T97
Chalkboard
Objectives for Pages 284-289 1.
To multiply
2.
To divide
3.
To add and subtract
8
•
fractions.
fractions.
fractions with the
MULTIPLYING FRACTIONS
same denominators.
(pp.
284-285)
Teaching These Pages
Compare how we
3-2 5-7
1.1 5
Compare an
multiply in arithmetic to
7
_6^ 35
3
X
y
.
multiply in algebra.
xk
k
m
y
'
m
arithmetic example to an algebra example to
often simplify the product of two fractions.
4
how we
4
3
ym show how we
Suggested Extensions 1.
Have
2.
To review
a
la
— 2b l(a - b) -l(b - a) 2a
work as well as emphasize the need to watch out have students discuss the "proof" that 2 = — 2.
= = = = =
for
b
lb lb
Multiply both sides by
— la l(b - a) lib - a)
-1=1 •
on page 287.
earlier
restrictions,
10
on the denominators of the
the students discuss the restrictions
fractions in the exercises
2.
Subtract the same values from both sides, (la Factor.
a
—
b
=
—{b
—
a)
Divide both sides hy b
—
a.
ADD, SUBTRACT— SAME DENOMINATORS
(pp.
288-289)
Teaching These Pages Point out that in algebra as well as in arithmetic the
same denominators by adding 1
5
2^ 1+2
x_
Ix
5
5
5
5
Chalkboard Examples Simplify.
1.
^+ a
^
5a
Ix
we add
fractions having
the numerators.
X
+
Ix
3£ 5
=
lb)
And
in algebra, to
add or subtract
fractions that
have different denomina-
we need to rename the fractions. This section provides practice in renaming fractions and the next section appHes this skill to adding and subtors
tracting fractions.
Chalkboard Examples
_ ^•3-6 -7
X
12
•
12 2)X
2a
?
4a
3.
8fl
ADD, SUBTRACT— DIFFERENT DENOMINATORS
Teaching These Pages
Compare
the steps of each example.
11_3 2
3
3-2
2_32 2-3
6
6
_6_
IOjc
16x2
8jc
3+2
5
6
6
(pp.
292-293)
\6ab
Mb
Chalkboard Examples Simplify.
X
-2
X
2{x
\
-2) -{x -
-2
JC
-
\)
-
jc
5(x
1
-
2)
-
3
4
4
4
2
X
-
2{x
-
3x
1)
2.
10
10
a
-
3.
+
-2 =
a
1
1
3fl
14 •
a
\
2(a
-
1)
+
3(a
1)
-
(a
-
2)
EQUATIONS WITH FRACTIONS
(pp.
=
4a
+
3
6a
6a
6a
2a
+
296-297)
Teaching These Pages Review the basic skills developed earlier following examples reviews a skill. X
+
6
=
12
-
for solving equations.
Each of the
Suggested Extension
Which ,
greater?
is
1
1
2
2
+ +
a
+
Let X X2 4.
•
b and
+/
+ + >>
2^2 (a
A:_y
15
1
2
1
+
-,4 —
1
2.
or
= +
4+1 5
1
a
+
_
5
+
3.4or3 +
1
2
1
Evaluate.
A.
2Z?2
^)(a
4+
1
(-y
+7)
2a
5.
b)
(a
WORK PROBLEMS
(pp.
-
b){a
+
b)
298-301)
Teaching These Pages Discuss different types of jobs where the job.
dents in class
may
deliver newspapers.
time to deliver the newspapers that the time
gether.
many
persons contribute to complete
(Building houses, making cars, delivering newspapers.)
is
Discuss
not reduced by
why
or
why
Ask whether
when another person
— when
2 students
it
takes
helps.
—
Some as
stu-
much
Chances are
do the paper route
to-
not.
Chalkboard Examples 1.
Steve delivers newspapers and it takes him 70 minutes. His sister Sandra can do the job in half the time. How long would it take Steve and Sandra to deliver the newspapers together?
Let n
= number
of minutes needed to do the job together.
l
2+1
3
+
1
2+1
Suggested Extension 1.
A
rectangular tank has dimensions 20
Which of
the following will
empty
A: 2 holes in the bottom
mx
m
10
and
is
6
m
deep.
the tank faster?
or
B:
hole in the bottom of
1
of the tank each with
the tank with radius
diameter 4 cm?
3
The
rate at
cm?
which the tank empties depends on the area of the
hole.
B Area of Holes
A — l"n
= Thus B
16
•
is
Stt
'1?
cm^
cm^
Area of Hole
A =
=
TT
•
977
32 cm^ cm^
the fastest.
BINOMIAL DENOMINATORS (OPTIONAL)
Teaching These Pages This section extends the work of Sections
Compare
these examples.
4
,
10, 12
and
13.
(pp.
302-303)
Chapter
Decimals and Percents
10.
As consumers, students need to use their skills with decimals, fractions, and percents. The first three sections of this chapter review the skills needed to work with decimals and fractions, and Section 7 deals with the topic of percents. Once the skills have been established they are applied to equations, which are then applied to real-life problems that involve interest and investment. In the
last section,
mixture problems are solved using the
skills
developed in the chapter.
Objectives for Pages 312-321 1.
To add,
2.
To express
3.
To
solve equations with decimal coefficients.
4.
To
solve problems using decimal equations.
1 •
subtract, multiply,
and divide decimals.
fractions as decimals.
DECIMALS
(pp.
312-313)
Teaching These Pages To motivate
the class,
show examples of
different
ways
that
we
use deci-
mals every day.
(Cottage
Cheese
QQ 9 -WW tt^I
Review the meaning of place value. This read decimals but also to understand points
when we add
to line
only to
up the decimal
or subtract decimals.
26.396 means 2-10-f-6-l
Compare place
will help the students not
why we need
+3--^-h9- -i- +
1
6
1000
value and face value as they relate to the example.
Chalkboard Examples
Add 1.
or subtract.
4.63
+
389.2
=
393.83
2.
0.57
-
0.3
=
0.27
3.
492.19
-
273.284
=
218.906
T105
Multiply. 4.
69.8
X
=
0.04
2.792
x
0.69
5.
46.21
=
31.8849
6.
7.32
x
0.12
Suggested Extension
Newspaper advertisements can be used
to
provide additional practice.
TOOTY-FROOTY shampoo, $1.42
1.
A
case of
cost? 2.
How much cost?
2
shampoo holds 24
bottles.
does a case of shampoo
would 6 cans of shaving cream and
3 boxes of facial tissues
$8.13
DIVISION WITH DECIMALS
•
How much
$34.08
(pp.
314-315)
Teaching These Pages Review that to rename 19.55 -=- 2.3 we can think of the fraction form. We rename by multiplying numerator and denominator by the same number, namely 10.
X 10 X 10
19.55
19.55
195.5
2.3
2.3
23
Before beginning the exercises, review the meaning of rounding to one dec-
imal place, to two decimal places. Introduce the symbol
=
for "is approximately equal to."
Chalkboard Examples Divide.
Round
to
one decimal place. 25.82
8^ 1.
2.3)19.55
-^
23)195.5
2.
0.17)4.39
-^
=
25.8
I7J439
Suggested Extension 1.
a nickel is about 2.15 cm. How many nickels placed edge to edge are needed to stretch 1 km? about 46,512
The diameter of
T106
=
0.8784
How many
Measure the diameter of a quarter.
2.
km?
stretch
1
Which
is
3.
quarters are needed to
about 40,816
worth more: a kilometer of nickels or a kilometer of quarters?
quarters
3
•
DECIMALS FOR FRACTIONS
316-317)
(pp.
Teaching These Pages Use the
sports pages to
rational
number. The
the best in the
show that it is useful to find a decimal form for a do not help us quickly to decide which team is standings. To compare the standings we need to find the ratios
decimal forms for the rational numbers.
Chalkboard Examples Express the fraction as a decimal. 1.
-=
0.75
43 = 0.666
2.
4
4
•
3.
.
^=
0.151515
33
EQUATIONS WITH DECIMALS
4.
4 = 0.375
318-319)
(pp.
Teaching These Pages
Have
the class
same?
How
compare the following two equations.
How
are they the
are they different?
8j
=
24
0.8/
Point out that to solve equations,
it is
easier to
=
2.4
work with whole numbers
than with decimals.
Review multiplying by
and 1000.
10, 100,
Chalkboard Examples
Would you
multiply by 10, 100, or 1000 to obtain an expression with the
smallest whole 1.
0.4x
4.
0.26x
number
10
-4.1x
100
coefficient? 2.
40.23/
5.
0.25/
+
0.5/
8.
0.127
+
0.037
100
1000
3. 4.036A:
100
-
6.
0.025m
9.
0.15(7-2) =0.3
0.25«
1000
Solve. 7.
0.9jc
9x X
= =
3.6
36
=4
12_y
+
= 3_y = y =
0-3
30 2
15(7
-
= = 7 =
2)
7-2
30 2
4
T107
Suggested Extension
—
1.
- as a decimal. Express ^ 33
3.
Use your answers 0.090909
.
for
1
0.030303
and 2
.
2.
.
to predict the
Express
——
decimal form of
as a decimal.
0.060606
^.
.
Express each fraction as a decimal. 6 ^^^0.1818...
4.
5.
5
•
^=
0.2424
6.
33
33
USING DECIMAL EQUATIONS
(pp.
4133
=
0.4545
1.
33
320-321)
Teaching These Pages Review the Guide
Problem Solving on page 54
for
as
you discuss the ex-
amples on page 320.
Chalkboard Examples 1.
On Monday
Jenny jogged 2.6 times as far as she did on Tuesday. The total distance she jogged on the two days was 14.4 km. How far did she jog each day? Let X
=
distance jogged on Tuesday; 2.6x
2.6x 26jc
+
+
X
IOjc
36jc
X She jogged 4
km
= = =
=
distance jogged
on Monday.
14.4
144 144
=4 on Tuesday and
10.4
km
on Monday.
Suggested Extension
The data below was taken from a
cereal box.
lems based on the data they find on nutrition
VITAMINS AND MINERALS
Have labels.
^=
students create prob-
0.9696
Objectives for Pages 322-331 1.
To express decimals
as percents.
2.
To express percents
as decimals or fractions.
3.
To
6
solve problems involving percents.
PERCENTS
•
(pp.
322-325)
Teaching These Pages Use examples from the newspaper used in everyday
to illustrate the
many ways
percents are
life.
^
100/0
^
Down Payment
Chalkboard Examples Express as a percent. 0.48
1.
= 48%
2.
0.375
6.
10%
=
37.5%
3.
44 = 0.25 =
25%
4.
^ = 0.2 =
8.
4.5%
20%
Express as a decimal. 5.
38% =
The
circle
9.
What
0.38
What
0.1
7.
=
46.5%
0.465
=
0.045
graph shows how Mike spends his time. percent of the day does
50%
he sleep and eat? 10.
=
percent of the day does
he spend in school?
25%
Suggested Extension 1.
2.
3.
What
percent of
about
61%
What
percent of
about
68%
What
percent of
about
67%
its
games has Rochester won?
its
games has
its
games has Nova
Springfield lost?
Scotia
won?
1
AMERICAN LEAGUE
7 •
USING PERCEN TS
(pp.
326-327)
Teaching These Pages Introduce this topic with an advertisement of a sale from the newspaper.
Save 250/0! was $16. ..now only $12!
Introduce the meanings of original price, discount, and sale price as they relate to the advertisement.
Chalkboard Examples Compute. 1.
4.
40%
A A
400
of 1200
pen
2.
35%
of 1800
set is regularly $8.
15% discount
is
5.
offered at a sale.
X
8
=
3.
75%
A
badminton set regularly sells on sale at a 20% discount. Find the sale price.
0.2
1.20
X
7.5
=
1.5;
$7.50
-
Suggested Extension
Have
the students collect sale advertisements
and create problems based on
the ads.
8
•
PERCENTS IN EQUATIONS
(pp.
328-331)
Teaching These Pages In this section, as well as the sections on interest, investment, and mixtures, the students will be solving equations involving percents.
of the equations that occur in the 1 1
to
show students
real-life
problems
Chalkboard Examples
The stadium holds 4500
80% of the How many 0.80
TllO
X
seats.
seats are filled.
4500
seats are occupied?
=
Point out
some and
in Sections 9, 10,
the need to learn skills for solving equations involving
percents.
1.
3600
of $23
$17.25
for $7.50.
It is
Find the discount. 0.15
450
3600 seats are occupied.
1.50
=
$6.00
A
2.
total
What
of 360 bottles were returned and 18 of them were broken.
percent were broken?
J^ = 0.05
= 5% 5%
of the bottles were broken.
360 After the game, the spectators were asked
3.
if
they enjoyed the game.
63 persons said they enjoyed the game. If
6%
0.06x
6x
X
of the persons interviewed enjoyed the game,
= = =
how many
persons were interviewed?
63
6300 1050 persons were interviewed.
1050
Objectives for Pages 332-339 1.
To
solve simple interest problems.
2.
To
solve investment problems.
3.
To
solve mixture problems.
9
INTEREST
•
332-333)
(pp.
Teaching These Pages Introduce this topic with a bank brochure showing the current rate of interest
paid on deposits. Collect different brochures to show that different rates
of interest /
may be
paid by different banks. Discuss the interest formula
= prt.
Chalkboard Examples 1.
How much money does $200 earn in one year at 6% interest? = 200 X 0.06 X = 12 The interest is $12 for one year.
/ 2.
1
George deposited $250 Find the 13.75
0.055 3.
= =
in the bank.
amount of
After one year, the
interest
added
to his account
250r
The
r
interest rate
5.5%
is
In January Jan put $150 into an account that earns
Four months
How much Interest
The
was $13.75.
interest rate.
later she
added $100
interest did the
on $150:
total interest
money earn
at the
7=150x0.06=9 earned
is
$9
+
6%
interest per year.
to the account.
$4
=
end of the year? Interest
on $100:
7=100x0.06x^ =
4
$13.
Till
10
•
INVESTMENT
(pp.
334-335)
Teaching These Pages Point out
some of
investment
ways money can be invested: stocks, bonds, Obtain advertisements that give the interest rates
the different
certificates.
for various types of investments.
that occur in the exercises
Relate these rates of interest to the rates
on page 335.
Chalkboard Examples 1.
The school band saved $2400
One
part of the
The
interest
How much Let X
money was
earned is
in
for their trip.
invested at 6%, and the rest at 8%.
one year
is
$180.
invested at each rate?
= amount
invested at 8%; 2400
—
x
= amount
invested at 6%.
Chapter
1 1.
Squares and Square Roots
In problems that apply the Pythagorean Theorem, the need arises to find square roots. This chapter provides skills for working with radicals as well as finding their decimal values from a square root table. Work on square roots leads to the introduction to irrational numbers.
The
skills
of finding square roots are also used in solving equations. The
Pythagorean Theorem provides an opportunity to solve a new type of problem, as well as apply the skill of solving equations of the type x^ = a^. The chapter concludes with developing skills to multiply and divide, as well as add and subtract radicals. An important skill for simplifying computation is to rationalize the denominator of a radical expression.
Objectives for Pages 348-355 1.
To recognize
2.
To use
3.
To recognize
4.
To simplify
1 •
perfect squares
and
find their square roots.
a square root table to find square roots.
and
rational
irrational
numbers.
radical expressions.
SQUARE ROOTS
(pp.
348-349)
Teaching These Pages Review the meaning of the square of a number and grams below.
yA
- 3^ or 9 units
relate
/4
it
to the dia-
= 4^ or 16 units
Point out the following. 1.
The square of
3
is 9.
3.
A square root of 9 is 3. — 3)2 is also equal to 9,
4.
The
2.
(
so
—3
is
also a square root of 9.
positive or principal square root of 9, represented
by
\/9,
is 3.
For example, we are given a measure of each side? We need to know how to find square roots to answer the question. Relate the symbol V^ to the example, pointing out that the side of the square would measure ^/\i cm. Point out the need to find square roots.
square region.
T114
What
is
the
A
=
12cm2
Summarize
that the next few sections will
Introduce the symbol
show how
to find a value for
r!z.
Chalkboard Examples
Which numbers 36
1.
are perfect squares?
\es
37
2.
no
3.
65
no
4.
64
Find the value.
-
Vm =
5.
\/36
=
8.
Fmd
the perimeter of a square
\/62
X VsT =
4
=
4
6
X
6.
9
=
36
if
-
\/l02
the area
The perimeter
is
= - 10 is
7.
V52
-
42
= V26 -
16
= V9 =
3
81 cm^.
36 cm.
Suggested Extension
The numbers 3, 4, and 5 have a special property, namely 3^+42 = 5^. Such numbers are called Pythagorean triples. Find other Pythagorean
Some
triples.
2
•
are
10; 9, 12, 15; 5, 12, 13.
6, 8,
USING A SQUARE ROOT TABLE
Teaching These Pages Introduce the use of tables to show values concisely.
For example,
in su-
permarkets the cash registers often have a table summarizing the sales tax posted on them. Thus the operator does not need to calculate the tax each time but refers to the handy chart.
To provide some background
V6
lies
between
3
and
for the square root table, point out that
2.
V9=3>\/6>\/4 = An
approximate value for \/6
table
is
is
—^—
or 2.5.
2
Note that the value
2.449.
Point out that a computer has calculated the square roots for
numbers
in the
^
all
the
in the table.
Chalkboard Examples
Use the
table to find the value.
=
1.
\/39
5.
-V21=
6.245
-4.583
=
2.
\/74
6.
-V53=
8.602
-7.280
3.
V98 =
9.899
4.
11
7.
\/80
=
8.944
8.
V28 =
=
-3.317 5.292
T115
Round
one decimal place.
to
=
\/35
9.
5.9
10.
\/86
=
9.3
Find the length of a side of the square correct
13.
\/T3
= =
4v'l3
9.8
- V67 =
12.
8.2
one decimal place.
to
3.6
Find the perimeter of the square correct
14.
V97 =
11.
to
one decimal
place.
14.4
13cm2
/^
-
/4
= 59 cm2
Suggested Extension the students find the length of a side of the square to one decimal
Have
place and then find the perimeter using that value.
Axl.l =
Now
\^ =
have the students find the length of a side correct to two decimal
places and then find the perimeter using that value.
4
X
7.68
Have
1.1\
30.8
=
\/59
=
7.68;
30.72
the student round 30.72 to one decimal place
and compare the an-
perimeter they found, 30.8. The difference will provide a useful source for discussion about adding numbers that have been rounded.
swer to the
3
•
first
IRRATIONAL NUMBERS
(pp.
352-353)
Teaching These Pages numbers are numbers that can be expressed as the two integers. Point out that the prefix "ir" in irrational means "not" and thus irrational means not rational. Have the students give other examples of words with the prefix "ir." (For example, irresponsible, irreleReview
that rational
ratio of
vant.)
Review that rational numbers have decimal forms that do not terminate, but repeat. Discuss the examples of bers on page 352.
or
either terminate irrational
num-
Chalkboard Examples Is the
number
rational or
is it
irrational? irrational
irrational
rational
1.
5.
1—
9.
Find the area
A
T116
6.
irrational
to
='7rr'^= 3.14
•
\/9
rational
one decimal place. 3^
=
28.26;
4.
y\2
8.
3.1515
irrational
-V5
V5
A =
28.3
7.
^/l5
irrational
rational
4
•
SIMPLIFYING SQUARE ROOTS
(pp.
354-355)
Teaching These Pages This section uses square roots of small perfect squares, such as to find the square roots of larger numbers.
Provide examples to illustrate the rule \/a^
=
\fa
•
4, 9,
16, 25,
y/b.
Chalkboard Examples Find the value. \/3600
1.
= V36 ^A00 =
60
•
V32 =
2.
VT6^ =4V2
3.
V^ =
\/9
Suggested Extension
Have the students investigate the prime numbers that are not perfect squares. 100
=
2
•
2
•
5
•
factors of perfect square as well as
125
5
=
5 -5 -5
Point out that the prime factors of squares occur in pairs and that write the square root by taking one
100
=
The students can use
member
Vm=2'5
2 -2 -5 -5
this
method
to
we can
of each pair.
=
lO
do Exercises 1-20 on page
355.
Objectives for Pages 356-361 1.
To
2.
To use
5
•
solve quadratic equations by taking the square root of each side.
the Pythagorean
Theorem.
SOLVING EQUATIONS
(pp.
356-357)
Teaching These Pages Review x^
=
tions
two square
roots,
25 and have the students
name
that 25 has
—5 and
5.
Introduce the equation
the solutions.
Point out that the solu-
can be found by taking the square root of both sides of the equation.
Chalkboard Examples
•
3
•/ ^ 3yV3
6
•
LAW OF PYTHAGORAS
Teaching These Pages
may have dealt with this topic in earlier courses. To review, draw diagram on page 358 on the chalkboard and have the students count the number of squares in each square region. Have the students studv the relationship among the numbers 9. 16. and 25. Thev will discover 9 + 16 = 25. Write this equation as 3- + 4- = 5- and relate it to the sides of the triangle and the squares drawn on each side of the triangle. Students
the
Chalkboard Examples Find the length of the third side of the triangle. 3.
?
4vl3
?
4.
A
wire
is
attached to a pole as shown.
Fmd
6\5
the length of the
wire correct to one decimal place. f2
c
= 16 + 64 = = v'^ = 8.9
80
8
4
Suggested Extension Discuss taking short cuts across corner
lots. Use the shown to see how much walk along AB and BC than to walk
dimensions of the comer further
it
is
to
lot
along AC.
Objectives for pp. 362-369 1.
To
2.
To multiply and divide
3.
To
4.
To simplify
T118
find the square root of a fraction.
rationalize the
radical expressions.
denominator of a radical expression.
radical expressions
by addition or subtraction.
m
m
7
•
QUOTIENTS OF SQUARE ROOTS
362-363)
(pp.
Teaching These Pages
^
\/a
Illustrate the rule
,
b
:^ 0,
with simple examples.
V^ . u
= V/74 =
,
2 but also
9
100
=
=
\/4
— = —V36 V9
/36
/
V
=
100
100
25
25
2 but also
25
2
9
_ ~
10
_ ~
5
Chalkboard Examples Find the square root.
—=
9
v/81
10
2.
9
^/9
1
36
3 1
1
4.
6
25
49
49
3^
'25/
\/25f2
5_^
3^2
100
VIOO
10
64
_ ~
5
7
aV3
5.
/225
225
V
15
Find the square root of
7.
/35
Express your result correct to one decimal
5.916
35
/25
25 •
35
25
place.
8
36
25
5
=
1.183
=
1.2
5
MULTIPLICATION AND DIVISION
364-365)
(pp.
Teaching These Pages
Show how
can be written in a different way to be useful in different + c) = ab + ac can be written ab + ac = a(b + c)
rules
ways. For example, a{b to
show how
to factor out a
In a similar
show
how
us
way we can
to multiply
y/ab
=
\fa
•
common monomial
factor.
rewrite the rules of the previous sections to
and divide with \/b
radicals.
can be written
^Ja
•
\[b
=
\fc^
can be written
V5
^Tb
Chalkboard Examples Simplify.
L V3
•
V27 =
\/f^ =
VsT =
9
2.
2\f5'A^/2
= %^/5~^ =
8\/l0
T119
3.
^96
3Vx'2lVx = S\Vx^ =S\x
4.
=
Vn
= 2V3
V8 '125
/T25
[5
/72x^
V72]
5.
50
9
•
= V2F = xV2
36
RATIONALIZING THE DENOMINATOR
(pp.
366-367)
Teaching These Pages
Have
the students find the value of
\A5 =
3.87
and
Vl =
one decimal place, using
to
V2
1.41,
15
.
3.87
V2 The
—VT5— =
2.1 A
=
2.7
1.41
calculation will take about 5 minutes of tedious work.
time that ate the
is
actually taken.)
method of
With
(Record the
this experience, the students will appreci-
rationalizing the denominator.
——
Introduce the method
\/l5
and have students
find the value of
again, recording the
v2
time they take. 15- y/2
Vl
Vl- V2
30
5.48
2
This calculation will take students only about
Chalkboard Examples Express
in simplest form.
=
2.14
2 1
minute.
=
2.7
amount of
A common error that should be pointed out y/6 = \/lOand3V6+6V3==9\/9.
if it
occurs
is
to write
V4+
Chalkboard Examples Express
in simplest form.
1.
3V6+2V6 = 5V6
4.
^/S
5.
\fl2y
11 •
2.
V4^
+ VYS =
3.
3V5+V5-3V3=4V5-3\/3
V9^
= 2V2 + 3\/2 = 5v^ - ^/3y - V25 3^ = 2 V3y - \/3y-5\/3y = -4^3^ = V4
+
V^
- V3y-
8V7-10V7=-2V7 •
•
3_>;
RADICALS AND BINOMIALS (OPTIONAL)
(pp.
370-371)
Teaching These Pages
—
Introduce this topic by having students consider the value of
we could
find the value
evaluate as shown. 1
may
Students
indicate there
we would
is
2
+
1
3.414
1.414
a simpler
is,
way
Point out
to find the value.
an expression that does not have a radical we would like to rationalize the denominator.
l-(2-
1
+ V2
(2
+
\/2)(2
To
+ v2
like to find
denominator, that
2
1
+ V2
2
that
2
\/2)
- v^)
2
- \^
2
in the
- V2
4-2
2
Chalkboard Examples Express
L 3.
in simplest form.
-
\/5(2 (2
-
=
V^)
3 \/3)(3
2
V5 - VSO
V2 -2)=6\/2-4-9\/6
2.
4V3(6 + V^) = 24V3
4.
(1-
\/3)(V3
-
1)
+ = -4
12
V2
Rationalize the denominator. 3
1
3
- V3 ^/3
1
_ V4
^3
+ V3
'
6
V3(l
+
V4)
-3
\/3+2\/3 -3
2
3\/3
-3
4 -
3(2
+ V5
VS)
= -6 -
3\/5
5
= -V3
Simplify. 8.
(1
+
V5)2
=l+2V^
+ 5=6+\/5
9.
(2V3-
3 \/2)2
=30 - 12V6 T121
Chapter
12.
Quadratic Equations
In earlier chapters the students have dealt with the graphs of linear equa-
The intersection of lines was related to the solutions of equations having two variables, and the meaning of function was introduced. In this tions.
chapter, the students erty: if fl^
=
solve quadratic equations using the zero prop-
first
^
then a
Q ox b
=
0.
Quadratic equations are then solved
using factoring, taking square roots, and finally using the quadratic for-
mula. These
skills
are applied to solving problems that require quadratic
The chapter concludes with a study of
equations.
the graph of this nonlin-
ear function, the quadratic function.
Objective for Pages 380-385
To use
1.
1 •
the zero property
ZERO PRODUCTS
and factoring
(pp.
to solve quadratic equations.
380-381)
Teaching These Pages Introduce this section by having the students complete each of the following examples.
4
3
Summarize that the product of two numbers, one of which is zero, must be Then ask the students what they know about the factors in the fol-
zero.
lowing examples.
5x
=
14^
y^ =
=
ky
=
if two numbers have a zero product, one of Extend this to equations with a binomial factor.
Students should conclude that the
numbers must be
zero.
Chalkboard Examples Complete. x{x
1.
If
2.
If (X
4)
-h
-
=
then x
l)(x -H 5)
=0
= _J_
then
or
jc
-|-
_2_ =
4
= _?
0;
or
^^ =
-
5)
jc-1;x +
0.
5
Solve. 3.
y(y
+
6)
=
^ = 0orv +
4.
6=0 y = -6
T122
3k(k
=
5.
k=0ork-5=0 k
=
5
(2x
+
8)(4x
-
16)
=
2x-H8=0or4x-16=0 X
= -4
x=4
Suggested Extensions
-
1.
Find _JL-
in (x
2.
Find
in (2x
3.
The
=
.)
the solution
if
.)(x
—
4) if the solution
solution to an equation
is
x
the form {ax
2
+
3)(x
-
+
-
b)(cx
=0.
d)
=— {4x
or
-
=—
jc
SOLVING QUADRATIC EQUATIONS
•
=
3 or
4 or
jc
jc
=
—7.
7
= —
Write the equation in
.
-
l)(3x
x
is
=
x
is
=
5)
382-383)
(pp.
Teaching These Pages
The
developed in
skills
this section
and
subsequent sections
in
will
be ap-
plied to solving problems that involve the quadratic equation (in
Section
7).
Have
the students find the product in each example. (x
-
4)(x
-
5)
=
x2
-
9x
(x
+
2)(x
-
5)
=
x2
-
3x
Point out that in this section
we
+ -
will solve quadratic
20 10
=0 =0
equations like
—
9x + 20 = 0, but the first step will be to factor them. Review the types of factoring before assigning the exercises, and remind students to always check for a common monomial factor first.
x^
Chalkboard Examples Solve.
+ 8x + + 6)(x +
x2
1.
(x
12 2)
= =
(a
a-S=0ora
x-f-6=0orx+2=0 X 3.
= —6
X
-5a -24 =0 - S)(a + 3) =
a^
2.
= —2
a
/
+ 12;; + 36 = + 6)(7 + 6) = y + 6=0 or y + 6=0 y = —6 y = —6
4.
=
+
3=0 a
S
= -3
m2 - 16 = - 4){m + 4) = m — 4 = or m + 4 = m=4 m = —4 (m
(y
Suggested Extension Solve. 1.
2/ +
3/
+
1
y
or
1
=
y = —
2.
1
2A:2
k
=
+
^ 3
_
3
=
or k
=
3.
1
6x2
_
13;^
+
6
^
2 3 x=jOrx=-
T123
3
SOLVING QUADRATIC EQUATIONS
•
(pp.
382-385)
Teaching These Pages Use the examples on page 384 side of the equation.
x^
X
—X =6 - =6.
into
jc(jc
to point out the
need
to write zero
A common error students make is — 1) = 6 and then concluding that
on one
factoring to change
x
=
6 or
\
Chalkboard Examples Solve.
-5x = (x - 8)(jc + 3) = X = ^ or X = —3 x^
1.
+ 2)iy - 13) = -50 y = 3 or y = ^
24
X
iy
-4 ^ -3 X
X
=
X 2 or X
+ = —2 1
Objectives for Pages 386-389
To
1.
4
solve a quadratic equation by taking square roots.
USING SQUARE ROOTS
•
(pp.
386-387)
Teaching These Pages 4, Chapter 11, in preparation which students will solve equations of the form (x + of- — c. Throughout the exercises, students should be encouraged to solve the equations by taking square roots.
This section
is
primarily a review of Section
for the next section in
Chalkboard Examples Solve. 1.
x2
X
5
•
= 36 = ±6
2.
X
= 12 = ±^/n
X
= ±2\/3
jc2
USING SQUARE ROOTS
(pp.
3.
a
388-389)
Teaching These Pages
Have
the students
compare the following equations. x2
/
= =
Use the above examples and
T124
36
^
(x
49
^^
(y
fl2
+ +
find the roots
1)2
3)2
= =
36
49
by inspection.
= 64 = ±i
4.
= 50 = 25 w = ±5
2m2 m2
Some jc
+
+ 3)^ = 16 as x + 3 = 16 and problem occurs, encourage those students
students will solve (x
= — 16.
3
If this
as 4^ before taking the square roots of
both
to write 16
sides.
Chalkboard Examples Complete. 1.
+
{X
2)2
+
X
2
= =
9
2.
-
(x
X
4)2
=
16
3.
+
(2x
-4 =^L
2x
1)2
+
1
= 25 = _1
Solve. 4.
{X
+ X
3)2
+
3 jc
X
= 16 = ±4 = -3 = —1
5.
-
(x
X
3)2
-
±4 or
3
X
= 12 = ±2\/3 = 3 ±2V3
6.
-
3(x
X
2)2
-
2
X X
1
= 12 = ±2
=2±2 =
or 4
Suggested Extension Students can solve any quadratic equation by the process of completing the square, illustrated below.
Solve x2 1.
2.
—
2x
—
2
=
0.
Write the equation with the constant
term
in the right
Add
to each
x2
—
2x
-
2x
=
2
=
3
=
3
member.
member
the square of
+
1
one-half the coefficient of x. 3.
Write the
left
member
as a perfect
(x
-
1)2
X
-
1
square. 4.
Take the square root of both members.
X If the coefficient
of x2 in a quadratic equation
lent equation in
which the
coefficient of x2
is
is
not
1,
first
= ± \/3 = ± \/3
find
1
an equiva-
1.
Students can use this technique to solve the quadratic equations in Section 6.
Objectives for Pages 390-399 1.
To use
2.
To use quadratic equations
3.
To draw the graphs of quadratic
the quadratic formula to solve quadratic equations. to solve problems.
functions.
T125
6
•
THE QUADRATIC FORMULA
(pp.
390-393)
Teaching These Pages So that students
them attempt
will appreciate the
6x2
The formula
Some
"power" of the quadratic formula, have by factoring.
to solve the following equation
is
+
i3;c
+
6
=
first Chalkboard Example. work with the formula if they record
applied to this equation as the
students
may
the values of a, b,
find
and
Chalkboard Examples
it
c for
easier to
each exercise.
Let
y =
the width;
-4) =
y(y
llXj
—
15)
The length
is
15
+
(y
7 — 4 =
the length.
165
=
(Is
cm and
possible to have a negative length?)
it
the width
11
is
cm.
The perimeter of a rectangle is 32 cm. The area of the rectangle is 55 cm^.
2.
Find the length and width. 21
+ 2w = / + w = /=
32 16 16
— w)w = w =
(16
The length
w =
use
is
-w 55 or
11 1 1
w =
5
cm and
the width
5
is
cm. (Point out
why we do
not
11.)
Suggested Extension
Two numbers
1.
The sum of
by
differ
4.
their squares
is
208.
Find the numbers. (x
2.
+
4)2
+x^ =
208
The numbers
are 8
A
m
garden
is
4
The diagonal of
and
12 or
longer than the garden
—8 and wide.
it is
is
—12.
20 m.
Find the length and width. (x
+
4)2
+
The length
8
•
x2
=
400
is
16
m
and the width
QUADRATIC GRAPHS
(pp.
12 m.
is
396-399)
Teaching These Pages
Have
the students
draw the graphs of the following equations.
y =X Discuss the graphs.
How
y
=
y = —x^
x^
are they the
same?
How
are they different?
In-
troduce the terms parabola and vertex. Introduce the meaning of a function as
it
relates to the
graphs of 7
=
j>c2
and /
=
—x^.
Till
X
Chalkboard Examples 1.
2.
+
+
Draw
the graph of v
What
are the coordinates of the vertex of
this 3.
=
parabola?
What
is
x'^
4x
4.
(-2,0)
What
the smallest value of y?
the largest value
of/?
There
is
no
value. 4.
How many
values of x
5.
How many
values
make 7 =
1?
two
=
0?
one
of/ make
a:
is
largest
"^
Basic
Algebra
Basic
Algebra Richard G. Brown Geraldine D. Smith
Mary
P.
Dolciani
Editorial Advisers:
Andrew M. Gleason Robert H. Sorgenfrey
Houghton
Mifflin
Atlanta
Hopewell,
New
Company,
Dallas
Jersey
Geneva,
Palo Alto
Boston Illinois
Toronto
The Authors Richard G. Brown
Mathematics teacher at The PhilUps Exeter Academy, Hampshire. Mr. Brown has taught a wide range of mathematics courses for both students and teachers at several schools and universities, including Newton, Massachusetts High School and the University of New Hampshire. He is an active participant in professional
New
Exeter,
organizations and the author of mathematics texts and journal articles.
Geraldine D. Smith
known
in both publishing and in mathematics Smith has done extensive work in developing and directing a wide variety of major elementary and secondary mathematics programs. She also serves as a special consultant for many
education for
Well
many
years. Miss
other educational materials.
Mary
P.
Professor of Mathematics, Hunter College, City Uni-
Dolciani
New
Dr. Dolciani is the author of many mathematics both elementary and secondary, and has developed programs of basic mathematics review for beginning college students. versity of
York.
texts,
The
Editorial Advisers
Andrew M. Gleason
Hollis Professor of
Philosophy, Harvard University. search mathematician.
National
Academy of
His
many
Mathematics and Natural is a well-known
Professor Gleason
affiliations include
membership
re-
in the
Sciences.
Robert H. Sorgenfrey
Professor of Mathematics, University of CaliforLos Angeles. Dr. Sorgenfrey has won the Distinguished Teaching Award at U.C.L.A. and has been Chairman of the Committee on Teaching nia,
there.
He
has been a team
member
of the
NCTM
summer
writing
projects.
©
by Houghton Mifflin Company part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage or retrieval system, without permission
Copyright
1980, 1977
All rights reserved.
in writing
No
from the publisher.
Printed in the United States of America.
ISBN: 0-395-27863-5
IV
Contents UNIT A CHAPTER
1
Working With Variables
Using Letters for Numbers 2 2 Parentheses, Order of Operations 3 Coefficients and Terms 8 4 Properties of Addition 10 5 Exponents and Factors 12 6 Properties of Multiplication 14 1
•
6
•
•
•
•
•
7
•
The
8
•
Properties of Zero
Distributive Property
Properties of
16
18
One
20 10 Introduction to Equations 9
•
22
•
•
1
Introduction to Inequalities
Reviewing Arithmetic Skills Career Notebook 27 Consumer Corner 28 Reviewing the Chapter 30
CHAPTER
2
24
26
Solving Equations
1
•
Solving Equations by Addition
2
•
Solving Equations by Subtraction
34 36
More Equations 38 4 Equations with More Steps 42 3
•
Solving
•
5
•
6
•
7
•
A
Mind-Reading Trick Combining Terms 48
46
Writing Algebraic Expressions
50
Applied Problems and Puzzles 54 9 Variable on Both Sides of the Equation 10 'Equations with Parentheses 60 1 1 Puzzles 62 Consumer Corner 64 Reviewing the Chapter 66 8
•
•
•
Cumulative Review
68
58
UNIT B CHAPTER
Positive and Negative Numbers and Negative Numbers 72 2 Inequalities and Graphs 74 1
•
3
Positive
•
3
•
Addition
4
•
Subtraction
5
•
Simplifying Expressions
6
•
Multiplication
7
•
Simplifying Expressions
8
•
Division
•
Solving Equations
9
76 78 82
84 86
88
90
10 Absolute Value (Optional) 92 Reviewing Arithmetic Skills 94 Career Notebook 95 Consumer Corner 96 Reviewing the Chapter 98 •
CHAPTER 1
•
4
Formulas
Perimeter Formulas
102
Area Formulas 104 Volume Formulas 108 4 Writing Formulas 112 5 Motion Problems 1 14 6 Cost and Money Problems 1 18 7 New Formulas from Old 122 8 Formulas from Mechanics (Optional) Reviewing the Chapter 130 Consumer Corner 128 2
•
3
•
•
•
•
•
•
Cumulative Review
VI
132
126
UNIT C CHAPTER
Working with Polynomials
5
1- Addition
136
2
•
Subtraction
3
•
Multiplying Monomials
138
140
142 4 Powers of Monomials 5 Polynomials Times Monomials 146 6 Multiplying Polynomials •
144
•
•
7
•
Multiplying at Sight
8
•
Square of a Binomial
9
•
Division with Monomials
10
•
11
148
152
154
156 Exponent Rules (Optional) 'Polynomials Divided by Monomials
12 'Dividing Polynomials (Optional)
Reviewing Arithmetic Skills 163 Career Notebook Consumer Corner 164 Reviewing the Chapter 166
CHAPTER
6
162
Factoring
1
•
Factoring in Arithmetic
2
•
Common Monomial
3
•
Using Factoring
170
Factors
172
174
4 Factoring Trinomials •
— Two Sums 176 — Two Differences
5
'Factoring Trinomials
6
•
Factoring Trinomial Squares
7
•
Factoring Trinomials
8
-A
9
•
Factoring the Difference of Squares
10
•
Many
Special Product
1
188
Types of Factoring
•
Reviewing the Chapter
Cumulative Review
180
182
84
More Difficult Factoring Consumer Corner 196 1
158
160
190
192
(Optional)
194
198
200
Vll
UNIT D CHAPTER
Graphs 7 Graphs You Often See 204 2 Graphs with Lines and Curves 212 3 Points on a Graph 1
•
•
208
•
216
4 'Solution Pairs 5
•
6
•
7
•
Graphs of Equations Slope of a Line 224 Functions
220
222
Reviewing Arithmetic Skills 228 Career Notebook 229 Consumer Corner 230 Reviewing the Chapter 232
CHAPTER
8
Equations with
1
•
The Graphing Method
2
•
No
3
•
The
Solution,
Many
Substitution
Two
Variables
236
Solutions
Method
240
244
246 4 • The Add-or-Subtract Method 250 5 Word Problems, Two Variables 254 6 Using Multiplication •
•
Cost Problems
256 Boat and Aircraft Problems (Optional) Consumer Corner 260 Reviewing the Chapter 262 7
•
8
•
Cumulative Review
viu
264
258
UNIT
E
CHAPTER
Working with Fractions
9
1
•
Fractions in Algebra
2
•
Simplifying Fractions
3
•
The -
4
•
Ratio
268
270
Factor 272 274 Problems Involving Ratio 1
5
•
6
•
7
•
8
•
Multiplying Fractions
9
•
Dividing Fractions
278 Proportions Applying
276
Proportion
10 'Add, Subtract
Renaming
280
— Same
Fractions
1 1
•
12
'Add, Subtract
13
"More
284
286
Denominators 290
— Different Denominators
Difficult Fractions (Optional)
Equations with Fractions 296 -Work Problems 298 16 Binomial Denominators (Optional) Reviewing Arithmetic Skills 304 14
288
292
294
•
15
•
Career Notebook 305 Consumer Corner 306 Reviewing the Chapter
CHAPTER
302
308
Decimals and Percents 1 Decimals 312 2 'Division with Decimals 314 3 'Decimals for Fractions 316 4 Equations with Decimals 318 320 5 Using Decimal Equations 322 6 Percents 7 Using Percents 326 8 Percents in Equations 328 9 Interest 332 10 Investment 334 1 1 Mixture Problems 336 Consumer Corner 340 Reviewing the Chapter 342 10
•
•
•
•
•
•
•
'
•
Cumulative Review
344 IX
UNIT
F
CHAPTER 1
•
2
•
3
•
4
•
5
•
6
•
7
•
8
•
9
•
Squares and Square Roots 11 Square Roots 348 Using a Square Root Table 350 352 Irrational Numbers 354 Simplifying Square Roots 356 Solving Equations Law of Pythagoras 358 362 Quotients of Square Roots 364 Multiplication and Division 366 Rationalizing the Denominator
368 Radicals and Binomials (Optional) 1 Reviewing Arithmetic Skills 372 10
•
Addition, Subtraction
•
373 Career Notebook Consumer Corner 374 Reviewing the Chapter
CHAPTER
376
Quadratic Equations 1 Zero Products 380 382 2 Solving Quadratic Equations 384 3 Solving Quadratic Equations 12
•
•
•
Using Square Roots 386 388 5 Using Square Roots 390 6 The Quadratic Formula 394 7 Problem Solving 396 8 Quadratic Graphs Consumer Corner 400 Reviewing the Chapter 402
4
•
•
•
•
•
Cumulative Review
404
370
EXTRA PRACTICE EXERCISES 406 TABLES 436 CUMULATIVE REVIEWS 438 ANSWERS TO SELF-TESTS 449 GLOSSARY 452 INDEX 457
XI
Symbols
Diagnostic Tests in Aritiimetic 1.
2.
What
is
the
denominator
in
—? 3
The area shaded
3.
in figure
D
is
repre-
sented by which fraction in row E?
e15' 15' 15' A
D
1
4.
Which of
the figures below repre-
sents the fraction
F
—'^^
Write the fraction represented by the
5. .
set
diagram shown below. ^
-
6.
711-^ 12-11755 88 54
Fractions
— Addition 3
1.
7.
Fractions
3- + 4- = ^L^
2.
3— 12
3.
— Subtraction
\4
1.
9
1
- + - = ^L5
i±
3^
2.
7
3.
.1
,1
Fractions— Multiplication
e:
4.
9.
Fractions
'
-
5.
6
.7
1
^4
11
3
^ 9
^
15
3— V4— ?2 .2 3.^X4-
^4
,
= 5.1^X2^ 5 2
'"'^To
Tpia
21
2—gX^-^4 V — — "^4
21
14^
6.
3
Z.
29
-J
= -^To 44x^ 6 5
3
9
6
1— V — — "^27" 1.3X^-^27
9 4A
5.
2
5I
i5 8.
- + - = _IAO
7I
4.
8
1
4.
— Division 1
5
1 '
5.
——= '9
45
2
-4=^^ 41
5-
_ 32°"^
8
10.
?
-
6— — = '"4 6.
?
^ J_ *6"2
3 J_
8
5-- 5-=
9
?
3^
=:
4
?
A^A— *9'7 5
?
9
28
9
6
8
^32
Decimals 1.
Write the decimal which represents "two and
2.
—5— = _J_ 1000
5.
Round
(Decimal)
3.
0.206
=
206 1000
0.005
thousandths." 2.0O6
six
(Fraction) ^_ 103
4.
3- = _i_
(Decimal)
3.80
5
500
40.5656 to the nearest tenth.
6.
1.033
+
0.1
+
10.066
=
_^11-199
40.6 7.
11.
856.175
-
20.05
310.1
^_ 836.125 =
71.60
8.
-
5QQ42
+
Percents
735.02
9.
^3-96
10.
6639
7.709
X.0031
727.311
^K^^9
625.98 2.
1.
0.69
= _J_7c^^
3.
- = -J-.7c 33^3 3
% 240.5
6.
289%
=
?
»
38-% = _?_ 2
9.
89
(Mixed numeral
4.
6- =
_^% 650
2 in simplest
form)
Too
1
7.
.07)21.707
11.
(Decimal)
8.
25% of 23 = ^^5.75
0.385
of 20 = 8 _^% 40
10.
20% of
= 340 _^ 1700 XV
Why
Study Algebra?
You've probably asked
this
question at least once. Here are a few
reasons to think about.
Through algebra
you'll learn to organize
and
ex-
press your thoughts concisely. That's important in
today's world.
You know
worked with a
calculator, for
this is true if
you've ever
you have
to
have a
plan for solving the problem before the machine
can begin
to help
you
at all.
You'll get a chance to view from a different angle all
those concepts you've already learned in math.
This will help you firm in your mind what math is all
about and
Algebra
and
will
how
it
works.
prepare you for further work in math
the sciences, if this
is
for you.
Also, hundreds
of careers rely upon a good, solid math back-
ground. Algebra can help give
this to
you.
Once you've learned basic algebra skills, you'll probably find yourself using some of them to solve many problems in your everyday life.
XVI
Here's what you'll learn in this chapter: 1.
2.
3.
4.
To find the values of expressions with variables. To simplify expressions. To find the value of expressions with exponents. To check whether a given number is a solution of an equation or inequahty.
Chapter 1
Working With Variables
1 •
Using Letters for Numbers
Imagine a calculating machine that adds it. It might look like this.
INPUT
5 to
any number you put into
When we
We
call
;>r
use a letter to represent a
+
5
and y
number we
—
3 variable expressions.
X
+
call the letter
a variable.
Here's a hint to help you
remember
,/-3
5
A
find the value of
EXAMPLE
1
If «
an expression
=
10, find
EXAMPLE
2
If
jc
=
and
14
in the following way.
+
the value of «
«
;;
=
+
3,
2
= =
10
+
2.
2
find the value of
x
- /.
11
Suppose you had a machine which multiplied by
is
.
12
=
There
another way to show multiplication, too.
7.
It is
with a dot. You'll
see this used very often.
7x4 < variable
is
means
the
same
as
and the dot are usually
In algebra, the times sign
>
left
7
•
out
use^.
EXAMPLE
3
If «
=
2nk
If
the value of the expression.
-
^
+
6 10
31. /7^ 96
1
2 • Parentheses, Order of Operations When you
have several numbers
to add, subtract, multiply, or divide,
helps to show which operation to do
Here we add
+
(3
Multiply
first.
5)-2
7
-
16
When
there are
Step
1:
Step
2:
Subtract
first.
I (2
•
and
divisions in order
EXAMPLES
from
multiplications
all
additions and subtractions in order from
how If
6
+ 5-2 +
+
5
V 10
=
-
+
5
left to right.
left to right.
2-3
V
8-6
13-6
V
when
to use these ideas
X
2)
steps to follow.
13
see
+ 11
all
3
let's
-
1
3
Now
(8
5
no parentheses, there are only two
Do Do
first.
I
+
5
3)
7-6.
8-2
it
Parentheses are used.
first.
4,
there are variables.
find the value of 2(x
+
+
and 2x
5)
5.
This means that you multiply here.
7 2{x
+
5)
Vi Classroom Practice
~1^
= 2(4 + = 2-9 = 18
2x
5)
+
5
= 2-4 + = 8 + 5 = 13
5
f/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/A
Find the value of the expression. 1.
If
5.
(7
x
-
=
2(jc
3)-2
3, find
+
8
2.
2
+ 3-4
14
+ ^ _
3.
3
7.
3(x
1)6'
-
4.
7
8.
3x
(3-2)
the value of the expression.
5) 16
6. 2jc
+
5
1
-
1) 6
-
1
8
1
)
,
^/i
Written Exercises
W/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/A
Find the value of the expression. 1.
(16
-
4) -2 24
4.
(16
-
4)
7.
3-4 + 517
10. 2
^
•
-
2
1
-
(3
1)
3
3
13. 8
+
(2-3)14
16.
4
+
8
19.
If
X
=
4, find
20. If
X
=
21. If
b
-
+
2.
16
-
5.
(8
+
8.
6
+
4) 8
•
-
+
(4
2) 2
4 38
+ 7-431
11. 3
+
14. (2
2 8
(4-2) 8
-
(4
+
2-2 +
+
17. 6
3)
25
1)
29
the value of 2(jc
+
6)
and 2x
+
6. 20,
7,
find the value of 5(x
+
1)
and 5x
+
1.
=
9,
find the value of S(b
-
4)
and Sb
-
4.
22. If 7
=
8,
find the value of
3{y
+
2)
and 3y
+
2. 30, 26-
a
=
5,
find the value of 4{a
—
2)
and
-
24. If 5
=
3,
find the value of 6(5
23. If
Tell
what
is
+
4)
4xy
xy
-
9.
-9
lab
— xy; = 5a
Ixy
-\-
f/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/A
Complete the statement. 1.
In the expression 5x the coefficient of
2.
In the expression 6x
3.
In the expression Aa
understood
to
x
?^
is
+ ly the coefficient of x + b the coefficient of ^ is
.
?
is
^
not written.
It is
be -JJL.
Simplify. 4.
X
+
X
S.
y +y
+y
r/A
6.
+
lab
3ab
Written Exercises
1.
3a
+
a
+
+
3
2
4a + 5
Sab
3y
2x
W/M/M/M/M/M/M/M/M/M/M/M/M/M/JF/M/M
Simplify.
A
1.
5.
9.
+
4x
-
6y
2x 9y 3«
2.
6.
^
5a
4-
6xy
+
4a
\1.
10.
8^-25-2
21.
3a
+
6x7 -^
14.
25. 7 y
a-4
18.
4a - 4
-
xy
+
5x/ +
-
2y
29.
3«
if
+
3 not possible
33. ar
+
4ar 5ar
3a
2jcy
1.
2a
+ +
6b
+
3b
-
3« 2a
+ +
b 86
a - 2 4a - 2
5a
+
11.
4
+
7a
15. 3
+
19.
6
-
22. 9c
1
-
2c
+
+ 6 7a + 6
+
5y
+
+
4y
4c
23.
5d
6y
1 1
9x
26. 2jc
+
6d
x
+
4y
If
27. 7aZ)
-
4s
+
3y
+
12.
2x
+
4x
+
5y
-
-
2ab
31.
16.
3a
+
+
3a
+
20.
7 + 2y -
+
3b
+
35. 8« 2 not possible
3
7
3/ - 7
3d
24. 8 y -f 3 y
+
5/?
-
^
28.
cd
-}-
4c(^ 5ct/
32.
2a
-
2a - 2a
+
5a 7a
36.
4v Ty
possible.
3a^
1
6a + 3
not possible
34. r 4s not possible
4y 12/
6x + 7
5a6 + 56
you cannot simplify, write not 30. 4^
5y
8e
om tvb uuuau.
THE eBLATlVS SPFEO-A VAUUe THE" FIFTH *^
POSA/Bfi K>\THF1?
THE WIZARD OF ID
TTf
TFUM CUSED.
iA4F«fiSlPtgI^F THAT VVEPB- TlJUe, VV& WOULD
BE
IN/
•V:fTAU
Cy\KKNE55
AT
by permission of John Hart and Field Enterprises,
THIS
Inc.
I 13
6 • Properties of Multiplication Multiplication has
some
properties which are like the properties of
addition.
THE COMMUTATIVE PROPERTY In multiplication, the order of the factors
(AT
•
3
=
3
•
difference.
Algebra Example
Arithmetic Example 2
makes no
ab
2
=
ba
THE ASSOCIATIVE PROPERTY In multiphcation, the grouping of the factors
•
3)
•
4
=
2
•
(3
(ab)c
4)
•
=
a(bc)
the addition properties, you do not need to memorize the names. Just be sure you can use the properties. They can make your
As with work
easier.
EXAMPLE
EXAMPLE
1
2
Evaluate 5
SimpUfy 2
•
•
82
•
2.
5
•
3
Simplify
•
{5x)
4
•
2 •
(5
82
•
2)
•
82
10-82 820
= =
(2
•
5)
•
X
\0x
= =
(2
•
4)(r
•
r)
8r2
Simplify 6 -a- a- 5 -a. 6
14
5
(2r)(4r).
(2r)(4r)
EXAMPLE
= = = =
2
•
{5x).
2
EXAMPLE
82
'
a' a' 5
'
a
= =
difference.
Algebra Example
Arithmetic Example (2
makes no
(6* 5)(a
30^3
'
a- a)
Vi Classroom Practice
f/m/M/M/M/M/M/M/M/M/M/M/M/M/M/M/k
Find the value of the expression. 1.
17 -25 -4
1700
Show how you use
the properties.
7 • The Distributive Property Jan at
is
buying two
$3 each.
Her
2($1
+
felt-tip
$3)
at $1 each and two books can be figured in two ways.
pens
total bill
f/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/A
Vi Classroom Practice
+
=
1.
Check
2.
Is
3.
Once you've checked to see you should be able to tell
——
2(4
to see if 3(2
•
=
3)
(2
•
4)
•
(2
1)
•
(3
•
2)
+
(3
•
true
1) is true,
3) true? no
the statement in Exercise 2
if
2{ab)
if
=
(2a)
•
(lb)
is
true.
is
true, Is it?
no
True or false? 4.
3(a
-\-
=
b)
3a
+
3b
5.
3(cd)
=
(3c)
•
true
(3^?) false
State the expression without parentheses. 6.
2(x
+
7.
5)
2x + 10 10.
^/i
3(x - 4) 3x - 1 2
1.
f
11. a(a a2
Written Exercises Check
to see
3(4
+
5)
3(x + 2) 3x + 6
if
(3
3)
- 3a
4(a + 1) 4a + 4
12. bib b^
-
3)
- 3b
9.
5(7
+
a)
35 + 5a 13. y(2
-
y)
1y - y^
W/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M
the statement
=
-
8.
is true.
n
8 • Properties of O
r
• Any number times
is 0.
7-0 = •
No number
fl-0
can be divided by
=
0.
• Zero divided by any number (except 0)
0=0
0=0.
6
to a
number
+
=
gives that
6
n
Adding and subtracting A2
+
5
—
5
the
=
same number
+
«
1
Let X
=
4.
Find the value of 9(x
%x -
EXAMPLE
2
Simplify 2
+ 2
3x
-
4)
= 9(4 = 9-0 =
=
= =
King Features Syndicate 1974
18
-
4).
4)
2.
+ 3;c-2 = 3x + 3jc
3x
+
number.
^^ -
Let's put these ideas to work.
EXAMPLE
Divisions are often written with a bar.
6
Adding
is 0.
2-2
«
is
like
adding
0.
T/i
Classroom Practice Find the value
if
possible.
If not,
say impossible.
8-0
3.^
2.0
5-0
1.
f/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/A
9
7 impossible
Let
jc
=
5.
Find the value of the expression.
5.0
6.
jc
X
f/A
Find the value
if
8.
35(x
2.
0-31
3.
.
7.
impossible
9.
Let X
10.
Let X
IL Let 12.
JC
Let X
Simplify.
= = = =
1.
2. 3.
0.
Find Find Find Find
5)
If not, write impossible.
18
+
35
-
18
4.
16
the value the value
5
•
— 1). of (jc 2) ^ 4. of {2x - 6) ^ 4. of x ^ 3.0
the value of 5(x the value
-5
8.
impossible
+ 16
35
16
-
11
W/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/A
possible.
12-0
5
7.
5
Written Exercises
1.
-
+
16
•8-0
.
9 • Properties of
1
Whether you add to a number or multiply the same result the original number.
number by
the
you get
1,
—
original original
The second sentence above
+ x
number number
= =
1
states a property
original
original
of
number number
1.
• Multiplying a number by
1
number.
gives that
6*1=6
ft'
• Any number (except 0) divided by
=
\
n
itself is
1.
^=\
1=1 4
1
1.
Remember, you can
Any number
divided by
write a division with a bar.
1
is
that
6 Q -=8
number. n -=«
V. Here
are a couple of examples to
EXAMPLE
1
Simplify 8
show
these properties.
-f
=
EXAMPLE
2
Simplify 4x
-
X
4.
This gives you 4;c
^4 =
4x
4
= 20
X
4 4
—X,
or
jc.
Vi Written Exercises Let X
A
1.
^
= 1
4.
W/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/A
Find the value of the expression. 2. 4a;
16
3.
(4
-
x)
•
1
1
6 Simplify.
10 • Introduction to Equations STRIKE THREE!
THE GAME 15
NEVER OVER
UNTIL THE LAST ^
MAN
(« OUT.'
ICAN5TILL BE
A
HERO.,
1959 United Feature Syndicate, Inc.
Consider whether the following statements are true or
Hank Aaron Charlie
He
is
is
Brown
a baseball hero is
true
a baseball hero
false
maybe
a baseball hero
(It
The sentences below 6
true,
maybe
+
4
=
10
true
8
false
=
1
maybe
-\-
1
(It
true,
maybe
false
depends on what x
is.)
These number sentences are called equations. Equations always have an = sign, but are not always true. Is
X
+
2
= X
7 true
+
2
=
if
7
x
=
6?
false
depends on who he
are like the ones above.
5+1 = X
false.
is.)
8
Vi Classroom Practice Tell
,
r/A
in color is a solution.
7 -
=
jc
+
3
=
9 6
7 or 6?
2.
3.
4
+
X
=
10 6
9 or 6?
4.
3n
=
\2 4
5.
3x
8 3
2 or 3?
6. 4jc
-
12
7.
m-
16 or 34?
8.
5a
=
35 7
-
=
1
=
9
25 34
Written Exercises
+
=123
9
3.
3n
=
36 12
5.
3«
+
7
7.
5(r
9.
;c
=
+
2)
9
=
+
5a
-
13. n^
=
13 2
=
30 4
4x 3
=
1
3fl
+
7
1 1
18 or 4?
18
4 or 9?
=
7 or 8?
20 8
30 or 7?
W/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M
which of the numbers shown
.1.x
11.
B
which of the numbers shown
1.
Tell
A
f/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/A
9 5
25 5
in color is a solution.
-
=
3 or 21?
2. jF
12 or 33?
4.
4x
=
24 6
2 or 4?
6.
2a
-
6
3 or 4?
8.
7(5
7
-
or 4?
10.
2b
5, 6,
or 7?
12.
4(a-
0, 5,
10?
14. a^
15.
Write an equation which has 9 as a solution.
16.
Write an equation which has
17.
Write an equation which has
18.
Write an equation which has no
=
4 or 5?
b
5 or 6?
21 6 -\-
1 2
= a+
5)
=
6 or 20? 45
=
5
-\-
\7
\^
Answers
may
Find at least one solution. You
=
3)
2, 3,
8 or 22?
15 22
to exercises
1, 2,
or 3?
6, 7,
or 8?
0, 1,
or 2?
15-1
vary.
as a solution.
1 1
as a solution.
number
may need
to try
as a solution.
many numbers
for
some
exercises. 19.
JC
+
5
=
23.
Zj
-
4
=
27.
C
31.
7 +
1
3a 7 •
=
13 8
2 8
41 40
20.
7 -
24. b
28.
X
^ +
=
17
5
5
20 37
= 525 =
25 20
21.
=
5«
30 6
-0 = Answers may
25. «
29.
x
+
=
3
=
42 2
32.
2m +
9
=
21 6
33. 2/?
+
4
=
6x
26.
x
+
5
=
116
30.
m -
9
=
27 36
34.
x^
36 6
vary.
+
x
Answers may
=
22.
3
vary.
4
=
x
0,
1
23
II Not
all
•
Introduction to Inequalities
number
sentences are equations.
Some number sentences are We call a number sentence
statements that two numbers are not equal. like this
an inequality. Here are some examples of
4
is less
than
7.
inequalities.
f/i
W/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M
Written Exercises
.
or
Al.
6?7
Tell
which of the numbers shown
in color are solutions.
9.
jc
13.
;c
>
14 20
15.
«
17.
X
< 9 4,
6
4, 6, 9,
10
18.
7
+
27. 83
5
•
-
29. (24 For
4) ? 7
more
-
•
8)
•
Tell
10. c
0, 1, 3,
7
12.
7
> 9 10,
14.
y
>a+
4«
7
4, 6, 10,
1
22. 4jk
0, 3, 5,
14
-
9
7
3, 5,
3^^
2, 5, 7,
2
13 7
6 5
4, 3, 6,
2
2, 4, 5,
7
1, 0, 2,
5
or =.
150 calories in a can of cola
200
->
200 calories in a piece of pizza
number of number of
Check:
r/A
Written Exercises
A
1.
An
= =
X
Answer:
calories in 3 pieces of pizza calories in 4 cans of cola
=
=
3
4
150
•
•
200
=
= 600 600 /
W/M/M/JF/M/M/M/M/M/M/M/M/M/M/M/M/A
apple has 30 more calories than a peach.
Five peaches have as
How many
many
calories as 3 apples.
calories are in each?
peach: 45 calories; apple: 75 calories 2.
A
donut has 50 fewer calories than a glass of milk. Four donuts have as many calories as 3 glasses of milk.
How many milk: 3.
200
calories are in each? doughnut: 150 calories
calories;
One number
is
7
more than another.
Twice the larger is 22 less than 4 times the smaller. Find the numbers. 18, 25 4.
One number Five times
is
t+ie
5 less
Find the numbers. 62
than another.
smaller 7,
number
12
is
1
less
than 3 times the larger.
!
5.
Maria has twice as much money as Paul. Paul has $8 more than Rocky. Together they have $104.
How much 6.
Rich
is
Ruth
is
B
7.
8.
Paul: $28, Maria: $56; Rocky:
$20
3 years older than Carla.
twice as old as Rich.
Their ages
How
does each have?
old
is
total 33 years.
each person?
Caria: 6 yrs. old; Rich: 9 yrs. old; Ruth:
Warren is 14 years older than Chuck. Next year Warren will be 3 times as old How old is Warren? 20 years old Francine
is
as
18
yrs. old
Chuck.
6 years older than Carol.
Last year Francine was three times as old as Carol.
How C
9.
See
old
if
is
Carol? 4 years
you can solve
msBsn
this
old
problem.
'A
MAN HAS A PAU6HTERANP
A
IS
THREE VEARS ." OLPER THAN THE DAUGHTER. SON.. THE SON
1972 United Feature Syndicate, Inc.
For
more
practice, see
The man
is
THE MAN WILL &E OLP A$THE PAU6HTEK IS N0W,ANP in TEN i/EAfiS HE WILL 5E fOmiEH 4EAK5 OLPEKTHAN THE COMBlNEP A6E5 OF HIS CHILPREN... WHAT IS THE MAN'S PRESENT A6Er" IN ONE VEAR SIX TIMES AS
41 years
I'M
SORRV,
kJE
ARE LWA5LE TO
COMPLETE Y0i;RCALL..PLEA5E OIECK
THE NUMBER ANP PIAL A6AIN
old.
page 411
iSELF-TESTi— Solve. 1.
4.
3x
=
48
Lou has
—
X 12
twice as
2.
6(a
much money
-
3)
=
12 5
3.
4(m
-
3)
=m +
15 9
as Jo.
Jo has $11 more than Sherry. Together they have $89.
How much 5.
does each have?
Jo: $25; Lou: $50; Sherry:
Al has twice as much money as Vic. Vic has $5 less than Connie. Together they have $125. How much does each have? Vic: $30;
Al:
$14
$60; Connie: $35
63
S ER CCRIVIER f ^^T
^^j
Metric units such as the milligram (mg), gram used to measure quantity.
The
prefix milli-
means
1
-^
->1000 milligrams
1000 grams
Here are some objects usually measured
Here are some objects measured
V 64
in
=
1
=
1
(g),
and kilogram
gram
kilogram
-4
f/m/m/m/M/m/M/M/M/M/m/m/m/m/M/m/k
—
—
2 • Inequalities and Graphs The number X
-
line
3
X
= =
is
useful for picturing a solution of an equation.
-3-2-1012345
2 5
Think about inequalities. Sometimes there are an unlimited number of solutions. Take the inequality x < 8.
Some
More
In order to
solutions: 7, 6, 5, 4, 3, 2,
solutions: all
show
0,
1,
numbers between
all the solutions,
we make
—1, —2, —3,
and
8
7,
.
.
.
means "and
so
on"
between 1 and
6,
\
a graph like
this.
x
x
H-4-3-2-1 ^ EXAMPLE
=
5
2
the solutions of
\
3
-2.
\
\
1
x
2
1
< -
1
\
h
2
3
4
2
3
4
1.
^
-4-3-2-1 74
1
3
4
5
r/i
W/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M
Written Exercises
Graph the given numbers. Write an
Sample
A
-2,3 > -2
1.
-2 -7
8. 0,
-4
-4
-4
number
the solution of the equation on the
Graph
-6
-6 < -1; -1 > -6
7.-1,-7 -7 < -1; -1
M
I
-1,
3.
> -5
-5
-7
_i; _1
4 4
'
-1
-7,
6.
.7
-3,1
compare them.
inequality to
^
12x« 17.
11.
(2x)(2x2) 4x^
23. (-;c4)(-3jcyz2)
24. (-6a^b^)(abc^)
3x^yz'^
25.
(a^b)i-5a^b^)
26.
(-x3)(-5x2y)
27. (;c73)(_2x3y2)
28. (-a^b^){a^bc^)
-2x4/5
141
4 • Powers of Monomials Study the examples below and see rule of exponents.
(y^)^
To
find the
can discover another important
if you
= y^
= y^^
power of a power of a number, multiply the exponents. (x^f
=
-y^ 'y^
=
EXAMPLE
1
(x^f
EXAMPLE
2
(ySy -^8-2 _^16
x4-3
Suppose you have an expression
=
x^^
jci2
like (xy)^.
You can
rewrite
it
following way.
(xyY
Make
=
ixy){xy)
a note of this rule of exponents.
(xyT
EXAMPLE
3
EXAMPLE
4
(ab)^
=
=
x"/"
a^h^
WARNING! 2^2 la'
2^2 and {laf are not the same.
^
(2^)2
xY equal 3^2^
2;>c^,
?
times. 2
^Jq^^
j^
equal
9a: 2?
or does
it
equal
Sjc^? Sx^
qj-
Qx^
Simplify. 5.
(^4)2 38
9.
(X2)3 x«
6.
(C2)5 cl«
10. (2a)2 4a2
7.
11.
(«6)4 a^b*
(4«2)2i6„4
8.
(5^)2 25a2
U2. (-3x7)2 9xV^
r/i
Written Exercises
W/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/A
Simplify.
A
1.
(;c2)3
5.
(C^)
3\5
2.
3\4 (^3)
6.
4M0 (n4) ,40
9.
(
5 • Polynomials Times Monomials Now
that
you can multiply monomials, you can put the
distributive
property to work.
You know
3{a
1
+
3b)
= =
(3
2
-\{2x -3y)
EXAMPLE
3
a(a^
EXAMPLE
^/i
4
y
+
2qb
-2x(9x3
Classroom Practice
+
-
3a
EXAMPLE
+
5(3
same way
In the
EXAMPLE
that
+
a)
+
5(3x
+
{3
-
4)
+
=
4y)
(5-3)
+
= =
3x)
+
+
207
+
(a
(5
•
\5x
(5-4)
(5
•
4y)
3b)
9b
= {-\'2x)-{-\' 3y) = -2x + 3y
b^)
3x^
= =
+
(a
a^
x)
•
a^)
+
+
(a
•
+
2a^b
2ab)
•
b^)
ab^
= {-2x'9x^) + (-2x'3x^) + {-2x'x) = -18x4 - 6^3 - 2x2
f/M/M/M/M/M/M/M/M/M/M/JW/M/M/M/M/A
Multiply.
L
2{a + 4j 2a + 8
2.
3(x + v) 3x + 3y
4.
x(x-2)
5.
«(«
x^
— 2x
a2 8.
+
3.
2b)
6. 2c(dz
_
2c)
9.
x(x2
-5a2 + 10c 10.
-x(x2
-
2x
+
4)
IL x2(x2 x^
144
-
+
b)
2ac + 2bc
+ 2a6
-5(«2
-l(2x + 3y) -2x - 3k
3x 3x3
X^
+
+
1)
x^
12.
+ +
2x
+
4)
2X2
+
4;f
a2^2(_^ _ fy^ -3^62 - 32^3 1
^/i
W/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M
Written Exercises Multiply.
A
1.
5.
2{x + 4} 2x + 8
-6(«
+
- 6n -
9.
2m)
+
4a(a
2b
19.
22.
+
23.
-
2jc
4.
5(^2 _^ ^j Sa^ + 56
-
Z?)
8.
x{x
- 1 5x2 -
+
5jc
+
2v)
12. 2;c(3x
^
Qx/
Bx^
^2)
-3c(2c2 _6c3 -
+
2x
+
ab{a^
18.
a^b
-
4c
I2c2
1)
+
+
z
/
laZ?
2a^b^
-
1)
- ab
-y\y^ -
2y2 + 4y) -/5 + iy4 _ 4-pi
21.
5) 15c.
-
- 2x
- \(2x + y + z) _ -
15.
4x2
_ 2v + 1) -4/4 + 8/2 _ 4j^^
3y)
+ 3xk
+
-4v(y3
20.
+
x^
-5;c(3jc
11.
-4 - 20x -
17.
4)
- 8x
y)
- ab
a^
4Z))
-4(1
14.
3)
-Jc2(x + 2jc2) -x3 - 2x4
A
b^
- 2a^b + 4ab^
2x3 - 4x2
is lOn centimeters long by (n + 6) centimeters wide. area as a polynomial. (lOn^ + eon) cm2
rectangle
Write
B
-
2x(x2
a(a
7.
b"^}
-
-«Z)(2a
10.
Z))
+
4x + 4y
-5a -
4a2+8a6+12a 16.
3. 4(jc
- l(5a +
6.
2m
- ca - cb
13.
b)
3a - 36
1
+
-c{a
\a -
2.
its
You have
collected (3«
+
1)
dimes.
What
their value in cents?
is
(30/1
+
1Q) cents
Solve.
24. 4(2« 26. 5;c
+
3)
-
2
-
2(2x
-
6)
-
+
28. 2i5x 30. 3(;c
-
4)
32. 6(1
-
3x)
'SELF-TES
+
3(n
-
=
+
6)
=
3(2;c
-
4)
2(2x
-
1)
2(2x
+ +
1)
-3,
25.
10
=
=
5)
,
4 2
=
^r
_^
40 -2
'
+
-(«
3)
-
27. (2^
-
3)
29. 3(1
-
2a)
^3i. 2(«
-
6)
33. 4(2a
-
+
3)
2(n
(7
+
(6
+
6)
-
5(2«
-
+
2(«
7)
=
2^)
=
-lU
63 72
= -7
+
4)
=
32 2
-
8)
=
22 3
1
H6
•
Multiplying Polynomials
To multiply by
a binomial
you
also use the distributive property.
use a vertical form. Your work
is
«
+ +
n^
-\-
n
+ +
3«
n
«2
1
3
Here are a few more examples
EXAMPLE
1
X X x^
x^
2
2y
+ + +
-
n(n
4«
Be sure
+ +
+
3(«
5x
5
14 14
1)
that like terms are lined up.
to study.
x(x
-
+
3
+
7)
Ix
2x
1)
3
7 2
Let's
a lot like multiplying in arithmetic.
-2{x
+
7)
5
2
f/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/A
Vi Classroom Practice Multiply. 1.
+
{x
4.
7.
{x
f/A
-
3)(;c
x2
- 2x -
+
{2a
5.
5)
3/)2
+
5){x 2x2 - X
8.
1
-
1
-
+
(a
6.
5) 5
+
4)(5jc
0x2
4)(a 1) + 3a - 4
a2
6){2a 4) + 8a - 24
2a2
1)
8x - 4
-
+
6){n 3) 3n2 - 3n - 1 8
11. (3«
2)
- 10
Written Exercises
-
{2x
+
1)(a:
-
+
{a
3.
7)
+ 11x +
2x2 3Z))
+
2)(n
+ 9n + 14
+
(2jc
1
+
Z>)(«
+
(«
/)2
+ 7a6 +
-
{2x
2.
4)
+ 6x + 8
+ 2a2
10.
+
2){x
x2
12.
-
(4;c
+
\){2x
-
8x2
3)
0x - 3
1
W/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/A
Multiply 1
+
X
4.
7.
+
3)(«
+ 8a + 15
-
;c
-
X
B
25.
6
+
^)(«
14.
+
n
b)
-
/,2
+
2)(x2
m—
(x
+
l)(jc
(«
+
+
a/)
(y
-
17.
2)(m^
6.
+
c)
9.
+ ac +
/)c
b)(a
-
3)(y
_ 7/ +
1
20. (a
-
+
6x2
-
2b)ia
23. (6jc
1
-
33.
+
28.
4)
2a-
30. {n
4)
3a/)2
+
a3
ti^
-
ab + 4b^)(5a^ -\- ab -\- b^) - 3a^b + 2^a^b^ + 3ab^ + 4b*
lOa" 37.
n
7^
For
1)(«4
-
4n'*
_
3«3
+ 4n3 -
moi''e practice,
24. (3>^
a
-
+ n^
+
-
8«
4)(w2
-
„2
+
+ n -
see page 417.
1) 1
-
-
38. {a
_
Z7)(a2
-
a2/7
-
7jc
+
-
+
-
b){a'^
+
1
1)
- 2
y
16)
Z72)
+
b^
12)(jc2
_
3jc
-
1)
4)(2x4 1
b) 6
2)(2>^
-
+
+
\){a
- 10x3 + 32x2 _ 29x - 12
3x4
1)
-
+ 2a6 -
X*
1
-
6x2
+ 48„ - 64
12/72
a/)2
34. (x2 36. (;c 2x5 -
-
4)
2x + 1) + 5x + 2 6y + 9) 27/ + 27
+
+
l)(4;c
6/2
+ 3X72 + 9/2 +
{y
n^
35.
+
-
2a2
2){x^ + 4x2
/3
2« + l)(n2 + 2« + 1) + 4/7=' + 6n2 + 4a7 + 1
+
n2
21. {2a
3
_ y _ 20 1
x3
32. (a
+
18. (4jc
5)
+
(;c
5){y
1)
+
4/7
1
2b)
-
5)(6jc
26.
2x - 8
— 2m +
3a2;b
(7
36x2 _ 60x + 25
31.
+
-
-
/2
- 462
4)
-
3)(«
/72
/73
a^
-
1
15.
3)
+ x/ +4x + 4/
12. (n
4)
+
9)(>^
+ 12/ + 27
+ y){x +
{x
x2
6)
+ 24
0/7
1
+
(J^
+
4)(«
+
k2
+ 8x + a2
4jc
+
7) 7
iy + 3)iy - 4) _ K- 12 (4x + l)(4x + 1)
2« + 1) + 3n + 1
-
6x2
+
+
{n n2
3.
+ 8x +
/2
2y^
-
1)(«2 + 3n2
-
X
3)(7 + 4) + 7/ + 1 2
,2
— x){6 — x) 36 - 12x + x2
x3
29.
11.
2y)
+
5jfj^
n3
27.
a2
2) 2
+ X -
a2
22.
8.
2) 2
+
1)(X
+
a
/2
- 3x +
+ y){x +
2x
+
(;;
x2
/2
-
\){x
2x2 19.
5.
5)
+ 2x/ +
x2
16.
+
+ y){x + 7)
X
x2
13.
2.
2)
+ 3x + 2
a2
a
x2 10.
+
l)(x
x2
-
3jf3
+
5x^ - 7x^ + 10) + 28x2 + 1 0x - 40
a^b
+
^2^2
_^
^^3
+ a^
^4)
-
b^
147
7 • Multiplying at Sight When
you multiplied binomials by using the vertical form, you may have noticed
a
pointed out
pattern.
The
at the right.
pattern
is
X
Be sure
to
watch the
EXAMPLE
EXAMPLE
2
3
signs!
Multiply (x
-
4)(x
-
2).
Step
1:
{x
-
4)(jc
-
2)
=
x^
Step
2:
(x
-
4){x
-
2)
=
x^
-
6x
Step
3:
(x
-
4){x
-
2)
=
x^
-
6x
Multiply
(;c
-
4){x
+
2).
(;c
-
4)(jc
+
2)
=
/
x2
-
2;c
T
-
\
.
.
+
.
4^ S
- 4^2 A =
29.
a
2) 2
6.
5)
9.
-
2)(«
-
+
5)(a
+ 2a - 15
+
b){a
b)
-
/)2
-
2^
-
+
1
+
3b) 3b'^
6x
- y)(4x + _
+
1
its
its
area. 6/^
Find the area. Use
+
l)(x
+ + +
27. (4x 4x2
—
2)
+
4)
+ 9x + 4
5)(2x
6x2
-
5)
_ 25
4)(3;c
-
4)
9x2
-
-16
2y)(2x + 3j) ^ 1 2xy + 6/2 3y)(x
—
5y)
_ 23x/ + 15/2
+ 2p){7n - /?) 21n2 + llnp - 2p2
30. (3«
2y) 2/2
3
+ 6x +
9
2a
x+3 x2
- 68 -
+
1
a+4 2a2
x) centimeters by {x
—
4) centimeters.
by (3/
+
1)
+ 5a -
x^
A rectangle measures (2^ + Find
+
X
0x + 24
area. 2ix
-
5)(;c
+ 3x - 10
33.
4
A rectangle measures (17 — Find
4)(/ + 1) - 3/ - 4
Iw.
x + 6 x2
3) 1
+
24. {3x
5)
-
Qjfy
—
2)
x2
21. {3x
32. x +
-
iy
3x - 5
- ^Oab +
b){a
6)(a:
- 9x +
4x2
3a2
3a
— x2
18. {2x
+
\){2x
6)(x
+ 8x + 12
2x2
3) 3)(27 4/2 - 9
-
3x
+
+ x2
15. (2jc
3)
-
(x
12. {x
1)
a2
(x
/2
+ n - 2
24x2 +
31.
35.
+
6x2
3a
Find the area. Use
34.
a
-
3)(a
a-
l)(2n 3) 10n2 + 13/7 - 3
3jc
n
-
- 8a + 15
- 4
7)(6a - 7) 36a2 - 49
+
-
3.
5)
- 3x +
n^
6)(v - 2) + 4y ^ 12
5n
«
+
l)(;c
a2
Y^
6fl
x2
1
+
x2
19.
3)(x
+ 8x + 15
X
+ 2a - 3
^+ X
5.
1)
+ x2
X
+ 3x + 2
x2
a-
13.
+
2.
3) centimeters
centimeters.
1/ + 3
A =—bh. 2
B
37.
36.
h = 4n
b = 2n -^ h=n + 3 (2n2
b 150
6n2
- 2n
+ 5n -
3)
1
2
Find the area of the side of the house
38.
shown
at the right.
1(2x2 - 3x -
9)
+
2x2
+ X -
3
x-1 2x
A
39.
shown at the found by using
trapezoid
may
be
A =
]-h{a
+ h
a b
Find the
^
.
right.
the
Its
3
area
formula
b).
= = =
2x 3x 3x
—
\
-\-
\
area. Sx^
The box shown
40.
^
is
+
hold a cup
at the right
and
saucer.
is
designed to
The
top,
the
bottom, and two sides are rectangles. Two sides are trapezoids. Find the area of the
whole surface of the box. 318x2 - iox
lOx
There are 3 red hats and 2 black hats in a drawer. Melba, Fran, and Brent line up in a single file, and a hat is placed on each one's head. They are asked to figure out what color hat they are wearing.
who could look ahead and see Fran and Melba, says, "I don't know." Fran, who could see only Melba, says, "I don't know." But Melba, who could see nobody, says, "I know." Brent,
What color is Melba's Melba's hat
is
hat and
how does
she
know?
red.
151
X
8 • Square of a Binomial You know
The same two
that
is
factors,
when you square
a
number you multiply
6^
=
6 '6
x"^
=
by
it
itself.
X'
when you square a binomial. All you do is to write the then multiply as you did in the last section.
true
{a
+ bf = =
{a
a^
+
b)(a
-\-
lab
+
The square of b)
-\-
b^
-
4)2
binomial
1970 1975
1980 (estimated)
It is
graph above
AUTOMOBILE REGISTRATION
is
The bar graph below
is
in the advertisement
misleading.
Do you
see
why?
At is
first
glance
it
looks as
if
Tru-Vision
twice as popular as Brand A.
This
TV
is
because the scale on the side of the graph does not start with 0. Instead, it starts with 18.
Perhaps the graph
at the right is
more
honest.
^/A
Written Exercises The graph
at the right
W/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/A
shows the
increase in sales for a newspaper in 1.
one year.
NEWSPAPER SALES FOR ONE YEAR
During which months did the sales decrease?
Aug
July. 2.
The bar
for
December
is
twice as long as the bar for
January.
Does
this
mean
doubled during the year? No that sales
3.
What was sales
the increase in
from
January
s^^ x
3jc
-9 +
8.
^'-Bx^B
"° 1
;;
-
2a;
=
9.
°"^
4
- 2x = -3 + j = y
2,
2x
^"
W/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M
Find the slope of the lines whose equations are given.
A
\.
y = Ax ^
5.
y -
9.
2x
Tell
2.
j =
y + 2x =
=
6 2
6.
-y =
3 2
10.
2>x
y
-j-
4jc
5x
+ 34
=
4 -2
-5
2
3.
/ = jc+1i
1.
4x
U. y
whether the equations share one solution
-y =
=
-
S
4
7x -7
no solution
pair,
1
1
4.
y = ^x +
S.
4x
12.
2
-^
+y =
y = x -
5
-4
^
pair, or all
solution pairs. '3
13.
y = 2x
14.
y =
15.
)-x
fio
y = 2x
11.
21.
X X
-y= +y =
y = -X
one
^
18. 2jc
y
6
=
1
+y = 2x = + y
9
j^
25. 2jc
3x
-^
+
y 4x A
-\-
2jc
26. no-
7
=
8
16.
-h
= =
y
2
9
19.
^ one -^ 6i
-\-
2x
=
+^^y = — 3x = y 3x
S
4 ^
20.
one
2
2x = 4)^^'-'^^'-4x = 9^„ y - 9 = 4a: / +7 = 8
= -3x + y — X = 5 y
-^
3
one
x
-/ =
x
—y
4
one
3x+y = 7^„22.^ + y
242
S
jc
rit)
3
-\-
7 +
27.
24.
-y = / — 4x = y 4x
_ no
J-
x+y=-3
-^
,
-\-
6
y - 2x = -6
-y = 4 28. y + one . / = 4 —3x y — y 3x
=.6
2x 2x
= =
4
-
^^^
B
29. a. Will the
graphs of these equations intersect? no
y = 2x y = 2x -\-
b.
Will the graph of /
c.
Can
30. a.
= —x +
1
3 \
intersect either graph? yes
the three equations share a solution pair? no
Are the
lines in color at the right parallel?
(Check
their slopes.) no b.
Will they intersect? yes
c.
Do
the equations of the lines share a solution
pair? yes
3
•
The Substitution Metiiod
There are more exact ways of solving a pair of equations than by graphing. Here is one method.
EXAMPLE
Solve
1
Step
1
Step
2:
y +
\
=
+y =
3x and 2x
Solve one equation for y.
:
9.
->
;;
+
1
=
3x
+y =
2x
Substitute the value of ^
2x
in the other equation.
+
-
(3Ar
Solve for x.
5jc
-
9
=
9
1
5x= Step
y = 3x y = 3-2y = 5
Substitute your x value in
3:
the equation in Step
The
Check:
Is (2,5)
y
-\-
\
5
+
1
6
solution pair
a solution of both equations? Yes!
2x
+y =
2-2 +
3-2
4
/
+
The
Solve x
2
first
=
2y and x
says that x
=
y = +7 = 3y = y =
X
equation
^
2y.
+y =
-\-
2>'
Substitute 2y for x.
9
5 5
9
EXAMPLE
1
is (2,5).
=3x 6
10
\
1.
Find the value of y.
Answer:
9
1) z=
/
6.
6 6
6 2
Substitute.
Substitute 2 for in
X X X
j'
one of your
original equations.
Answer: 244
The
= 27 — 2y = 2-2 =4
—J x
15)
(5,
+y =
3.
7 X
20
- y = 2 (5, 3y + X = 14
X
3)
7.
=X +y =
X +7 = 5x = 3y
2
4.
(7, 5)
\2 (1,2)
3
-
^.
1
= 2>y 5y - x =
X
x - 2y 3x + y
(12, 4)
%
= 5 = -6
(-1, -3)
Ua Written Exercises
W/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M
Solve by the substitution method.
A
1.
y = 2x
(3, 6)
x+7 = 5.
7 = X + 4
x+7 9.
=
2.
9 (9,
13)
6.
22
+7 = 2 3x + 7 = 8
X
10.
y = 5x 7 — x =
X = 57 3x = 7y
10)
= 2 - 7 (-5, 27 + x = 9
2x 4x
3.
+7 = -7 =
x
=7+
x+7 =
X
(3,-1) 13.
(2,
8
7) 7.
4x+7 =
5
11.
1
3x 2x
14.
16 (10,2)
4.
(5, 2)
7
7 = 3x +
1
(I- 4)
8.
8
- 7 = 13 + 37 = 16
(1,3)
+
3
12.
7 = x +
1
x+7 =
5
7 = 2x -
3
4x+7 =
9
2x 5x
15.
(5,1)
^ + 2Z> = 7 2a = 3b
(3,-1)
(2, I)
-7 = 9 + 27 = 27
(5,2)
X - 57 = 8 4x + 2y = 10
(2, 3)
16.
p - 5q = 3p ^
-
2q
6
=
5
(1,-1)
(3,2)
Solve.
Sample
D= r
=
Find
B
17.
D
bh 2h Find A in terms of h. A = 2/»2
20.
/.
18.
=
Solve for x, y, and
C
in terms of
^ = b
D= D= D=
Answer:
rt
5t
V=
Bh
B =
2h^
Find
V in
\9.
rt
(50/
^
Substitute.
5/^
V=
Iwh
= 3/2 w = 2h
/
terms of
V =
h.
2/)3
Find
V
in terms of h.
V =
6/|3
z.
+ 7 + z = 180 = 3xx = 20 7 z = 5xK = 60 X
z=100
21.
X x
+ 7 + z = 62 = 2z — 5x=i9
^^3^_5y, =
3i
z=12
22.
x + 27 2x + 7 3^ +
+ =
3z
=
6 x = 3 = o
^^8y
Z5=-1 245
4 The Add -or- Subtract Method •
When
add or subtract the one variable.
solving a pair of equations, you can often
equations to get a
EXAMPLE
new equation with
Solve:
1
just
+y = -y =
(2x
[2x
l S
5jc
+7 = -7 = + =
5x
=
2jc
Step
1
Add. This makes the y
:
term drop
^
3jc
out.
new
equation.
Step
2:
Solve the
Step
3:
Substitute 3 for x in one
X
7 8 15
15
=3
+y = 2-3+7 = 6+7 = 2x
equation. Find the value
of J.
Answer:
EXAMPLE
2
Solve:
1:
The
solution pair
+ +
[5^
67 27
= =
make
Subtract to
is (3,1).
You may check
the
5>:
x
+
new
equation.
*
4y
47
3:
Substitute
—2
for
7
3
5x
in
value of X.
+
67
5;c
246
solution pair
is
(3,-2).
=
+ 6(-2)= 5x - 12 = X
The
j
= -8
y=-2
one equation. Find the
Answer:
11
= -8
5x Step
=
67
^. 5x + 2y =
+ Solve the
it.
1
term drop out.
2:
7
3
-^
Step
7
7=1
l5x
Step
7^
= =
3 3 3 15 2>
You may check
it.
f/A
f/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/A
Classroom Practice
Would you add or would you does the y term drop out? 1.
5.
+ 7 = 6 add; jc— 7 = 2/ X
= + 7 =
2x 2x
6 sub; 2 Jf,
3y
-\-
2x Ix
2.
Does the x term
subtract the two equations?
+ -
X
6.
+y = —y =
3x
2y 2^
5 add;
3.
2>y
= =
7 add;
7.
5/'
5x 3x
+ +
2j 2y
= =
X 2x
+ -
3^
= =
3/
9 sub;
13.
f/i
X =
0;
K = 2
14.
X =
3;
/ = 2
15.
9K
3;
-\-
2y = - 7 = -\-
2.
x =
1
;
/ =
1
16.
x =
1;
/ =
1
1
/ = -1
= 5 sub;y = 2x
Ay
-\-
4x 4x
S.
add;
;
x =
X x
4.
1y
9-16. Solve the pairs of equations in Exercises 1-8. 1 1 x = 1 / = 2 1 0. X = 2; / = 1 9. X = 4; K = 2 .
or
6 sub; 3
W/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M
Written Exercises
Solve by the addition method.
A
I.
5.
X X
+y = —y =
2a4a +
b b
2
2.
10(6, -4)
= =
3
+ —
5;c
3x
+7 = —7 =
6. 2;c
9
{2, A)
= =
47 47
3x
4x -1) 2x 3.
1
7(1,
10 5
7.
(3, 4)
-7 = +7 =
3;c
-
x
-\-
8
—2(1, -4) 2p
—
3q 3q
4r
-
7^
4r
-\-
= =
4.
27
2y
8
8.
'
8(4, 2)
5p
-\-
= =
\0
4
(2, 0)
= 13 = -29
Is (-2, -3)
Solve by the subtraction method. 9.
X X
+ +
67 27
= =
10
(-2, 13.
3x
X
+ +
27 27
10.
2
X X
- 7= - 37 =
3x
11.
+7 =
18
14.
14
-
2x 2x
= =
57 37
(2,
(2, 6)
7
12.
-2x + 7 = -8
10
(25, 5)
2)
= =
20
-2)
(3,
14
15.
2a
2a
10
+ -
=7 = -5
3b b
-2)
-
3x 2x
(-1,
47 47
= =
(3,
-
16. 2s
6s
18
-3)
= =
5r
5r
21
17 \
(-4, -5)
3)
Solve by either the addition or subtraction method. 17.
+7 = —y =
X X
9
18.
+ —
X 3x
5
27 27
3x 4x
+
27 27
= =
(3,
25.
-X 4x
27 27
13
22.
8
4x 2x
-
37
37
-2)
= =
3a
-
\2b
19.
2x
+
2x—
5
37
^9
= =
9
23.
3
-4x + 7 = 4x
+
30.
2x
-
1
=
=
8
7=—
20.
8
a
-{-
2a
+
(-2,4)
3x 6x
+7= -7 = (2,
(3, 1)
10 26.
(2,-1) 29.
7
(3,2)
(7,2)
21.
= =
7 27. 8x
37 (-1-3)
5
X
6x-ll7=10
(-1, -1)
(-2, -2)
37
24.
18 5
5
(3,5)
6 9
4a 2a
- 7b = -lb =
5x 5x
+
13 3
(5, 1)
28.
4 37 (1,-1) 8
= =
(3,1)
-6)
= =
- 27 = 5x-27 =
17 '= 18 31. 6x
a-\2b=\\
+ -
3b 3b
32.
4x
^-4x
27
= =
7
2 37 (1,-1)
- 7 = 15 + 37=-5 (5,5)
247
Jf
Can you work
with fractions?
The
following exercises have fractions in the
solutions.
2x Sx
Sample
-\-
—
4y 4y
Sx
X
= = =
=
2
^2-i +4^ =
2
2x
2 2
Substitute.
4
+
4^
2
=
1
+
4;;
4y
= =
2 1
1
/ = 7 Answer:
B
33. 6jc 6jc
+ +
9^ 3y
= =
(4.
(hi)
4
5«
34.
10a
+
^ Z)
= =
3
35. Sa
3
8a
+ 6b= - 4Z> = 4
(I--)
1)
Pass the center of one string under the string circling the other person's wrist, over that person's
A
String
Duet
hand, then back under the string again.
Here is a stunt you might of your friends.
like to try
with one
Tie a piece of string to your wrists.
Tie
another piece of string to your friend's wrists so that the two pieces of string interlock as
shown. Now try to separate yourself from your friend without cutting the string, untying the knots, or taking the string off your wrists. It can be done! is the branch of mathematics that can explain why this stunt can be done. Topology deals with figures and how they can be bent and
Topology
stretched.
248
a:4)
r/i
Mixed
f/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M
Practice
Solve by the graphing method. 1.
j=-jc +
3
2.
= 2x-A^
y
_.
;c+7 = 3x + y = -S
7 + 3a; =-7 y - 4x = 1
3.
x +y = A 2x -\- y = 4
(-2,-1)
(-4,4)
(1^, if)
4.
(0,4)
Solve by the substitution method. 5.
= 3y + y=
X x
6.
y = X + X
12(9,3)
+7 =
x X
1.
I
5(2, 3)
2y 3/
-{-
+
= =
2
H.
13(-20, 11)
y - 2x = 4 / + 4x = 16 3,8)
Solve by either the addition or subtraction method. 9.
X 2x
-\-y
-
y
= =
7 5
10. (4. 3)
= =
2a-\-3b 2a + b
3(i,
-
3x
11.
5
2^ 4y
-3x +
I)
= =
12.
1
7
(3, 4)
-
4x 6^
= =
ly 7y
9
3 (-3, -3)
Solve by the method which seems easiest.
=X +2 + y = 11
n. y
2jc
-^
14. (3, 5)
X
+/ =
4(7,
+7= - yy =
2x 3x
15.
8
x-3y^ =
1)
9
6
16. (3, 3)
= =
x + 2^ 3x + y
5
10
(3, 1)
17.
-
2x
X
= 8 18. = lib.-\
3/
-\-3y
^
21.
jc
x
= +
2/ 3y
19.
=
5(2,1)
3/
3a; - >' = 8 X + 2>^ = -2
22.
/ - 2x = 7 7 + 3x=2(-i,5)
2a 4b = 6 -a-3b = l
23.
x + / = -2 2x + y = 4
(-1,-2)
practice, sfee
page 424.
24.
x
-
2x
= y =
5/
-\-
' '
'
2
4
(2,0)
((6.-8)
^ more
y - x = 3 7 = 2^ + 4 (-1,2)
-
(2,-2) For
20.
V
c
\J
y?
iSELF-TESTi Solve by the substitution method. 1.
y = 2x X
-{-
3y
=
2.
\
U^^'
X
=
3.
3y
2x=y +
^^
5^^''^^
4p
=
3q
p-q = 2 (-6, -8)
Solve by the addition method. 4.
3x
X
-/ = -\-y =
2 6^^''^^
5.
2a
+
3b
= -I
a-3b =
_
6.
^'^'
^^
4
3r
+
2^
=
4
4^-2^=10 (2,
-1)
Solve by the subtraction method.
'7'^^y = ;c-37=
^
,0 2) -6^°'^^
^-^ +
2^
+
2/
35
= ^-2 = 4^
5) '
*
9.3x-y=\0
x-y =
4
(3,-1)
249
5
•
Word Problems, Two Variables
You know how
to solve
Now you
problems by using one variable.
can
use two variables in problem solving.
EXAMPLE
Art and Lynn were partners in a bowling tournament. In the first game, Lynn's score was 10 more than Art's.
1
Their combined score was 330. Find each person's score.
= =
Let a Let b
Art's score
Lynn's score
Lynn's score was 10 more than 4^
4'
b
=
4^
sb
10
+
Art's. •i'
a
Their combined score was 330. a
Now
solve the two equations.
Take your second equation.
=330
b
-\-
We'll use the substitution method.
'b
=\0
^a
-\-
-^
a
b'= 330 "x Then
a
+(10 + 2a
+
fl)
10
2a
Now
a
use one of your
original equations.
10 10
substitute.
= 330^ = 330 = 320 = 160 + +
Then 160
substitute.
x
+7 = 8 -2y = 5(3,
+ 5/ = 3 + 2y = 13(-59,
3x
2.
X
2)
2x 3x
3.
36)
+7 = 5 - 2y = 4 (2,
4. l)
- ^b = 5 3fl + 26 = 4
a
(2,-1) 5.
+ = - 2/? =
3« fl
/>
4
+/ = 7 3x - 2j =
6.
6(2,
-2)
_c +
3c
7.
jc
11(5, 2)
= -7 5^ = 6(-i,
g.
4^-
2a
2
•
—
f
a^
1
4
«
,0/, s
-^-^ - 2a^ 4
x2
24.
8m 2^m
2
-5
*
5
6
5
(ab)^
/^
—-
2
^
-^i^y
9„ 9x - 9
^ -2a 8^2 b
7^-77+12 •
d
/ — 3
^"^
29
a
——~ ^
4ab
C
2
y + 3
^: Ix
2
^ ^-^— — 9
——
-^"^ X
2
a
3/2
15r^
(-^y)^
^^
''^-^
fif2
£
d^
^
^~
7^
2
^
_
fl^
1
-y
,
I
^^^ +
1
^^±J-
14.
3ib
16.
47 "
+
4
_ -
'
;
7- ^ ^— L^^ 1
4
19
'4 + _?:
^
-
2
_
-4
r
16
8
4
6x
17.
1
4x •"
+
2
3
^^
8
8
~
'
" 2
-
_
3
^
4
-
^
12.
7x + 9 12
6 ^
-
^
+
4
x-2-
::
1
3a - 11
15.
^^^^ +
5
+
12
4
*
3x +
1
8
^L±^ 3
b
-4
3
-2
2b
^^-^5 s^^
18^
~"4 3~A +
^
20
4
16
+
3fl
6
2
21. 5
,^
.
3y
-^
9
4
2
+
lOx
3
- 3a
3
-
2/
+
7^ '
^^±1 - 2 ^1^ 5
2
^ 12f- 7
8
+
-
3r
5 1.
12
4
2
-
_ 15
iijf
2
-
4/c
18
3
8
5x + 7/ ox
1 ^
~r
2x
^
4
8.
1
9
5.
3
13.
~
^
6
12
10
+
^
1
6
_ 3
a
-
2A:
2.
T5~~
5
3
7.
x +
4
X 8x + 10
2
-\-
1
15
22 "
^
"^
8x
_ 1^
3
,
18
^
-
^
3x
—
1
2x
1
5a + 33
36
12
1
6x
3x 8x
1
295
5
14
•
Equations with Fractions
Sometimes
how
see
want
you'll
these equations can be solved.
^+^= 2
Think about the first.
them. Let's
to solve equations with fractions in
Rename
left
10
3
side
3x
the fractions
,
2x
10
same
so that they have the
denominator.
3x
+
2x 10
Now
you have a proportion. All you have to do is to
6^^^^1 5x
X
il + iA
Check:
2
60 \2
10
3
6
+
4
10
/
10
It is
J:
= =
cross-multiply.
easy to do just one step at a time, and simplify one side of the
equation
That way you can get a proportion which
at a time.
not
is
difficult to solve.
10
5
10
+
3
4
]_
^+ 4
20
3a_
4
2
+
20
1_
a
«
296
= =
70 10
always wise to check your answers.
,^a
^^^^2
+ 20) = 2a + 40 = 40 = 4 =
2ia
In
2
4
n
10
3a
n
n
10
4
It is
3^
a
EXAMPLES
4(3«)
\2a \0a
a
f/A
Written Exercises
W/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/A
Solve.
1.
4.
10.
4-4=
1+ 4
4 30
^=^ 12
3
2
3
7
5
10
«
+
i?-
^=
2.
:^
+^=
6
7
4
6
8.
4_A=6^ 5
11.
7 6
3.
£+^= 6
3
5
3
«
_
15
X
2«
^1
14.
i« -
=
«
467 1
_9
12. -^
15.
4 42
+
9 16.
^^Ltl
+ ^Lizl =
5
17.
^
~
^
-
^^
4
X 19.
+
X
3
—
=
1
22.
6x
-
4
^
=5
18.
3
1
2c
6
15
2_A^23 3x
--
-^
4
9
+
^
=
6
-92
3
-2 J_
20.
2c
C
~ 4
-7 2
^=1
X
%
B
15
9.1_A = l-60
c
- = l^ «-f An + n
612
1 = ^2
5.
1
13 42
2=ii^^ + ;c4
c
+
3
_
5
X
4c
23. 2
7x-l ^
3^
6
Diophantus was an ancient Greek mathematician. According problem was written on his tombstone:
3x
_
19x
+
3
3
_3
4
to legend, this
HERE LIE THE REMAINS OF DIOPHANTUS. HE WAS A CHILD FOR ONE SIXTH OF HIS LIFE. AFTER ONE TWELFTH MORE, HE BECAME A MAN. AFTER ONE SEVENTH MORE, HE MARRIED. FIVE YEARS LATER HIS SON WAS BORN. THE SON LIVED HALF AS LONG AS HIS FATHER AND DIED FOUR YEARS BEFORE HIS FATHER.
How
old did Diophantus live to be? 84 years
297
15
•
Work Problems
Suppose an long
it
office
manager wants
will take to get a report
who work
to figure
how
typed by two typists
at different speeds.
Suppose your mother helps you to paint a room and you wonder how long the job will take.
Algebra can help solve both problems. Let's consider the painting
EXAMPLE
problem.
and her mother plan to paint the living room. it would take her 10 hours alone. Her mother says she could do it herself in 5 hours. How long would it take them to do the job together? Ella
Ella thinks
r/i
A
Written Exercises 1.
W/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/A
Mr. Macy's two grandchildren offer
to paint his kitchen.
Sandy thinks it would take her 10 hours alone. Meg says it would take her 15 hours alone. How long would the job take if they worked together?
6 hours
5.
Taddy and Rose deliver groceries for a neighborhood store. Today Taddy needs 4 hours to deliver the groceries alone. 2— Rose says it would take her 5 hours. ob together?
hours
B
9.
Judy Rogers needs 6 hours correct
some
papers.
to
When
her assistant helps, the job is How finished in 4 hours.
long would
take the assis12 hours alone? tant to work it
B 16
•
Binomial Denominators (Optional)
When you want to add
or subtract fractions with
more than one term
in
the denominator, there will probably be quite a few steps.
EXAMPLE
Think:
+
1
which x(x +
x{x 4{x
+ +
+ 2) 2)
+
x{x
+
I
is
must
new denominator by both x and x + 2.
find a
divisible
2) will work.
2)
3jc
+ 2) 4jc + 8 + 3.x x(x + 2) 7jc + 8 x(x + 2) xix
Think:
which
EXAMPLE
2
4 2(«
+
+
a
^
+
'
C2>
-3
X
+
Ci>
4){a
+
jc
302
-
4){a
+
2) will work.
2)
Think: I must find a new denominator"^ which is divisible by both x — 3 and X + \. {x — 3)(j>c + 1) will work.
1
5(.x + 1) 4(x - 3) {X 3)(x + {x 3)(x + 1) 5(jc + 1) - 4{x 3) U - 3)(x + 1) 5 — 4x + 12 5x {X - 3)(x + 1) + 17 {X 3)U + 1) -I-
(a
new denominator by both a — 4 and
find a
2)
- 4){a + 2) + {a2(a + 2) + ?>{a - 4) (« - 4)(a + 2) 2fl + 4 + 3(3 - 12 (^ _ 4)(« + 2) 5fl - 8 {a - 4){a + 2) 3
2.
must
divisible
2
{a
EXAMPLE
+
I
is
1)
Exercises W/M/M/M/M/M/M/M/M/M/M/M/M/M/M/M/A -54 x2 + lOx - 21 4(a - 6) + 2a(a + 6) — :^79- -: 15. (b + 2)(b (x + 3)(x (a + /))(a - b) 3) 3)
r/i \Nritten
-zrT-.
Simplify.
A
1 '
3
_)_
k
k
+ +
8/c
^\
/c(/f
5
A
2 •
1)
/
/
+
+ +
5(f
3 "^
'(f
5
2)
3.
4.
r
+
5
r
(r
+ 7
(y
7.
3
5){,r
(3
ja +
5.
"
a
4
c
—
a
A
-\-
—
a
C
5
—
4
X
y
X
3
+
X
6
;c2
—
+
6
-y
r
1
19.
^2
_^2
3(x
^ X
6^^
-
/7
r
+
4- 3
-
+
1
a
+ 3x + 3/ + y)(x -
^^
3
a
—
3
r 2a
1)^^
-
a
1
+ 2s - s)(r +
4 - a
a
r
_|_
—
s
'
r
(r
2)
4
j^
4
a
2
-\-
8
(c
5
+
+
+
la
_^
—
a
b
c^)2
b
4_^
_12
jg '
+
+^
c
6^)2
/
2c - 3
l(c
c
c—
c
+
+ 3)(c-
3)^^
21.
3
3
23. JC
+
1
.X
+
^
+ 5 4-^2 +
j^
X
5
1
5
,
+
ts
-\-
t"^
2«
2_^ J_
—
- 1) + 5)
4(3(
5f/
(a
^ = l3 +
X
a
Solve for the variable.
a
3
11)
^
_^
-
5)
2)(a
-
3(a
22.^
—
7
-
*
(c
—y
+
6)(£)
8 + 5c
36 1
+
2Z?(ib
3x^
16.
b
3
r2
14
11(^
+
X
^
- X X + / - 29 + 6)(x -
5)
5 6(3/)
Z)
—y
+ 20 - 4)(c +
b
9.
+
c
Z?
\
X -\-
2)
5)
_
9
4)^^ '
(c
7
13.
-
4)(a
4
2)
3(a + 4)(a- 1)^^
+
+
^(>f
3
c2
7
6
jQ
2)
+ 3/ - 1 + 3){y - 2) g
3(3a +
_
8a
+
2 /^
7 +
+
-3(x +
5 ;C
-3r r
2.
5)
2 2)
303
'/a
Reviewing Antiemetic
Recall that decimals are another
tens
way of
SIciiis
W/M/M/M/M/M/i
writing fractions.
CAREER IMGTEBC9K Environmental Sciences
O
Society is becoming more and more aware of the need to protect our environment against pollution, land abuse, and destruction of wildlife. Here are some careers in the field of pollution control.
o Air pollution engineers study the effects
of
air pollution,
controlling
and develop plans
for
it.
Chemical engineers can recycle waste materials, converting old newspapers
o
into sugar, for example.
Water pollution technicians anaand bacterial
lyze the chemical
content of water samples.
305
g ER CCRIVER f Tipping
V)
People in a restaurant
2
who
wait on tables
usually get tips. When you are the customer, do you know how to figure the tip? The amount you leave depends on you, but people usually leave about a 10%, 15%, or the figuring
First think
is
done
in
20% tip.
All
your head.
about percents and what they mean.
100%
50%
25%
10%
the whole
half
one fourth
one tenth
Suppose you want to give a 10% tip. All you have to do is round the amount on your restaurant check, then move the decimal point one place to the
left.
Your check
Once you know how your
to figure 10%,
result.
Your check
then you can find
20% by
doubling
7i
Reviewing
{See pp. 268-273.)
Simplify.
lAl
2^-J5x •
3
24
A 3k
187 ^'
VM/M/M/M/M/M/M/A
C/iapter
t/ie
_27>'2
3 •
I5x
^
7'•
18a:/
1
8
6
•
ll£
"L
22
2
4
-"^^
^1
12«
3
'
^^ ^
•
2.^
10
Sx^
'
2/7
^^
^^^'^'
9
b
-ab^
iM 5„
5
'
28«^2j_ 46
3^-15 a
11.
.
—
— y ^ y - X X
a 20.
1
a-3
1—
21.-
-1
2a
2 jc^
3m/7 - 2/nn
^m^n^
1
19.-
1^-
r>
Ai -1
^2 a - b /?
- 3/2 — y^2~
3^2
12m2^2 17.
-^
_ +
^
—a ^2-49 1
-22
22.
1
Solve.
23.
^
{See pp. 274-277.)
The
ratio of blue to
How much
yellow in a paint mixture
blue paint
ratio of
How many Solve.
coaches to players
—X = «.
'3
33.
—=— 2n
4
26.
7
_
n
+
1
^
2Q
6
*4+ 3a
18 oranges cost $2.70.
Simplify.
^ 21
is
3 to 2.
10 cans of the mixture? 6 cans
is
2 to
9.
•
\2
6
How much
35.
make
{See pp. 278-283.)
14
2o
to
players are there? 18 players
2
25.
needed
team has 22 members, including coaches.
24. Central High's soccer
The
is
^
"^
will 36
^
'
21
27.
1
_
-5 = —10A
9
a
—
2
_
^
^.
6 34. 3
oranges cost? $5.40
{See pp. 284-285.)
63
4
x
=—^ 7+11 3y
10
28.
_L 4
32
20
—
2
_
*x+23
dozen apples
How much
2/?
b
'
2
_
J_
-1
cost $2.25.
will 2
dozen
cost? $1.50
4
^
2
_
1
Simplify.
43
47
(See pp. 286-287.)
1 ^ _L x^
—
y^
44
9
L- ^
X
+
y
-^
^-A
X - y 1-
45
3
b a __I^
5
a
-\-
4«
^- - ^
^
—
b
a
^!-
+ b
46
3a
49
-^ - ^- — —
a
3 :
—
\
3 —
Here's what you II learn in this chapter: 1.
2.
To solve equations with decimals. To use equations with decimals to
solve
word
problems. 3. 4. 5.
To To To
solve equations with percents.
solve
word problems involving percents. and mixture
solve interest, investment,
problems.
Chapter 10 Decimals
and Percents
1 •
Many
Decimals
equations contain decimals.
and how
to
work with them.
Fraction
First, take a
new look
at
decimals
EXAMPLE
4
Multiply: 0.42
x
6.65
^ 2.7930
2 • Division Witii Decimals In dividing by a decimal, the
first
step
is
make
to
the divisor an integer,
or a whole number.
EXAMPLE
1
The Problem
First Step
Then Divide
1.6)1.76
1.6)1.7 6
16)17.6
1.1
16
The
16 16
division could be written in
Moving
fractional form.
the dec-
1.76
imal points one place to the right in the division
like multiplying the
is
numerator and denominator by
EXAMPLE
2
^
17.6
16
1.6
10.
Then Divide
First Step
The Problem
170 0.02 JXia
0.02 Jl4
The
2J34O
division could be written in
Moving the mal points two places to the fractional form.
in the division
is
^xioo-^
deciright
3.4
like multiplying the
numerator and denominator by
^
340
0.02
2
100.
Often divisions do not come out "even." In those cases, you will have to
round the answer.
EXAMPLE
3
Find 3.75
^
Round
0.07.
to
one decimal place.
Keep 53.57
=
5.4
5.4 5.4
last year.
Let X
=
Then
1.2jc
5.4
28 1.4
4
mg mg
iron in the chicken
iron in the spinach
/
5.4
ball player's batting average
average
X
10
54
What was
is
0.330 this year. This
is
his average last year?
batting average last year.
=
batting average this year.
1.2jc
1000
X
\.2x
1200JC JC
= = = =
0.330
1000
X
0.330
330 0.275
\fi
X X
EXAMPLE
^
1!
2>
root.
Some
quadratic equations have no real-number solutions.
EXAMPLE
3
+
;c2
= = -9f
2.
(IOjc
-1 + 5
5.
-
+
8.
+
6x + 10 (4x - 7)
-
3)
/— -^4
0.9m
6.
5 \/x
9.
3m2
—
3.
0.03m o.87m
4 V^ 20x
•
16
Factor. 7.
10.
a^ H- la a{a
+
x2
13.
/
+
7x
+
(X
+
7)
12
3)(x
+
127
+
11.
4)
/
-
(y
_
6)2 14. ^2
36{K +
+
8^3
X^ x^(8x +
-
3/
Z)2 (a
6)(a
+
15. ;c2
6)
16. (4x
-
16jc2
-
25
+
5)(4x
17.
+ 6m —
m^
40
(m + 10)(/n -
5)
18. x^
+
ii„
-
-
{x
3m(m +
1
(n
3)
-
+ 2m
_
12. n2
18
+
6)(K
1)
-
5x
8)(x
-6x
4)
18
-
9){n
4)
2)
24
+
3)
-\-9
{X
-
3
=4a - 9-4
3)2
Solve.
+
4
=
22.
5(2m
+
3)
25.
-(x -
19. 6;c
28 4
=
-7 = -28-7
20. 3y
35 2
23.
+
-{a
=
3)
3fl
+
1-1
+
21.
7a
24.
-4{y- \)=y-6 2
10)
=
4
+
X
26.
-3x +
12
=
-jc
-
3 28.
—=15
^^ ^^'
34.
— y = \0 X = +y = \\y=
is
=
O.Oly
2.4
^+^21
30 30.
5
(3x \4x 12
27.
8
29.^ =
3
4
OAy -
3:
^-
-^
'^^'
[2^ [ a
— +
b
3b
= \0 a = -2b
30% of what number? 40
= 4; = -2 35.
f + 1 = 910 4x
33. 3jc
What
+
3y 27
= =
\5x =
percent of 18
3;
= -
7 /
12*^
is
661^% 36.
-
x(x
4)
=
00, 4
37.
6m(m +
3)
=
Oo,
-3
38.
(y
-
4)(y
+
3)
=1 -3
4,
39.
m^
- 5m =
=
42. «2 _^ « 45. 2x2
+
3jc
Oo, 5
40. x^
-\-
6-3, 2
43. 2jc2
-
46. ^2
2
=
4x
4S.
X
2y
=
=0-^. -3 41.
+
5;c
+
5
=
y'-y -6 = 3,
-2
44. 4(x
+
1)2
=
362,
47. 4x2
-
6x
+
2
-5 ± VB
49.
-4
= 1
''2
2
the graph of the equation.
+
3
= 50±5
f-^ ^ Draw
-\-
Check student's graphs.
y =
x^
404 A more extensive cumulative review
-2 for
50.
7 =
x2
-
8x
+
16
Chapters 1-12 can be found on page 444.
Extra Practice Exercises CHAPTER
1
For use with pages 2- 7 Find the value of the expression.
A
1.
7
6 42
•
2.
If
/I
+
2(3
=
13.
9n 27
If
=
jc
2) 10
3, find
11.
X
-\-
If
=
2, find
j;
21.
+
1)
24
4 24
7 63
4.
6
7.(6-l)-3i5
8.
(4
+
2)
12.
10
+
(6
9
3.
•
8
-
(2
15. 9
-
« 6
11.
3) 2
•
•
-
32
1)16
•
the value of the expression.
y =
5, find
y^5
+
6(y
40
8
10. 4(5
14.
10 and
•
6.5-2-64
5.2-3+410 9.
5
+
9
« 12
16. 9
^
« 3
x
+
5
the value of the expression.
18.
-7
^
5
ig^
x
23.
(j
3.
2n
7m + 3m lOm
x
-\-
-\-
25
y
20.
-7
i
the value of the expression.
2)
24
22.
+
6y
2 14
-
1)
-\-
n3n
-
4 4
-
24. 2/
3
1
For use with pages 8-11 Simplify.
A
n2n
\.
n
5.
3x
-
X 2x
9.
4a
+
5a 9a
13.
3x
-\-
Sx
2x
-
17.
-\-
-
+
5
X ^0x
3a:
+
+
«
«
3/7
2.
/?
6.
5/
-
2y 3y
7.
10.
6x
+
3x 9x
11.
4a
14.
n
-
15.
a
-\-
5x - 5
6n
"7^-71
18.
6«
-
6
+
+
-\-
2a
2a
-
-
+
4.
«
8.
lOx
2«
-
3/?
x 9x
+
5x
b^a - 616. 3n
-
n + 2n +
2« 8n - 6
19.
12.
5m +
2
+
7«
-
6a?
+ 7n
+ 13
21. 8
+
6y
+
5y
-
7i1k +
1
22. 7
+
3;c
+ 1
23.
406
5x
+
3/
-
y + 1 Ix ^6x + 2/
24.
2x
+
5y
-
y + 9a: 11x + 4/
25.
4«
+
r
3x 7x
m
m + 6m 1
20. 6
-
5x
3a 3a
+
Im +
+
9x
2x + r
n
+
9 1
6
3n 2r
1 1
For use with pages 12-15 Find the value of the expression.
A
1.
329
2.
42 16
3.
224
4.
62 36
5.
41 4
6.
2^8
7.
33 27
8.
5225
Simplify. 9.
yyyy^
10.
X'X
S' S
14.
a
'
a' b b
18. 2
•
c
22. 6
'SwiSn
13. r
17. 3
21.
'
'
rs^
a' a 3a^
2'{4r)8r
x2
'
a-b-
c 2c2
•
11.
a' a' a
\5.
X
'
y y y xy^
19. 5
•
2
•
23. (4«)
•
ly
25. (2r)(4r)8r2
26. (10;c)(5;c)50x2
27.
29. (2a){4b)8ab
30. (3jc)(47)i2x/
31. 2
•
12. c
a^
«
•
n
1
0n2
4 1 6n
Sy 56/2
'
«
5
•
•
a 1 0a2
c
'
c
a
'
b
20. 6
'
a^a'
\6.
6«
24.
28. 65
•
•
'
b
-
c c*
'
c • c ab^c^
'
b sa^b
30n
5
5^ 30s2
x-x-A-xax^
32.
For use with pages 16-17 State the expression without parentheses.
A
1.
3(;c
+
3)3x + 9
2.
l(y
+
1)7/ +.7
3.
5{a
-
l)5a - 5
4.
3{n
+
2)3/7
5.
8(«
+
4)8n + 32
6.
6(c
-
2)6c - 12
7.
4{x
+
3)4x + 12
8.
aia
+
l)a2 + a
9.
4x -
2)x2 - 2x
y(y
+
7)/^ +
n.
«(«
+
2)n^ + in
12. c{c
+
73,
6{2n
-
8)12/7
4(5;c
-
4)20x - I6l5. 3(2a
-
17. 3(2«
-
3b)6a - 9618. 2i4x
-
2y)8x - 4/19. ai6a
+
13.
Use
B
10.
- 4814.
the distributive property.
21. 2(a7
+
3)
+
1
24. 4{x
+
3)
+
2x6x + 12
27. 4{2a
30. 5(2
+
+
3)
3x)
-
2/7
+
7
7« + 1 a + 13
-4x -1 1
1x
+
3
Then combine
+
1)
+
25. 2{x
+
4)
- 2x8
28. 3(5x
+
7)
3 4/
+
2x
+
-
20
17x + 31. 6{a
+
1)
6a - 3 16. 5(3^ + 2) 1 5x + 10
-
b)6a^ + ab20. x(4x 2y) 4x2 _ 2x/
+
23. 3(«
7
4{a + 2) 10a + 14
+
1)
+
83/7
+
26. 8(3«
+ 0-224/7
29. 2(37
+
5)
-
1
32. l{n
+
3)
+
11
+ 6
57 + 4 / + 14
2(« 9/7
-
6
+ 9c
9)c2
like terms.
4(7
22.
1)
+
6)
+ 9 407
For use with pages 18-21 Find the value
A
1.
if
possible.
6-0o
2.
If not, write impossible.
5-15
8-0o
3.
4. -^ o
4
5.
—
6.
—
7.
6
11.
^
15.
3x
impossible
8.
1
-t-
—
impossible
7
9.-^-^0
10.
—7
3
=
I.
Find the value of the expression.
13. l{x
-
1)0
Let X
"*"
^
impossible
12.
-
12
impossible
1
14.
X
XI
--
--
4
x3
16. -7-^1
4 Let a 17.
=
(fl
Find the value of the expression.
6.
-
1)
-
5
2x
-
3 2x
18. {2a
1
-
2)
-
10
1
19.
^-^
23.
4«
-
•
li
5 24. 2(x
+
20.
i
Simplify.
21. 3
+
22. 7
+
47
-
7 4/
+
5
+
-
2a
6a 25.
-|-«/7
26.
I
-cc
^•3x
27.
28.
/or use with pages 22-25 Tell
A
which of the numbers shown in color
3=4
a solution.
UorO
2.
x-2 =
4.
«
_
4
6.
«
-
5
1.
«
+
3.
5
+7 =
8
2' 2,
or
5.
6
-
4
4, 3,
or 2
a
=
Tell which of the 7.
9.
11.
408
is
«
+
1
>
1,
4, 5
16,
19,
3, 4,
10
15,
3
1
numbers shown
1^
9
12.
2c
>
3
10
6, 7, 9,
6, 8,
10,
4^,
1,
9
12
^
n
4)
-
8
2x
CHAPTER
2
For use with pages 34-37 Solve.
A
1.
x-2=46
2.
7-4 = 26
3.
«-3 =
5.
7-4
=
711
6.
x-3=69
1.
n
-1 =
9.
«
+
2
=
6 4
10.
jc
+
1
=
76
11.
;c
+
7
=
8
1
12.
7 +
13.
X
+
5
=
20 15
14.
7 +
6
=
14 8
15.
A7
+
9
=
15 6
16.
m +
17.7-5=2025
18.
X
-6 =
1319
19.
JC
+
6
=
1913
20. «
+
7
16 21
23.
jc
+
14
21 7
24. r
-
11
25 7
27.
jc
-
6
=
7 +
3
=
36
4.
r-4 = 48
6^3
8.
m-3
=
=
6
8ii
104
=
12
18 6
Solve.
+
21. «
+
25.
^
29.
15
9 16
= =
=7 +
17 8
22. «
-
5
22 6
26. «
+
18
30.
-
4
87
For use with pages
JC
= =
= 04
31.
=
18 24
27 24
=
14 7
=
22 33
28. 30
=
«
-
5 35
16
=
jc
-
1430
32.
For use with pages 42-45 Solve.
A
,
1.
2x
+
4
=
6
1
2.
4y
-
5.
5/
+
5
=
15 2
6.
6;c
+
9.
3fl
+
6
=
3610
10.
In
-% =
13.
7a
+
14
=
211
14.
2m +
17.
6x
-
12
=
307
18. 4fl
+
=
3
18
=
=
1
=
2
136 461
5«
-
3
=
22 5
8.
2«
-
6
=
20i3
137
12.
8x
-
5
=
355
51i2
16.
9«
-
40
=
328
20.
5/?
-
13
=
176
3«
-
7
=
7.
4«
+
5
=
45io
11.
3jc
-
8
=
15.
4«
+
3
=
19. 7jc
+
12
36 3
4l7
4.
3.
93
1 1
=
6
75 9
For use with pages 48-49 Solve.
A
1.
2x
+
3x
=
15 3
4.
2x
+
3x
-
5
=
7.
2«
-
5
8«
=
-
+ 6
-
10.
15«
13.
28
=7 -
16.
46
=
2;c
4
+
7«
+ 5x
2.
4a
-2a =
307
5.
5c
-
2c
45 5
8.
7x
+
3
-
11.
8/
-
3
+7 =
24 3
3j 8
14.
40
=
13x
-
6x
-
-
17.
19
=
8a
-
6a
=
34
5
37
+
10 5 8
= =
2;c
+
7
204 18 3
67
+
3/
6.
9«
+
a
-
2
9.
4«
+
7
+
3a2
1.
Three times a number plus
What 2.
is
the
number?
Cynthia's father
3.
How
old
Amy
has twice as
is
total 42.
Cynthia? 8
yrs. old
much money
as Barb.
Together they have $15. How much does Amy have?$io 4.
410
Our class has 90 students. There are 4 more boys than girls. How many boys are there? 47
=
7a
-
16
+
2a 8
=
6x
+
3x
-
4
6
18.
21
26 years older than Cynthia.
is
Together their ages
21 2
30 8
56
64.
= =
15.
is
28 3
7a
2 6
number
5 times the
=
-
10«
8
18 2
6
12.
For use with pages 54-57
A
=
3.
+
-
4x 5
For use with pages 58-59 Solve.
A
1.
2«
=
«io
2.
4x
=
X
4.
\6
-2y = 6y2
5.
5x
-
20
7.
7x
-
8.
3/
+
9
=
4^ 9
10.
67
-
13.
2«
+
10
12
+
= 4x4
=
2
1 1
4^
=
7
+ =
85
11.
3x
+
1
=
X
+
6«i
14.
5x
-
8
=
10
2.
l{a
-
+
73
+
2x6
+
2x
8x
6.
lOx
9.
14
-
2x
12.
5fl
-
4
=
8
15. 7c
-
4
=
2c
3x lo
+
=
3.
15 5
-
61
=
14
=
3x
2
5x2
- ^2 +
62
For use with pages 60-61 Solve.
A
B
1.
2{n
-
4)
=
I612
3)
=
^al
3.
5(z
+
2)
=
10z2
4.
6(7
-
3)
= O3
5.
8(x
-
2)
=
O2
6.
5(m
7.
4(x
+
3)
=
6x6
8.
6(7
+
3)
=
I80
9.
3(2
-
x)
=
10.
4(x
-
2)
=
3x
+
11.
3(x
+
1)
=
2x
+
10?
12.
2(5
-
x)
= 3x2
13. 2(c
+
3)
=
3(c
-1)9
14. 4(r
-
=
2(r
+1)5
15.
4(x
-
4)
=
money
as Bob.
2io
2)
+
2)
=
,
253 3xi
2(x
-
1) 7
For use with pages 62-63 Solve.
A
1.
2.
Al has 3 times as much Bob has $10 more than Together the three boys How much money does
Ed.
have $65. each have?
Ed: $5; Bob: $15; Ai:
$45
Jo, May, and Sara save dimes. Jo has twice as many dimes as May. Sara has 4 more dimes than May.
Together the three have 100 dimes. How many dimes has Jo? 48 dimes
B
3.
Ned is 1 years older than Rob. Next year Ned will be twice as old How old is Ned?21 yrs. old 1
as
Rob.
411
CHAPTER
3
For use with pages 72-75 Compare
A
the numbers. Write
-2
1.
5.6
-7>
?
Graph the
/or
-5>
?
9.
«
+
1
13.
X
3.
0> line.
3
=
2
7.
?
-4>
4.
-6
?
-1
8.-7
?
-3
)4(4«4M)
23. i4rsf (2s
-xV
36x4/4
(3m«)3(-3m«)
26. (2x2^2)(2;c3y3)2
-8I/774/74
8xV
wiV/r
(xY
12.
14. (x7)2 x2/2
-a^b* 25.
11. (j;5)2 yio
27.
8000/r7i2ni2
-BAa^/b^
pages 144-147
Multiply.
A
+
1.
2>{n
4.
— 2(a —
1.
-3{2a
5)
3/7
3Z))
+
- 6a -
2.
-4(x
-2a + Gb
5.
«(x
8.
- \(5x +
+
46 1
6c)
26 -
1
-
-
a)
20 - 9a +
a2
a)(4
-
16. (Sa
B
19. (a
-
+
\)(a
3a2
l)(a^ a^ +
+
+ n^
+ a
—
_
s^^s
-
5)(«
+
5;c2
-
jc ^^2
-
y)(x^
-
\){d^
9.
4x(x^y + xy + Sy) 4xV + 4xV + 1 2x)^
+
1) 1
-
3)
+ 2mn +
+
4){x
«)
8)
-
6)(n
9)
- 3n - 54
-
n){m'^
-
n^)
m^ — mn^ — nm- + 24. {n
-
3/) -Bx^ - Qxy
+ 12x + 32
n^
(m
e/,^
+ 2m^n + m^n
m'^
21.
/72)
m2(w2
j^
18. (n
2xy -^2)
+
a4
— 3j>c(2x +
+
x^ — 3x^y + xy^ + y^
\
+
6.
15. (x x^
3)
g^a
6{a^
1)12.
+
+
3.
z
+ 2n - ^5
-
20. (x
1)
+ 7)(x2 + 2xy + /) 23. {d^ x3 + 3x2/ + 3jf^2 + ^
22. {x
z)
{5x - 2)(5x - 4) 25x2 - 30x + 8
17.
1
—
+
2y
-3x4 + 14. (n
2a
a"^
ax - a/
_
-x{3x^
11^
1)
+ 2ai-
-4x + 4/
-5x - 2y -
^2 ^ _^ 1) 10. 5a(a3 Sa^ + 5a^ - 5a2 + 5a 13. (5
— J^)
8c
_
+
- y)
+ 15
+
2)(«2
+
n3
+
+
6a72
4„
n^
+
12/1
4)
+ 8
For use with pages 148-153 You may use
Multiply.
A
1.
(«
+ /j2
4.
(3x
3)(«
+
-
7/7
(2jc
FOIL
method. 2.
4)
+ 12
+
2)(2x
6x2 7.
+
the
+
5jf
+y)(4x
_
x2 5.
3)
A
- y)
8.
is its
-
\){5y
area?
(2«
6/72
+
+
17/7
1)
+
3.
+
b)(a
-
2b)
(2n
+
5)
5)(m - 2) - 7/77 + 10
6.
(la
+ 3)(3a - 1) 21a2 + 2a - 3
9.
(5x
- y)(3x - y)
1
- 3a6 - 2b^
cm by
-
(m
/n2
/
1)
25/2 _
2a2
rectangle measures (3«
What
(5/
6)(;c
+ 3x - 18
6^
8x2 + 2x/ - /2 10.
+
(x
15x2
cm.
- 8x/ +
/2
v
+ 5 cm2 417
Express as a trinomial. 11. («
2)2
12.
+ 4n + 4
n^
15. {X x^
+ +
^2
- \{n +
— 0^ —
a2
2)2
23.
/72
a2
+7)2
-2(4jc
A =\bh
/I\ \. /
^
=
A =
or
+
8/7
+
+
7)2
-
+
21. 4(2r
+
16rs
(;c
-
10)2
- 20x + 100
x2
\Obf
- 20a6 +
16r2
14.
+ 49
14a)
17. («
- 4a6 + 46^
20.
Find the area. Use
B
13. {n
-32x2 _ i6xK - 2/2
— 4
4/7
3)2
- Ibf
16. (a
37)2
+ 6xy + 9y'
19.
-
(7
- 10/ + 25
18.
(3m
+
2«)2
9,772
+ '\2mn +
4/72
5)2
22.
-3(2x -
3^)2
+
- 1 2x2 + ^Qxy - 27/2
100/>2
4s2
Iw.
24.
1
/
=
3«
+
1
1/7
1/1/
I6/72
H
/br
iis^ M'lVA
+
/:
I8/72
2/7
pages 154-155, 158-159
Divide.
3
1.
3.—.
2.
4
X
M 4/j
6x^
5.
+ 6n
-?^
n
3x2
-49a^b^
63/2^
2j!c
4x
-
9.
Sy —^2x-4/
10.
-
6a
—
6b
x^y^
xy
6 20/2^
10^2
_
13.
5^
"I
J
^
xy ~|~
^X
8x2j
+
- 2n -
12x^2
_
-X -
1
iDX
3x2
4.
5^3
16^3
16.
.^^
17.
4y 2x2 + 2xy - Ay^
-
1
+
6a2
-
18.
8a4
Simplify.
B
-
— x'^
5/2 4aj2
Zxy
15/72
12.
-a + b
-
—
\2xy
IO73
-2\ab
4^2
4x
5/^
lb
a
19.
+
30/72/2
CHAPTER
6
For use with pages 170-173 Find the prime factors.
A
1.
20 2
40
2.
5
2,
2, 2, 2,
5
2502,
3.
5, 5,
5
4.
no
11
2, 5,
140
5.
2, 2, 5,
common
Find the greatest
6.10,25 10.
factor of each pair.
5
90 15
15,
7
7.32,284
8.
40,48 8
12.
11.
26,
39
9.63,14?
1
70,105 35
13. 55,
242
ii
Factor.
-
14.
14
17.
2^2
20.
lOxy
2x
+
-
2(7
6x
2x{x
15.
20n
18.
y^
21.
1
X)
+
- /2 3,(iox
3)
-
y)
-
—
4 4(5n -
3y y(y -
_
1^2
i)
3)
44n ^^n(n -
4)
-
-
16.
12
19.
14^^
+
7x 7x(2x +
22.
6ab^
+
24^2^
36jci2(1
6ab{b + 4a)
_
23. 3n2
+
9„
12
24.
B
26. 5
28. a'^b
-
30.
+
a^b^
ab(a^
-
a^b^
a^b
+
-
aA^
-
+
4^2
- 3n + 4) - 25« + 75«2 + \25n^ 5(1 - 5n + 15n2 + 25^3)
12^7 + 24/2 + 3x/ + 6/2)
4(x2
3{/72
'
6xy
27.
25.
7a
+
7(a
+
+
\4b
+
2/)
70c 10c)
+ 42^2/ - 66jcy + llx^
+ 7x/ - llxV^ + 12xV) 36^^2 ^ 43^2 29. 12^2^ 1 26(a2 - 3a6 + 46) 6x/(1
_
ab^ b^)
lOO^y _ 75^:^ + 50^2/ _^ 25xy 25xy(4xV - 3xV + 2x/ + 1)
15m2«4
31.
_|_
45m3«5 - 60m^n^ + 3/77/7 - 4m^n^)
15/772/74(1
For use with pages 1 76-181 Factor.
A
1.
j2 (y
4. ;c2 (X
+
7^
+
3){y
+ +
+
1)(x
33
+
rt2
5.
^2
+
+
\9b
(b
+
9)(b
10. ^2
+
14^
(a
+
12)(a
+
2)
13. jc2
+
14x
+
13
90
8.
^
24
(x+13)(x+1)
13a:
+
11.
14.
24
+
4)
+
36
+
4)(x
12)(/77
n2
+
i9„
(n
+
17)(/7
+
lOy
72
(y
+
8){y
+
+ +
36
+
+
3.
6.
9.
3)
34
12.
2)
+
16ai
+
6)(/7
+ +
n^
+
16«
(n
+
^^)(n
+
13«
+
12)(/j
n^ (/7
60 10)
+
55
+
+
5)
12
+
1)
m2 + 17m +
16
{m + ^6)(m +
2)
16
n^ (/7
9)
m2 + 15m + (m +
+10)
6){n
+ (x
+
io«
+
(n
+3)
b^
7.
2.
4)
+
\4x 1
12
+
15.
x^ (x
+ +
16a:
14)(x
+ +
1)
28 2)
419
3x)
1)
-
16. b"^
19.
_
y2
+
\0b
-
(b
14^
16
17.
20. «2
45
_^
(K-9)(K-5)
-
22. ;c2
-
(X
B
25.
+
(x
_
^2
80
+
40)(x
+
i/s^
with pages
Many
Factor.
A
1.
4.
_
j2
5^
-
36
-
9)(n
-
1)
_
i8„
+
-
93
29.
-
10)(n
21.
24.
100
+
(/I
5)
_
{-jy
+
60
27.
x^
2)
{y
40« + 144 - 36)(n - 4)
33. n^ (/?
^^){y
-
101;c
+
lly
9
_^
_
(y
x^
-{-
+
5„
-
4
(6
(„
13.
-
y2
7y
^2
+
lOx
Watch
Factor.
1)
72
-
18)(/
-
4)
+
21«
+
68
+
4)(n
+
-
\4n
+
49
(n
-
7)2
17)
182-187
3)2
-
2.
x)^
5.
x2
4-
+
«2
+
+
4)(„
1)
8.
-
^2
+
16jc
64
+
20n
(x
100
(5
-
11.
x)2
m2«2
+
13x
+
(n
3.
S)^
+
10)^
42
-
7)
_ 4^„
_^
-
(X
10. 25
100
- 100)(x -
are trinomial squares.
12;c
+
«2
4)
+
6.
n^
-
81
18x
+
x^
-
x)2
(9
7.
5)
44
-
-
-
y^
+
15^
(x
30.
-
^2){y
_
4)
(m - 30)(/n -
_
-
-
/^
15
-\-
3)(n
(/
m^ - 32m + 60
32. «2
^^
Sn
-
{y
8)
+
-
n^ {n
gO
29a: (X + 25)(x
2)
(K
/or
9
+
26. jc^
24y - 31)(K - 3) 31. ^2 + 20x + 64 (x + 16)(x + 4) 28.
+
{n
9)
+
42x
23. «2
18.
5)
-10/2
(n
90
-
10)(x
+
x^
+
19x
m2 - 9m + 20 {m - 4)(m -
-2)
8)(b
6)(X
(mn -
9.
_
j2
22/ (y
4
_
12. /,2
+
121
-
11)2
+
iqZ? {b
2)2
-
25 5)2
the signs.
-
-
8 (K
8)(k
+
i)
14.
-
n^
-
2n
\5(n
-
5){n
+
3)15.
+
y^
6y
-
40
f/-4)(/+10)
_
16. «2 {n
19.
-
_
n2 (n
_
8„
-
17.
m2 - 5m - 50 (m - 10)(m +
2)
_
8«
9)(n
20
+
10)(n
9
20. n^
+1)
(n
13«
-
-
^)(n
+
-
«)(m
+
18.
-
^2 (jf
5)
-
_
21. «2
14
-
7;c
12)(x
3^
60
+
_
5)
4
{n
-
3.
(1
-
5x)(l
+
^
(^ _^ 2b){a
-
9.
(4«
+
3)(4« - 3) 1 6n2 - 9
12.
(8x
-
3)(8x + 3) 64x2 _ ^
14)
4)(n
+
1)
For use with pages 188-191 Multiply at sight.
A
+
-
4)(x
1.
(jc
4.
Oy +
7.
(5x
+
1)(37
4) x2
-
2;;)(5jc
1)
-
- 16 9/2
2y)
-
1
2.
(m
5.
(2x
8.
(8m
(1
-
6a){\ 1
420
+
6a)
- 36a^
11. (9
«)
m2 -
- y){2x + 7) 4x2
25x2-4/2 10.
+
-
-
«)(8m + n) 64m2 - n2
2Z7)(9
+
81 -
2b) 4/)2
n2
_
yt
5x^ 2Zj}
Factor. 13.
^2 -
(X
B
-
25
5)(x
+
5)
CHAPTER
7
For use with pages 204-207
The
table below
shows the average lifespan of several animals.
Bat,
6
yr.
Find three solution pairs for the equation. Answers may vary.
(-1, -4);
=
13.
7
(0,
-3);
+
17. 2jc
(-1,
—
X (1,
y
3
-2)
-1,
=
2); (0, 0); (1,
-2)
Guess an equation
B
14. (
y + X = 4x
18.
(-1, 16); to
5
(-1,
6); (0, 5); (1, 4)
(0,
+
=12
y
12); (1, 8)
go with the
15. y
+
3); (0,
1); (1,
-
y
19.
(-1, -2);
2x
=
1
-1)
=
table.
y = 3x -
y = 2x
For use with pages 220-223
Draw
the graph of the equation. Check student's graphs.
\.
y = 5x
4.
X
1.
y = —Ix
What 10.
-\-
is
y =
1
the slope of the line
y = 1x1
11.
when
2.
y = X +
5.
y — 2x =
6
8.
7 = 3x —
1
its
graph
7 = -Zx -3
is
5
drawn? 12.
y
=
\6.
^
y + 4x =
(-I, 11); 20.
(0, 0); (1, 2)
X
21.
y = X — 5
A
2x
y
(-1,
1
(0, 7); (1, 3)
-5x =
\0
5); (0, 10); (1,
15)
CHAPTER
8
For use with pages 236-239 Solve by the graphing method.
A
\.
B
5.
y = -X 7 = x + 2(-i,i)
7=x-3 7 =
3a:
+
1.
6.
y -Ix 7 = 3x-3(3,
2x+7 =
X— 7 =
2
3. 6)
3
7.
7
+7 = 3 — 2x +7 =
x
4.
-3(2,
3x-4 = — — 3x
/ = j;
+ —
3;c
1)2a:
x+7
8.
1
=7
1
=7 (-2, =
3x-7 =
_1_
6
2
(-!-¥)
(f-¥)
(l-l)
(l-l)
For use with pages 240-243 Find the slope of the line whose equation
A
1.
Do
7 = 3x
3
1.
is
given.
y = 2x + \2
3.
7 =
-2>x
- 1-3
4.
7 =
1
1
^x + 5^
the equations share one solution pair, no solution pair, or all solution
pairs?
S.
y = 2>x 7 = 3;c— 2no
6.
x
y =
-\-
X— 7 =
d
1.
2 one
x -\- y = Q 3x + 37 =
8. all
^+7
3
X
12 no
= +7 =
For use with pages 244-249 Solve by the substitution method.
A
1.
7
=
3a: (3, 9)
a:
-f
7 =
2.
12
7 = Ax (2, 8) 7 - a: = 6
3.
- 2y = -4 3a: + 7 = 2 (0,
X
4. 2)
-7 = 3 + 7 = 5 (2,
3a: a:
Solve by the addition or subtraction method. 5.
a: a:
(
+7 = -7 =
4 6
6. (5,
-
1
)
2a: 2a:
+ +
27 37
= =
5
7.
4
3a: a:
- 27 = - 27 = 4
- 27 = 18 5a: + 27 =
8.
x
12.
x
Solve by the method easiest for you 9.
3a:
4a:
+
7
=7
2=7
(-9, -34)
424
10.
7
=
3a:
4x
11.
-7 = (0,0)
a:
=7 + 57 =
3a:
-
=
47
3
8
a:-67=1
(1, 1)
(7, 1)
3)
-5)
B
13.
-
X X
5y
4y
= =
14.
3x
+
4/
X
-^
4y
9
(45. 9)
= =
6
5x
15.
\5x
7
(4-f)
-/ = -\-y =
5
16.
10
+
4x 4x
(l-f)
= 3 = -2
\2y \0y
\44
22
/
For use with pages 250-257 Solve. 1.
= 5 + 37 = 4
4-
jc
2jc
2y
2.
-7 = 4 + 27 = 6
2x 3x
(-7,6) 5.
3a: — j^ = X + 2/ =
9.
a:
- 47 = 2 - IO7 =
3x
6
6.
+
2a;
X
2
37
=
7.
1
+7 = (11,
4 -7)
4.
x - 37 = 3a; — 7 = (1.
57
8.
9
-1)
(1-
4 4
=
+7 = - 57 = 2 — 27 = 5
3jc a;
(-21, -13)
-1)
variables in solving these problems.
The sum of two numbers is 15. One number is twice the other.
What
are the
numbers?
5.
10.
10
Two hats and 3 ties cost $15.50. One hat and 2 ties cost $9.00. What does each item cost? hat: $4.00; tie:
11.
-
4jc
X
4 (-2, -1)
(2.0)
(2,0)
Use two
3.
Two bats and ball cost $10. One bat and 2 balls cost $8. What does a bat cost? $4.00 1
13. Al's score is 6
Their scores
Find their
more than
Jo's.
12.
total 16.
scores. Jo: 5; Ai:
The sum of two numbers is 15. One number plus twice the other Find the numbers.
14.
1
$2.50
5,
25.
is
10
One egg and 2 coffees cost $1. Two eggs and 1 coffee cost $1.25. What does one egg plus one coffee
cost? $.75
Solve.
B
15.
2x 3;>c
18. 3;c 2a:
21.
+ + -
37
47
47 57
= = = =
16.
6(18. -12) 5
19.
1(3, 1)
4x 3x
—
37
2x
-
47
3a:
27
57
= = = =
1
0(-2, -3) 4 l(-8, -5)
17. 2a: 5a:
20. 4a: 3a:
at 25C each and some pens at 75c The total number of pens plus pencils was 25 and the total was $6.75. How many of each did the club buy? 24 pencils;
+ —
27
= =
+
37 4y
= -2 =
57
(0, 0)
1
Tom's club buys some pencils each. cost
1
pen
425
CHAPTER
9
For use
pages 268-273
after
Simplify.
l6 4
6.
p^ 25a 5a -y^
12x2
3x
+ 2« +
^
^
x'y'
30;cv
6
*^-
1
^^
17 *'•
2
2
SxY
iO;c
-
1
«
21.
-1(-1 +
r
7^2
"-"•
-
9.'
r
8
5„3^2
„5
-^
18 *•
10
;,2
^'^
20 5
3
18
—-20xy^_^ -^—
10.
:r
lOm^n 5m
'
'^''*'
_
40^25
^
^^y + lOy
^^
66^ •
->^
_^2
1
24x4;;3 •
1
24^4^4
i2;c3y2
10
X + y
•
^^-^^ I X2a - Mb 3
— 1.
Write the expression with a factor 20.
—
8.•
7
14a:
^-
4
12
7.*
-4xy^
"^
16
•^-
5
15
X -i(-x)
-
22. 6
4y
-1(-6 +
n)
23.
-^2 +
^2
-1(n2-z2)
4k)
Simplify.
m— 24.
4a?2
20
—
l-«-
-An
g
a:2
^y
•;c2
/or Ms^
+ +
wiV/r
25.
1
—m
n
z
4
29
26.
a
—
_
3
4^ — 8 48 _ i2„2_
^Q
1
3
+ x
2a:
6x
— y2 — y ~ ^
a:2
27.
-,|
1
—« _3
3«2
1
-x -
y
1
3n + 3
6 + 3n
+ x + 5^ 1
+ +
1
^^
x^
—
Ix
-^
12
x - 3 + 2
•;c2-2jc-8'
5
— 3a: — 10 x - 5 •^•;c2-6;c-16^-8 ^^
a:2
pages 274-283 in simplest form.
18 points to 12 points
3.
$6 to $24
3 ^
1
^
ratio of girls to
How many
2.
4 days to 40 days jq
4.
9 students to 36 students ;J;
There are 21 students
The
— 3y -3 y ~ ^
3a:
-1
4
1.
5.
426
-^
z
9-a:2
Write the ratios
A
— —
d
n
in a
boys
is
math
class.
3 to 4.
boys in the class?
How many
girls? 12; 9
Solve the proportion. 6.
^ = -^ X
10.
—=—
7.
X
+
«
Five apples cost
How much
X + 2 -— —=— X
4
1
15.
30 < 22.-2 ? ,
,
or
=
in
place of the
-4
23. 3
5x
+
25
15.
3n
-
n
Can you
-
(2x
18
-e
=
-5
= - 16 -
1)
=
-8 X
-
1
^^=-39 —
21.
J
?.
-(-4)
> 25.
3
12.
9
Write
= 3o
x
10 8 18. 4
-81
+
6.
?
-3(-4)
24. 6
-(-2)
?
(-2)(-4)
< give a value for
—?
for
—? U
O ,
State the value if possible. 0; impossible
Problems 1.
A number is doubled. the
2.
3.
5.
is
added. The
sum
is
Find two consecutive whole numbers whose sum
Mary has dimes.
4.
Then 4
80.
What is
number? 38
twice as
is
65. 32,
many dimes
How many
as Don. Together they have 105 dimes does each have? Mary: 70, Don: 35
South High School played 32 games. They won three times many as they lost. How many did they win? 24
Tom
says "If
subtract
5,
I
I
33
triple
get 25."
the number I'm thinking What is the number? 10
as
of and then
439
2
Cumulative Review for Ciiapters 1-6 Basic Skills Simplify.
1.
15.JL0
2.
-(-3)2-9
3.
6.
-4a +
a -la-'iOa
\5x 4.
2(x
7.
(jc
-
-
8)
+
2x 4x - 16
5.
10x2 _\x2 9^2
27)
+
(3x
+
8.
(x2
27) 4x
-
+
2;c
-
1)
(^2
^
(_^)2
+
2x
-
+
4a
-4x +
1)
\
a^
+ 4a +
2
Multiply. 9.
Aa-a^Aa'
13. {x
+
10.
-
6){x
1) x2
(-3xv)(4x2y2)-i2xV H. (-3^6)3
+ 5x -
14. {n
6
-
4)(« «2
+
15.
4)
-7)
-27a3/,3 12. 2x(5
- 16
lOx - 2x/
{y
+
2){y
-3>)y^-
y-Q
Express as a trinomial. 16. {X x2
-
3)2
17.
- 6x + 9
x2
(jc
+
+ 8x +
4)2
18. (2;c
4x2
1
-
1
-
3)2
19.
(3x
-
9x2-1 2x/ +
2x + 9
2^)2 4/2
Divide.
-24^2^2
8«3
6«4
-^^^
24.
-
—5x
a
+ 3x - 5x2
1
^
26. -^^^
25.
— x'^
-72xv2
5a _ ^2 _ ^3
_j,
+ 2xy^
Factor.
27.
3x/
-
6x2j2 3xK(i - 2xK)
28.
a:2
(X
30.
jc2
+
4x
+
4
(X
+
2)(x
+
2)
+ +
5x
2)(x
+ +
6
+
72
+
7^
3r'je2
-
9^2
29.
6
(/
_|_ 9„ _ 22 (/7+11)(/7-2)
31. «2
(x
+
Solve.
440
X
-
5
=
20 25
2.
40
=
20
-
1)(k
+
-
3/)
3/)(x
.^
Equations and Formulas
1.
+
3)
«
-20
3.
4fl
+
1
=
13 3
6)
^
4.
-=
-4-16
-
6.
5x
+
30
8.
3x
-
{2x
Solve for 10.
6jc
2(x
7.
-10 = -3x + 24
9.
{2a
1)
3;c
+
2)
-
1
=
5x
+ 6-2
+
1)
-
14 8
(5a
-
1)
= -11
x.
=
Solve for
=
=
5.
4
24 4
=y^
11.
6x
\4.
y + X =
12.
kx
=y
j;.
13.
7 +
16.
Solve
7
=
10 3
P =
^x for
x.
22-
17.
g
Solve
15. 4jc
D=
rt
1.
Find the perimeter of a 4-cm by 4-cm square. 16 cm
2.
Find the area of a rectangle with length 2x and width Ax.
3.
A
triangle has
an 8-cm base and a 4-cm height. Find
its
for
5.
Find two consecutive numbers whose sum
6.
Wendy bought some cupcakes at did she buy? 20
Her
profit
\2
3.
3
'2(3a:
2.
8
3.
+
3x 5v)
4. 6.
2m« 9mn^(2
— m)
7.
Page 187
U
1.
+ 9)(x + 2) 2. (>' + 12)(_>' + 2) - 4)(fl - 2) 4. (m + 13)(m - 2) (x - 6)(x - 6) 6. (x + 12)(x - 4) {a - 3)(fl - 9) 8. (/ - 6)( F - 7) {b + 27)(^ - 3) 10. (m - 14)(m - 4) (jc + 10)(x + 10) 12. (m + 8)(m + 8)
3. (« 5. 7.
9.
11
,
Page 193 169 2. 9 - 16x2 3. 25^2 _ 49 - 8)(fl + 8) 5. (m - 6n){m + 6n) {Ax - 3v)(4x + 3v) 3(x + 3)(x + 2) 8. 2( V - SXj + 3) 2(m — 2«)(m + In)
1.
m2 -
4.
{a
6. 7.
9.
CHAPTER
Page 257
(-ll ll^
2.
(2,0)
4.
Yvette: 15; Lauren: 18
95c
milkshake:
:
(-2,1)
3.
1.
hamburger:
5.
60
21.
1.
23.
7
33. 5ar
Page
23
Page 23
Not possible 31. Not possible 35. Not possible
29.
+
31. 8x
Impossible
7.
5.
13.
27. Subtract 9.
1.
3.
17. 4fl
23.
II.
20 and 14
40 and 68 23. 12 and 18 lx,2x + 7 29. 11,22
18
15.
Page 9 I. 6x
17. 9
9.
21.
4
+ 5y +4
206
Page 21
14
II.
43.
8.
35
3.
I.
31. 96
21.
13.
27.
17
ba
37.
29. 36
35.
Page 19
11.
21.
29. 26
10
27.
39. Subtract 4.
Page 7 1. 24
24
35. In
7
10
7.
17.
10
25.
+
3
5.
24
15.
+ 15 + 32x
27. 7jc 33. 8
lOfl
+
37.
9.
6
19.
+
2
3
35
11.
21.
24
29. 6c
+
3
b 16
39.
125
\0x 47. \6b 53^ 6^3 55^ 30^2
45.
49.
6P
57.
2x
+
10
59. 20jc
61. 8c
-
\2d
63. 3a
51. 6x2
- \0y - \W 1
65. 3«
+
+
15m
69.
bb
8x
81.
83. a
93. 4
91.
41. 6
43. 3
45. 5
28A:
53. 4
55.
+
77. Impossible
79.
57. 2n
89. 5n
59.
+ +
18a:
36^
20w +
71.
5/j
75. Impossible
-
73.
67.
6x
85.
95. 3
10^
87.
97. 8
101. 2
10
99.
^n
16
2
23. 8
25.
43. 9
13
35. 6
45.
12
5.
13
7.
24
9.
11.
17. 9
19.
30
21. 25
27. 5
29. 35
31. 51
15. 9
10
33.
10
3.
15
37.
39. 21
11
41.
19
4
12
8
5.
17
15.
12
17.
5
23.
13
25.
10
27.
13
Mixed
18
7.
14
9.
29.
15
11.
12
21.
17
19.
31. 8
15
18
/?
11.
5
27.
17
29. 27
31. 46
25. 4jc
11. 3
2x 31. 2x
+ + +
12
25.
180
33. 54
35.
12
17.
5
7.
9
21.
12
27. 25
37.
10
9.
19.
18
29.
39. 56
15
44 41. 48 31.
35. 27
14
45. 3 55. 21
19
53.
5.
14
25. 2
23. 21
43. 5
5
15.
3.
23
4
7
7.
20
9.
6
19.
1
27. 21
29.
10
17.
«
Ax
c
37. 51
47. 21 57.
39.
8
5.
= 63; 9 = 73; 34 = 24; 8
first
equation
is
5 8
in
them. Multiplying by
amount,
2,
+
jc
3
5
3.
y
+
11. 3jc
+
6;
21.
X X
Answers may
2
13.
+ +
5. jc
1.
a:
9.
+
+
D
61. 6
same amount or doubling the
= 2 X 3b = h =\ = 9 7. 4f = 12; c = 3 = 3 11. 3 13. 3 15.
5
3>;
4
17. 2
19.
5
21. 5
23. 6
25. 5
27.
29. 5
31. 2
33. 5
35. 3
37. 3
39. 7
10
Curto
7.
15. r
+
2; c
4
vary.
13
results in a full glass in either case.
5x = 10; X la =\%\ a 6« = 18; «
I
19. c
11 2;
+
Z?
+
Annie
Pages 43-44 5.
2x
29.
Puzzles and Things, page 53
true only in the sense
that both glasses have the
13. 2
1
23. 7
3x
Puzzles and Things, page 41
The
11.
5
9.
21. 5
41. 30
51.
59. 28
to
doesn't matter
It
8
7.
19. 5
31. 5
18
49. 20 19
11. 6
21. 21
1
+
1.
9.
17.
Practice Exercises, page 40
II
20.
=
9
Pages 51-53
13
43. 72
33.
3.
27.
14
1.
4
25. 4
5.
+
n
always obtain 20.
will
17. 2
23.
13.
20
6
1.
Page 39
Mixed
+
15. 2
7
12
5.
with.
21. 33
3
9
3.
number from
29
13.
80, \QQn
Subtracting the original
19.
7
172
11.
what number you begin
19
15.
4329
5.
Page 48
6
9.
is
17.
5.
+
I00«
4,
17
3.
302
3.
you use algebra to solve this problem, you will notice that the next
7.
15.
23. 30
4
+
11
3.
13. 6
1.
15
If
7.
Practice Exercises, page 37
9
1.
2>
Page 47 5n
15
47. 25
15
13.
3.
19
25.
the last step
10
\1;
51.
12
Arithmetic: 40, 44, 880, 800; Algebra: 5«,
Page 37 1.
16;
=
«
2042 27. 500 Down 7. 27 9. 41 15. 3204 3. 3921 19. 46 21. 925 23. 10 17. 1120
Page 35 13.
5
1.
865
13.
1.
= =
4
49. 6
47. 7
= 32; « = 6 n =
8
Page 45 Across
CHAPTER
2a7
3x
+
1
1;
25; 8
1
15. 2
17.
25. 7
27. 3
29. 2
37. In
=^n +
13.
19.
1
1
31. 9
21. 2
23. 2
33.
35.
+
39. 2«
12; 3
1
=
24
23.^
-3-2-1
10
1
2
3
3
4
5
1
2
3
2
3
8«; 4 25.
Page 61 1. 6x 7.
+
24
15
3.
48/?
16jc
-
2
11.
9.
17. 2
19.
4
29.
31.
15
39. c
41. a
21. 3
40 3
33. 3
4
13.
23.
+
48/
5.
-3-2-1 29-
c
42,.
.._4
27.
27. 5
37. 4
14
35.
10
15.
25. 3
-10 12
24
4
^
I
I
I
-3-2-1
I
1
Pages 62-63 calories
31.
-4 -6 or -6< -1 5. 5 > -4 or -4 < 5 7. > -7 or -7 < - 9. 8 > -8 or -8 < 8 11.-17. Check to be sure you I.
-2
7.
>
3.
=
2
>
5.
^''-
4-hH -3-2-1
13
37.
15.
>
7.
= -6
a:
—h-^H—
h-^
\
19.
-13
27.
-6
35.
-13
43.
71. 17.
83. 91.
-3
23.
1
-11
31.
18
21. 29.
-12
37.
39.
3
2
1
25.-11 33. 4
-5
41.
1
45.-15 47. Ix - y 49. W^ -6a: + 7>y 53. -4 55. -9 -7 59. -17 61. -24 63. 24 -72 67. / - 5/ 69. -2a: - 6 -2,a + 3b 73. -x^ 75. -y + 1 X + 9 79. / 81. -y -6x + 9 -9 85. -4 87. -10 89. -4 -5 93. -2 95. -8 97. 9 99. 2 -3a:
101. 5
CHAPTER
4
Page 103 I. 22
9x
II.
15
3.
5.
The width
3a:
is
10
16
m2
+
3
lla
7.
cm and
9.
14a:
the length
is
cm.
7.
49cm2 16cm2
b.
lOx;;
23.
31. 6
15.
-2 25. 11 or -6 33. 14
tamer must be bald; since Geraldine and Amos have hair, Christopher is the lion tamer. Then, by lion
a
7^
_,_
_,_
1
11.
_4x'y
17.
-Ax'^y
Melba and Fran are both wearing black would know he must be
hats, then Brent
wearing red (since there are only 2 black hats). Since Brent does not know, either Melba or Fran (or both) must be wearing red. If Melba is wearing a black hat, Fran
23. 3r3/2
-6x
25.
21.
27. 4fl^c2
'f29.
-Id^b
1
or 59 minutes.
Page 159 a
8/?7
21.
-\-
17.
25.
Page 153
+ 4r + 4 9 7. x2 - 4x + 4 5. x2 16 11. / _ I4y + 49 8x 9. ;c2 15. m2 + 14m + 49 13. x2 4- 18x + 81 19. 9^2 + \2a +4 17. «2 _ 10/7 + 25 23. 9x2 ^ ^^y +^2 21. 16x2 - 16x + 4 25. 100^2 _ 20ab + f^ II. 16x2 - 24x7 + 29. 2fl2 + 4a + 2 31. 3x2 _ i2;c + 12 33. -2x2 + 8x - 8 8x 6x
+ + +
16
3.
3/) 5. 4x - ly 3. 2x - _y + 9/7 9. a + 9 11. b"^ - ab b'^ 15. 'imn^ — a^ + ab + 2m X - 3x2v + 2 19. Irs + 3^2 - 6^2 -Ad^b + fl2/,2 _ 2^3 23. -2 - 6« 3x2 _ 8;c_y + 9/2 -3x^ + 4x3 _ 6^2
+
1.
7.
wearing a red
hat.
number doubles every minute,
and the basket is full after 1 hour, the minute, basket was half full in 1 hour —
13.
+ + -
3jcyz
19.
31. 27fl3c4
would thus know she must be wearing red. Since Fran does not know, Melba must be
x2
——
7/74
Since the Puzzles and Things, page 151
I.
i5_
Puzzles and Things, page 157
39. 6x2
If
6m2
2x3
13.
27.
1
/•2
V
Page 161 1.
x
9.
a
+ -
2
3.
5
11.'
v
+4 -6
x
+
2
13.
X
-
5.
X
7.
x-4
2
Puzzles and Things, page 161 Yes; "Nothing split" really
is
better than a
means "There
than a banana
split."
is
banana
nothing better
If there
is
better than a banana split, cannot be better than a banana
nothing
then crackers split.
1
Page 162 1.
^2
^4
3.
13. 4^
^3
15.
33.
^2
^2
7.
17.
^7
5.
4 23.
-^
-^
35.
^5
9
^ 4
41.
11.
29. (x
33. (2
Green Mouthwash: $.12/fl. oz.; Fresh Mouthwash: $.115/fl. oz. 3. Dazzle shampoo: about $.2l/fl. oz.; Dew Drops shampoo: about $.20/fl. oz. 1.
Pages 166-167 7.
-
2
3«
9.
23.
+ 8^2 _a2 + 8 Am - 5n
29.
flS
13. 6^2 19.
+ +8
2
5x2 11. 4«2
^ 7^
2x
-\2y
5.
15, 5
i-j
Ix
25.
-
2y
33, 28jc4
-a^
35.
I.
31. {n 35. (x2
+ y){m - x) - A){y + 2)
_ Zj2)^ +5(x + 3) 3.
7r)r2
5
1,
3.
II. 3, 7
-56/
41.
-jc^/
-lOx
_,.
1
-6ab^
75.
+1
83. a^
+ 5a + 6 + 4/ + 5
3x
b) b)
(a2
5,
72
Pages 178-179
37.
43.
4I.
79. 6n
87.
x2(l -tt)
-7x
73.
65.
7.
4
2,
5.
13.- 3, 8
4
3,
7.
3,
5
9.
3,
6
15. 6, 6
+ 6)(fl+ 1) 19. (X + 6)(x + 3) + 3)(r + 7) 23. {n + 1)(« + 16) {a + l)(fl + 4) 27. (x + l)(x + 10) {n + A){n + 2) 31. (x + 7)(x + 2) + 3)(^ + 9) (7+ 1)(7+ 11) 35. (x + 4)(x + 8) 39. (x + 3)(x + 10) {a + l){a + 10) 43. (x + 25)(x + 2) (m + 25)(m + 3) 47. (a + 21)(fl + 3)
29.
69.
57.
61.
(4
33.
+ 5n 18x - 27/ 51. n^ -5n +6 6x2 _ 5;^^ _y2 55, loa - 31a «2 _ 9„ 18 59. 30«2 _^ i6„ + 2 3/ + 13^ + 4 63. «2 4. 8« + 16 67. 4x2 + 4x7 + 72 x2 + 2x + 9m2 - 12mw - 4«2 71. 3n
53.
-
1.
25.
27.
m^n^ 47. ^2 + 4a
39.
45. 30 49.
+
2){a
a){a
21. (r
- 2/
-jc2
21.
31. ^4
37. a^
+ -
17. (a
7«
3.
-lax
-I-
Page 175
Page 165
2x
7)
25(-2a2 ^b'^\zab) 8x^7x2/ - 9xy - 8)
25.
^3
+
2x
1(5/
23. 6(x2
^2
31.
29. |-
1
21. 4(fl2
^2
21.
39.
^3
11.
7
37. 42
13
^3
9.
19.
27. I5
25. 4?13
12
1.
- 3;;) 17. 3(x2 _ + 27 + 4) + ^ab - 4^2) X + 4xv + 7)
15. 9x2(1 19.
81.
77.
-
2x
11«2
1
85. 4a/?
-
3
(Z,
45.
Puzzles and Things, page 179
No,
not possible.
is
it
14
Page 181 I. -3,
-2
3.
-6,
-3
5.
-6,
-4
9. (x - 3)(x - 2) 7. (x - 5)(x - 2) 13. (x - 3)(x - 8) II. {y 1) 5){y 17. (« - A)in - 1) 15. {n + 6)(« + 3) 21. iy - l){y - A) 19. (X + 4)(x + 2) 23. (« + 5)(« + 5) 25. {y - e){y - 5) 27. (x - 7)(x - 7) 29. (m - 5)(m - 8)
Page 183
CHAPTER Page
6
171 7
3.
1.
2, 3,
7.
2,2,2,3,5
13.
7
23.
14
33. 20
15.
2, 2, 3,
1
25. 25
35. 6
3
5.
2, 5, 3
2,2,2,2,3, 13
9.
17.
11
19.
27. 4
37.
16
29. 8
21.
11.
3
12
31. 4
100
+ x) 3. 2(x - 5) 5. x(3x - 3) 9. ln(4n - 1) 4x(x — 2) 13. 5mn{5 — mn)
I.
3(3
2x(x
II.
yes (X
+ +
yes
4)2
5.
7. (a + - 2)2 13. (x - 9)(x (1 - 10x)2
no
11. (x
9)2
17.
19. {a
-bf
21.
5)2
{y
+
7)2
10)
Page 186 1.
-5,2
9.
7,
17. 1)
3.
15. {n
13.
Page 173 7.
1.
9.
21. 25.
29.
-4 (x + (b (X {b + (X +
3.
-4,
11.
14,
- 3) + A) 5)(x + 4) 6)ib - A) 14)(x - 2) 7)(x
3)(b
5.
1
-2,
1
7.
-10,3
-1
+ 9)(x - 2) - 6){n + 3) 23. V - \)iy + 15) 27. {b - 10)(^ + 3) 31. iy + S)iy - A) 15. (X
19. (n (
33.
(>'
35.
(m
39. iy
Mixed
+ 9)(;^ - 3) - 4)(m + 16) - 6)(y + 1)
Page 193 37. {n
-
9){n
+
7)
Practice Exercises, page 187
3. (« - 6)(« - 2) + 4) 7. (y - l)(y + 3) 9. (X 3)(x + 7) 11. (b- \)ib + 5) 13. {b -l){b - 1) 15. (y - A)(y + 5) 19. (n - l)(n - 8) 17. {X + l){x + 5) 21. (x + 9){x + 6) 23. (n + 9)(n + 5) 25. (« - \0){n + 5) 27. (7 + 26)(/ - 2) 29. Ax -A 1.
(jc
5.
{y
+ +
31. Ay
+1)
9)(jc
3){y
7.
n2 JC^
- 100 - 4x2
/2
3.
9.
_4
m"
5.
9^2
11^
21.
- «2 25^2 36 19. 25x2 _ gyz 100x2 - 25/ 23. fl^ - 4
27.
2496
13. 17.
25/
35. 3575
15.
37.
— A2^ _4
81m2
31. 8091
29. 896
6384
39.
25.
33.
399
864
2496
Page 191 1.
yes
9. (;c
13. 15. 17. 19.
21.
23. 25. 29.
31. 33. 35. 37. 39. 41. 43.
3.
no
5.
yes
7.
no
- 4)(x + 4) 11. (« - 3)(« + 3) a - 3b)ia + 3b) x - 2y)(x + 2y) b - Sc){b + 8c) 2a - 5b)(2a + 5b) X - ll)(x + 11) 8x - 37)(8x + 3y) 2 + x)(2 - X) 27. (8 - x)(8 + x) X - y)(x + y) 2x3 _ 2)(2x^ + 3) 12 - llx)(12 + Ux) 6bc - 2a)(6bc + 2a) / + 9){y - 3){y + 3) 4^2 + \){2a + \){2a - 1) 16 + x2/)(4 + xy){A - xy) xy + l)(x2;; + \){x'^y - 1)
a-b
b
1.
(3x
5.
{2b
Page 197 1.
1.
3.
19. ab(b
37. 41.
49. 53.
+b
57. 59.
63. 67.
5.
4
+
a)
14
7.
6
17. 2(n^
21.
-4x)
3y
9.
+
11.
x2
2)
2y\y +
25. 6(6x
A)
+
1)
+ 4) 29. (n + 3)2 31. (x - 5)2 (n + A)(n + 5) 35. (y - S)(y - 1) {n - \\)(n -2) 39. (n + 7)(« +5) (a - 9)2 43. (m + 4)2 {n + 25)(« + 3) 47. (y - 5)(_y - 8) (y + 6)(7 + 3) 51. (X + 6)(x + 5) (n + A)(n + 16) 55. (x + 3)(x + 9) (x - 4)(x - 20) (X + 16)(x + 2) 61. (y - 9)(j - 7) (n - 9)(n + 2) 65. (m + 8)(w - 2) (y - \2)(y + 2) 69. (y - 6)2
27. x(x
33.
3
15.
23. 2x2(1
a-b b
98-199 3
13. 6
45.
a
186,487 kg
Pages
a-b \
45.
5(a
+ l)(x + 1) 3. (2a + 3)(a + 3) + 3)(b + 1) 7. (3a - 5)(a - 1) 9. (3x - 4)(x - 2) 11. (3^ + l)(y - 2) 13. (2x - l)(x + 3) 17. (Aa + 9)(a - 1) 15. (2x + 3)(x - 3) 19. (3b + 5)(2b - 5) 21. (Aa - 5)(a + I) 23. (2x - 5)(2x + 1) 25. (Ar + 3)(2r - 3) 27. (4x + 3)(2x - 1) 29. 2(2c + l)(c + 1) 31. 3(3/ + 2)(y + 1) 33. 3(2x + l)(x + 2) 35. (5/ - 2)(2_y - 5) 37. (3x - ll)(2x + 11) 39. (3x + 4)(4x - 3) 41. (5n - l)(3n + 4)
-A
_49 — J'^
3{y
Page 195
Page 189 1.
+ A){y + 2) 3. 2{y - 5)(y - 3) - 6)ia + 2) 7. 4(x - 3)(x + 5) 11. 8(x - 2)(x + 2) 9. 2(x - 4)2 13. 4(x - 3)(x + 3) 15. 6(x - 2y){x + 2y) 17. -l(x -3)(x - 1) 19. 2(x - 9)(x + 9) 21. -A(a - 2)(a + 1) 23. 2(a - I0b)(a + lOb) 1.
5.
+ 5){a - 2) 73. {b + 6){b - 1) _ 16 77. - 4^2 79. 16«2 _ 81. 9x2 _y2 83_ ^2y2 _ 85. 16jc2 - 9/ 87. {a - b){a + b) 89. (5 - «)(5 + n) 91. (2x - y){lx + /) b){Aa + 93. (4a 95. {la - b){la + 6) 97. (1 -ab){\ -^ ab) 99. 3(a + 1)^ 101. 4(jc - 2){x + 2) 103. 3(a + 3)(a + 2) 105. 3(x - 4)(x + 3) 107. 2{n + 7)(« - 3) 109. a{b - 2c)(b + 2c) 111. 2(a + 4)(fl + 2) 71. (a
75.
/
1
l
1
Z?)
Page 200
= 14 3. m = 9 5. X = -5 = -1 9. X = -3 11. X = 6 17. 2^2+15 a = -5 15. a = 3^2 4x + 21. 3^2 - 4 25. 4m2 — 2«2 5jc2 + -2m2 +3m -3 29. a - 3b a2/,4 33^ 2x^ + 3x2 _ 4^ «2 - 14/7 + 48 37. 3^2 ^ ^ _ 14 6/ - 77 - 3 41. 25fl2 _ b^ a{a + 6b'^) 45. 3mn(\ + 4m) (a - 2){a - 1) 49. (« - 3)(« - 4) (x + 8)(x - 2) 53. 4(jc - 3)(jc + 2)
1.
jc
7.
jc
13. 19.
23. 27. 31.
35. 39. 43. 47. 51.
1
.
1
jc
CHAPTER
7
Pages 206-207 1.
July,
3.
2,500
August
AVERAGE WEEKLY PAY OF IN MANUFACTURING
PERSONS
11.
Pages 218-219 I.
a, c
9.
3; 5;
3.
5.
c
-1;
5.
8
II.
y = J =
25. 2jc
7.
11.
+X - 4jc — 5 =y 6 9
6;
2;
1;
10; 0;
14;
16
4
=9 19.'/ = 5 - 4x 23. 7 = 9x - 7
15.
13. 6; 3; 9;
21.
b
v
jc
Note: Answers to exercises 27-37
may
vary. 27. (1,0), (2,1),
(-1,-2)
29. (1,-1), (2,-2),
(-1,1)
31. (0,6), (1,7), (-1,5) 33. (0,4), (2,8), (-2,0) 35. (0,7), (1,11), (-1,3) 37. (0,2), (1,0), (-1,4) 41.
7 =
Page 221 1.
4jc
+
1
39.
43. a, b
y = 4x 45. a, b
13.
Page 231
NEWSPAPER SALES FOR ONE YEAR
1.
33
17.
Page 249
'(
23.
15. (3,3)
(9,3)
33.
17.
(5,
13. (3,5)
11. (3,4)
9. (4,3)
(2,-2)
21.
5.
3
(-20,11)
7.
(-2,-1)
3.
3
29.
41.
(-1,5)
19.
1)
45.
(6,-8)
23.
37.
47.
Pages 252-253 4
17,
1.
24, 19
3.
Porky: 65 kg 9.
Bugs: 35 kg;
5.
Bud: 170; Jan: 158
7.
11. 5c
length: 18; width: 6
Puzzles and Things, page 253
Let b = the number of balls, c = the number of cups, and p = the number of pennies. Then b + c = \2p and
Ab
-\-
=
c;
b
-
c)
+
2/7
=
50/7
= 12/7 — c, so = c; 48/7 - 4c + = c. 10 pennies
2p
4(12/7
5c;
10/7
2/7
=
c;
49.
Page 255 (-2,3)
1.
(6,3)
3.
9.
(3,0)
11.
(2,4)
7.
(-1,-1)
(-2,6) 19. (1,-1)
13.
(-5,-3)
17.
15. (3,2)
5.
(2,-3)
(-1,1)
21.
Pages 256-257 hotdog: 50C; cola: 40C
1.
student: $2
milkshake:
X
=
5.
ice
65'
+72
-
3.
17.
25x2
5^
1.
4^2
.
.
- 9 11. 81 (x - 5)(x + 5) 9(x - 27)(x + 27
1
15.
5(fl
19. A{n
-
23. /(jc2
27. 29.
b)(a 6)(«
+ +
- j)
4(x
b)
17.
4)
21. x{\
25.
10(fl
+
(Sa - 2b)(3a + 2^) 4(r - 25)(r - 4^) 31.
33. (X ->^)(x
+y)(x^
-
3)^ 3jc)
2b){a
5(x
-
-
15)(x
+/)
Page 422 (pages 204-207)
AVERAGE LIFESPAN
1.
50
OF SEVERAL ANIMALS
40 30
>
20 10
==
3b)
+
4)
5.
9.
1
1
Page 429
13.
(pages 318-321) 5.
-355.4
1.
10
6000 9. 2000
3.
20 and 30
7.
_4,
5.
(pages 322-327)
1.
80%; 0.80
0.05;
9.
$0.63
11. $3.07
13.
'
'''^-
$3.10
15.
M
7.
3.
\^, 4
- V2
-2\/2 -
1.
1.20% 3.350 5.60
,
2y%
3,
V^
1
+
1
+ V33
-
1
5.
$150
9%
at
1.
500 student
tickets
ii
6
15.
7
1.
5
3.
5.
7.
1
25. irrational
23. irrational
Page 432 (pages 354-355) 5.
7V3~
10
1.
27
7.
V5"
9x
9.
3.
80
x^yV^
11.
Page 432 (pages 356-357) 3.
6 or
9.
±4.5
-6
5.
1.
7 or
-7
2V7"or -2y/T
±4.2
7.
Page 433 (pages 358-361)
1.7.2
cm
1.
3.
3.7.5
cm
Page 433 (pages 362-367)
-|
^
5.
10 7.
6
9.
12
17. 0.79
13.
11. 90jc
15
5 15.
x^V^
19. 0.26
Page 433 (pages 368-369)
1.
-6\/r
V2
9.
-2^/3^
(pages 380-385)
1.
5.
2V6
7.
3.
3\^ H.
-^—
Page 434 5.
4,4
5,2
7.
0,
-1
9.
0,
-5
0,9
3.
3.
0,2
11.
-3,
-5
1,
_
1 '
4 17.
1,2
-8
1.
Page 435
l.a.
y =x
7
and 2 or -14, -15
14, 15 or
2,3,4 or -2, -3,
(pages 396-399)
-3.464 13. -1.732 -8.718 17. 9.9 19. 7.9 11.
21. rational
11.
4 15.
-7 and -2 5.
Page 432 (pages 348-353)
4,
03
(pages 394-395)
(pages 336-339)
5.
Page 435
Page 431
9.
-2,7
13.
$390 3. $6000 and $50 at 18%
2
^/2\
1
4
Page 43 1.
±2\/2
2
9.
(pages 332-335)
1
i±^. A^^ 2
9.
no solution
1.
+
4
1
(pages 390-393) 3.
$0.98
I
Pages 434-435
$10
$140
(pages 328-331)
17. 6,
11.
Pages 430-43
7.
-5,2
9.
m
0.09;
5.
8
2\^20
3.
15.
1
(pages 386-389)
Page 430
17.
2,
Page 434
-4
3.a.
5.a.
b.
7.a.
y =
5.-1 15. -8
6
7.
-6
9.
8
17.
13. 9
7
11.
19.
21. 9
1
15.