School District of Palm Beach County Summer Packet Algebra Readiness Summer 2013 Students and Parents, This Summer Packet for Algebra Readiness is designed to provide an opportunity to review and remediate foundational skills in preparation for success in Algebra 1. These materials include instruction and problem solving on each worksheet. The focus of each selected worksheet is a foundational skill designed to prepare students for Algebra 1. The source of the worksheets is the Glencoe Middle School Mathematics series. All of the contents of this packet have been copied with permission. We hope you are able to utilize the resources included in this packet to make your summer both educational as well as relaxing. Thank you! NAME ________________________________________ DATE _____________ PERIOD _____
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Reteach
A
Rational Numbers
To express a fraction as a decimal, divide the numerator by the denominator.
3
Write −
as a decimal.
Example 1
4
3
−
means 3 ÷ 4.
4
3
The fraction −
can be written as 0.75, since 3 ÷ 4 = 0.75.
4
Example 2
Write -0.16 as a fraction in simplest form.
16
-0.16 = - −
0.16 is 16 hundredths.
100
4
= -−
25
Simplify.
4
The decimal -0.16 can be written as - −
.
25
−
Write 8.2 as a mixed number in simplest form.
−
Assign a variable to the value 8.2. Let N = 8.222… . Then perform the operations on N to
determine its value.
−
N = 8.2 or 8.222… .
Example 3
10(N) = 10(8.222)
-N = 8.222…
9N = 74
9N
74
−
=−
9
9
2
N = 8−
9
Multiplying by 10 moves the decimal point 1 place to the right.
Subtract N
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
10N = 82.222…
Multiply each side by 10 because 1 digit repeats.
= 8.222… to eliminate the repeating part.
10N - 1N = 9N
Divide each side by 9.
Simplify.
−
2
The decimal 8.2 can be written as 8 −
.
9
Exercises
Write each fraction or mixed number as a decimal.
3
2. −
2
1. −
5
7
3. −
10
2
5. - −
3
16
4. 2 −
8
2
6. -1 −
9
25
2
7. 6 −
3
3
8. -4 −
11
Write each decimal as a fraction or mixed number in simplest form.
−
−
12. 1.7
9. 0.8
10. -0.15
11. 0.1
Chapter 1
12
Course 3
NAME ________________________________________ DATE _____________ PERIOD _____
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B
Add and Subtract Rational Numbers
Fractions that have the same denominator are called like fractions. To add or subtract like fractions,
add or subtract the numerators and write the result over the denominator.
( )
1
4
Find −
+ -−
. Write in simplest form.
Example 1
5
5
1 + (-4)
1
4
− + -− = −
5
5
5
-3
3
= − or - −
5
5
( )
Add the numerators. The denominators are the same.
Simplify.
Fractions with unlike denominators are called unlike fractions. To add or subtract unlike fractions,
rename the fractions using prime fractors to find the least common denominator. Then add or subtract
as with like fractions.
5
1
Find -3 −
-1 −
. Write in simplest form.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Example 2
2
5
7
1
11
-3 − -1 − = - − - −
6
6
2
2
7 3
11
= -− · − - −
6
2 3
11
21
= -− - −
6
6
-21 -11
=−
6
32
= -−
6
16
1
= - − or -5 −
3
3
6
Write the mixed numbers as improper fractions.
The LCD is 2 · 3 or 6.
Rename
- −72 using the LCD.
Subtract the numerators.
Simplify.
Exercises
Add or subtract. Write in simplest form.
4
2
1. −
+−
7
7
( 6)
6
( 2)
1
1
- 2−
10. -1 −
5
Chapter 1
7
3
5. - −
+−
5
4
- -−
6. −
3
1
8. 2−
+ 1−
3
1
9. 3−
-1−
4
9
10
11
8
8
7
1
7. - −
- -−
10
5
1
3. −
+ -−
10
5
1
4. −
+ -−
( 9)
5
1
2. −
+−
4
8
(
4
)
4
1
11. -2 −
- -1−
9
3
17
(
11
)
3
3
2
12. 3 −
-2−
5
3
Course 3
NAME ________________________________________ DATE _____________ PERIOD _____
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C
Multiply Rational Numbers
To multiply fractions, multiply the numerators and multiply the denominators.
3
4
Multiply −
. −
. Write in simplest form.
Example 1
8
11
1
3
3
4
4/
−
· −
=−
· −
8
8/
11
Divide 8 and 4 by their GCF, 4.
11
2
3·1
3
=−
or −
2 · 11
Multiply. Then simplify.
22
To multiply mixed numbers, first rename them as improper fractions.
3
1
. Write in simplest form.
Multiply -2 −
· 3−
Example 2
5
3
3
7
18
1
-2 −
· 3−
= -−
· −
3
5
3
Rename
5
18
-2 −13 as −73 and 3 −53 as −
.
5
6
7
18
/
= -−
· −
3/
5
Divide out common factors.
1
7·6
42
or - −
=-−
1·5
2
= -8 −
5
Example 3
Multiply. Then simplify.
5
Write the result as a mixed number.
What is the probability of tossing heads on a coin and rolling a 4 on
a number cube?
1
P(rolling a 4) = −
2
6
1
1
1
P(tossing heads and rolling a 4) = −
· −
or −
2
6
12
Exercises
Multiply. Write in simplest form.
3
2
1. −
· −
7
1
2. - −
· −
9
2
3. −
· −
2
4
4. - −
· -−
1
2
5. 2 −
· −
1
1
· 1−
6. -3 −
3
5
7. 3 −
· 2−
7
2
8. -1 −
· -2−
3
5
7
7
9
2
( 3)
8
5
10
3
6
8
3
(
5
)
2
3
2
9. 2 −
· 2−
3
7
10. What is the probability of rolling an even number on a number cube and tossing tails
on a coin?
Chapter 1
22
Course 3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1
P(tossing heads) = −
NAME ________________________________________ DATE _____________ PERIOD _____
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D
Divide Rational Numbers
Two numbers with a product of 1 are multiplicative inverses, or reciprocals, of each other.
3
Write the multiplicative inverse of -2 −
.
Example 1
4
3
11
-2 −
= -−
4
Write
-2 −3 as an improper fraction.
4
4
3
11
4
4
Since - −
-−
= 1, the multiplicative inverse of -2 −
is - −
.
4
11
4
11
(
)
To divide by a fraction, multiply by its multiplicative inverse.
3
6
Find −
÷ −
. Write in simplest form.
Example 2
8
3
6
3
7
−
÷ −
=−
· −
8
7
8
6
7
6
7
Multiply by the multiplicative inverse of −
, which is −
.
7
6
1
3/
7
=−
· −
8
6/
Divide 6 and 3 by their GCF, 3.
2
7
=−
16
Multiply.
Exercises
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Write the multiplicative inverse of each number.
3
1. −
5
8
2. - −
1
3. −
2
6. -1 −
2
7. -5 −
9
3
5. 2 −
5
3
1
4. - −
10
6
1
8. 7 −
5
4
Divide. Write in simplest form.
1
1
÷ −
9. −
3
2
4
10. −
÷ −
6
5
5
3
11. - −
÷ −
6
1
1
12. 1 −
÷ 2−
4
(
5
1
2
13. 3 −
÷ -3−
7
7
3
)
4
4
14. - −
÷ 2
9
6
÷ (-4)
15. −
1
16. 5 ÷ 2 −
Chapter 1
27
11
3
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NAME ________________________________________ DATE _____________ PERIOD _____
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D
Percent of Change
To find the percent of change, first subtract the original amount from the final amount to find the
amount of change
amount of change. Then write the ratio −− as a decimal. Finally, write the decimal as a
original amount
percent.
Example
Step 1
Two months ago, the bicycle shop sold 50 bicycles. Last month, 55
bicycles were sold. Find the percent of change. State whether the
percent of change is an increase or a decrease.
Subtract to find the amount of change.
55 - 50 = 5
Step 2
Write a ratio that compares the amount of change to the original number of
bicycles. Express the ratio as a percent.
amount of change
original amount
5
=−
50
percent of change = −−
= 0.1
Step 3
final amount - original amount
Definition of percent of change
Substitution
Divide. Use a calculator.
The decimal 0.1 written as a percent is 10%. Since the percent of change is
positive, it is a percent of increase.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Exercises
Find each percent of change. Round to the nearest tenth if necessary.
State whether the percent of change is an increase or a decrease.
1. original: 4 hours
new: 5 hours
2. original: 10 bottles
new: 13 bottles
3. original: 15 books
new: 12 books
4. original: $30
new: $18
5. original: 60 days
new: 63 days
6. original: 160 miles
new: 136 miles
7. original: 77 years
new: 105 years
8. original: $96
new: $59
Chapter 1
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NAME ________________________________________ DATE _____________ PERIOD _____
Reteach
Variables and Expressions
To evaluate an algebraic expression you replace each variable with its numerical value, then use the
order of operations to simplify.
Example 1
Evaluate 5m - 3n if m = 6 and n = -5.
5m - 3n = 5(6) - 3(-5)
= 30 - (-15)
= 45
Example 2
3 + ab
3
Evaluate
Replace m with 6 and n with
Use the order of operations.
Subtract -15 from 30.
3 + ab
(−
) if a = 7 and b = 6.
3
3 + (7)(6)
3
45
=−
3
−=−
Replace a with 7 and b with 6.
The fraction bar is like a grouping symbol.
= 15
Example 3
Divide.
Evaluate 6(x + y) - 4 if x = 8 and y = 3.
6 (x + y) - 4 = 6(8 + 3) - 4
Example 4
-5.
Replace x with 8 and y with 3.
= 6(11) - 4
Use the order of operations.
= 62
Subtract 4 from 66.
Translate each phrase into an algebraic expression.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
a. twelve dollars less than Tamika has
Let d represent the money Tamika has. The expression is d – 12.
b. three less than twice the number of students
Let n represent the number of students. The expression is 2n – 3.
Exercises
Evaluate each expression if a = 4, b = 2, and c = -3.
1. 3ac
2. 5b
3. abc
4. 5a + 6c
ab
5. −
6. 2a - 3b
ac
7. −
8. 6a - b
ac
9. 20 - −
+2
10. 2bc
ac - 3b
11. −
b
b
8
b
6 + 3b
12. −
2a - 2
Translate each phrase into an algebraic expression.
13. four more than the number of DVDs Joan has
14. six less than five times the number of miles
Chapter 2
16
Course 3
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NAME ________________________________________ DATE _____________ PERIOD _____
Reteach
Ordered Pairs and Relations
Example 1
Name the ordered pair for point A.
"
4
3
2
1
•
•
Start at the origin.
Move left on the x-axis to find the x-coordinate of point A,
which is −3.
• Move up the y-axis to find the y-coordinate, which is 4.
So, the ordered pair for point A is (−3, 4).
Example 2
•
•
-4-3 -2
#
y
0
1 2 3 4x
-2
-3
-4
Graph point B at (5, 4).
Use the coordinate plane shown above. Start at the origin
and move 5 units to the right. Then move up 4 units.
Draw a dot and label it B(5, 4).
Example 3
Express the relation {(2, 5), (−1, 3), (0, 4), (1, −4)} as a table and a
graph. Then state the domain and range.
(2, 5)
y
The domain is {−1, 0, 1, 2}.
The range is {−4, 3, 4, 5}.
(0, 4)
x
2
-1
0
1
y
5
3
(-1, 3)
0
x
4
(1, -4)
Exercises
y
Name the ordered pair for each point.
1. A
2. B
3. C
4. D
"
#
0
x
$
%
Express the relation as a table and a graph. Then state the domain and range.
y
5. {(−3, 1), (2, 4), (−1, 0), (4, −4)}
x
y
x
Chapter 2
22
Course 3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
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NAME ________________________________________ DATE _____________ PERIOD _____
Reteach
Analyze Tables
Use a table to write an expression that can be used to find the nth
term of each sequence. Then use the expression to find the next three
+1
+1
terms.
+1
###
Example 1
6, 12, 18, 24, …
Term number, n
1
Term
6
2
3
4
#12#18#24
Make a table to analyze the pattern.
+6
+6
+6
The terms have a common difference of 6. Also, each term is 6 times its
term number.
An expression that can be used to find the nth term is 6n.
The next three terms are 6(5) or 30, 6(6) or 36, and 6(7) or 42.
+1
+1
+1
####
Example 2
3, 7, 11, 15, …
+1
Term number, n
1
Term
3
2
3
4
5
#7#11#15#19
+4
+4
+4
+4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
The terms have a common difference of 4. Start with 4n. Then subtract 1 to
get the term.
So, the expression to find the nth term is 4n – 1.
The next three terms are 4(6) – 1 or 23, 4(7) – 1 or 27, and 4(8) – 1 or 31.
Exercises
Use a table to write the expression that can be used to find the nth
term of each sequence. Then use the expression to find the next
three terms.
1. −3, −1, 1, 3, 5, …
3
1
1
1
2. −
, 1, 1−
, 2−
, 3−
,…
3. 7, 13, 19, 25, 31, …
4. 4, 1, −2, −5, −8, …
Chapter 2
4
28
4
2
4
Course 3
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NAME ________________________________________ DATE _____________ PERIOD _____
Reteach
Analyze Graphs
A relation is a set of ordered pairs. If the ordered pairs form a straight line on a graph, the relation is
said to be linear. You can use the graph of a relation to write an algebraic expression that describes it.
Example
A plumber charges a flat fee to visit a home and
then charges an hourly fee for the amount of time
worked. The total amounts of plumbing bills for
different numbers of hours worked are shown in
the graph.
Total Amount ($)
Plumbing Bill
a. Write an algebraic expression to
represent the data in the graph.
Write the ordered pairs in a table. Look for a pattern.
Hours
0
1
2
3
4
Amount ($)
50
85
120
155
190
+35
+35
+35
200
175
150
125
100
75
50
25
0
(4, 190)
(3, 155)
(2, 120)
(1, 85)
(0, 50)
1
2
3
4
Number of Hours Worked
+35
b. Use the expression to find the total amount of a plumbing bill for
8 hours of work.
35h + 50 = 35(8) + 50 Replace h with 8.
= 280 + 50
Multiply 35 and 8.
= 330
Simplify.
The plumbing bill for 8 hours of work is $330.
Exercises
Pool Draining
1. POOLS The graph shows the amount of water
in a swimming pool after a plug is pulled.
a. Write an algebraic expression to represent
the data in the graph.
b. Use the expression to find how much water
is in the pool after 15 minutes.
Volume (gal)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
The common difference is 35, so 35h is part of the expression. To get the
total amount, add 50. The expression is 35h + 50.
500
450
400
350
300
250
200
150
100
50
0
(0, 500)
(2, 460) (
3, 440)
(1, 480)
(4, 420)
1
2
3
4
Number of Minutes
Chapter 2
33
Course 3
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NAME ________________________________________ DATE _____________ PERIOD _____
Reteach
Translate Tables and Graphs into Equations
Example 1
MONEY Malik earns $8.50 per hour washing cars. Write an equation
to find how much money m Malik earns for any number of hours h.
Let m represent the money earned and h represent the number
of hours worked.
The equation is m = 8.5h.
How much will Malik earn if he works 4 hours?
m = 8.5h
Write the equation.
m = 8.5(4)
Replace h with 4.
m = 34
Multiply.
80
Make a table to find his earnings if he works 7, 8, 9,
or 10 hours. Then graph the ordered pairs.
Hours, h
7
8
9
10
Earnings, m
59.50
68.00
76.50
85.00
+ 8.50
70
60
50
40
30
20
10
(10, 85)
(9, 76.50)
(8, 68)
(7, 59.50)
+ 8.50
0
1 2 3 4 5 6 7 8 9
+ 8.50
Exercises
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. CARS A car dealer sells 12 cars per week.
a. Write an equation to find the number of new cars c
sold in any number of weeks w.
b. Make a table to find the
number of new cars sold
in 4, 5, 6, or 7 weeks.
Then graph the ordered pairs.
Weeks, w Cars, c
2. WRITING An author writes four pages per day.
a. Write an equation to find the number of pages p
written after any number of days d.
b. Make a table to find the
number of pages the
author writes in 1, 2, 3, or
4 days. Then graph
the ordered pairs.
Chapter 2
Days, d
38
Pages, p
Course 3
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B
NAME ________________________________________ DATE _____________ PERIOD _____
Reteach
Functions
A function is a relation in which each member of the domain (input value) is paired with exactly one
member of the range (output value). You can organize the input, rule, and output of a function using a
function table.
Example 1
Choose four values for x to make a function table for
f(x) = 2x + 4. Then state the domain and range of
the function.
Substitute each domain value x, into the function
rule. Then simplify to find the range value.
f(x) = 2x + 4
Rule
2x + 4
2(-1) + 4
2(0) + 4
2(1) + 4
2(2) + 4
Input x
-1
0
1
2
f(-1) = 2(-1) + 4 or 2
f(0) = 2(0) + 4 or 4
f(1) = 2(1) + 4 or 6
Output
f(x)
2
4
6
8
f(2) = 2(2) + 4 or 8
The domain is {-1, 0, 1, 2}. The range is {2, 4, 6, 8}.
Exercises
1. f(1) if f(x) = x + 3
2. f(6) if f(x) = 2x
3. f(4) if f(x) = 5x - 4
4. f(9) if f(x) = -3x + 10
5. f(-2) if f(x) = 4x - 1
6. f(-5) if f(x) = -2x + 8
Choose four values for x to make a function table for each function.
Then state the domain and range of the function.
7. f(x) = x - 10
x
Chapter 2
x - 10
8. f(x) = 2x + 6
f(x)
x
2x + 6
44
9. f(x) = 2 - 3x
f(x)
x
2 - 3x
f(x)
Course 3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Find each function value.
NAME ________________________________________ DATE _____________ PERIOD _____
2-3
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C
Linear Functions
A function in which the graph of the solutions forms a line is called a linear function. A linear function
can be represented by an equation, a table, a set of ordered pairs, or a graph.
Example 1
Graph y = x - 2.
x
0
1
2
3
Step 1 Choose some values for x.
Use these values to make a
function table.
x-2
0-2
1-2
2-2
3-2
y
(x, y)
-2
-1
0
1
(0, -2)
(1, -1)
(2, 0)
(3, 1)
y
Step 2 Graph each ordered pair on a coordinate plane.
Draw a line that passes through the points.
The line is the graph of the linear function.
y=x-2
(3, 1)
(2, 0)
O
x
(1, -1)
(0, -2)
Exercises
Complete the function table. Then graph the function.
1. y = x + 3
x
x+3
y
y
(x, y)
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
-2
0
O
1
x
2
Graph each function.
2. y = 3x + 2
3. y = 2 - x
y
O
4. y = 3x - 1
y
y
x
O
x
O
x
Determine whether each set of data is continuous or discrete.
5. the size of airmail packages
6. the number of boxes in an airmail shipment
Chapter 2
50
Course 3
NAME ________________________________________ DATE _____________ PERIOD _____
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D
Linear and Nonlinear Functions
Linear functions represent constant rates of change. The rate of change for nonlinear functions is not
constant. That is, the values do not increase or decrease at the same rate. You can use a table to
determine if the rate of change is constant.
Example 1
Determine whether the table represents a linear or a nonlinear
function. Explain.
+2
+2
+2
x
3
5
7
9
y
7
10
13
16
+3
+3
Example 2
As x increases by 2, y increases by 3. The rate
of change is constant, so this function is linear.
+3
Determine whether the table represents a linear or a nonlinear
function. Explain.
+1
+1
x
1
2
y
-3
-6
+1
3
-10 -15
As x increases by 1, y decreases by a different
amount each time. The rate of change is not
constant, so this function is nonlinear.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
-5
-4
-3
4
Exercises
Determine whether each table represents a linear or a nonlinear
function. Explain.
1.
3.
x
3
5
7
9
y
7
9
11
13
x
3
6
9
12
y
2
3
4
5
Chapter 2
2.
4.
56
x
1
5
9
13
y
0
6
8
9
x
-2
-3
-4
-5
y
-1
-5
9
8
Course 3
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Reteach
Constant Rate of Change
Relationships that have straight-lined graphs are called linear relationships. The rate of change
between any two points in a linear relationship is the same, or constant. A linear relationship has a
constant rate of change.
Example
The height of a hot air balloon
after a few seconds is shown.
Determine whether the
relationship between
the two quantities is linear.
If so, find the constant rate
of change. If not, explain
your reasoning.
As the number of seconds increase by 1, the
height of the balloon increases by 9 feet.
+1
+1
+1
Time (sec)
Height of Hot
Air Balloon (ft)
1
9
2
18
3
27
4
36
+9
+9
+9
Since the rate of change is constant, this is a linear relationship. The constant
9
rate of change is −
or 9 feet per second. This means that the balloon is
1
rising 9 feet per second.
Exercises
Number of Tables Cost($)
1
10
2
18
3
24
4
28
Number of Cards Total Cost($)
1
1.50
2
3.00
3
4.50
4
6.00
3.
Donuts
Dozens Bought Cost ($)
2
3.25
4
6.50
6
9.75
8
13.00
Chapter 3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Determine whether the relationship between the two quantities
described in each table is linear. If so, find the constant rate of change.
If not, explain your reasoning.
1.
2.
Party Table Rental
Greeting Cards
4.
12
Running
Time (min) Distance(mi)
15
2
30
4
45
5
60
6
Course 3
NAME ________________________________________ DATE _____________ PERIOD _____
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Reteach
C
Slope
The slope m of a line passing through points (x1, y1) and (x2, y2) is the ratio of the difference in the
y-coordinates to the corresponding difference in the x-coordinates. As an equation, the slope is
given by
y -y
2
1
m=−
x - x , where x1 ≠ x2.
2
Example 1
1
Find the slope of the line that passes
through A(-1, -1) and B(2, 3).
y2 - y1
m=−
x2 - x1
3 - (-1)
m=−
2 - (-1)
(x1, y1) = (-1, -1),
(x2, y2) = (2, 3)
4
m=−
Simplify.
3
y
#
(2, 3)
Slope formula
(-1, -1) O
x
"
Check When going from left to right, the graph of the line slants upward.
This is correct for a positive slope.
Example 2
Find the slope of the line that passes
through C(1, 4) and D(3, -2).
$
y2 - y1
m=−
x2 - x1
Slope formula
-2 - 4
m=−
(x1, y1) = (1, 4),
(x2, y2) = (3, -2)
3-1
2
(1, 4)
O
%
x
(3, -2)
Simplify.
Check When going from left to right, the graph of the line slants downward.
This is correct for a negative slope.
Exercises
Find the slope of the line that passes through each pair of points.
1. A(0, 1), B(3, 4)
2. C(1, -2), D(3, 2)
3. E(4, -4), F(2, 2)
4. G(3, 1), H(6, 3)
5. I(4, 3), J(2, 4)
6. K(-4, 4), L(5, 4)
Chapter 3
18
Course 3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
-6
or -3
m=−
y
3-1
E
NAME ________________________________________ DATE _____________ PERIOD _____
Reteach
Direct Variation
When the ratio of two variable quantities is constant, their relationship is called a direct variation.
Example 1
The distance that a bicycle travels varies directly with the number
of rotations that its tires make. Determine the distance that the
bicycle travels for each rotation.
Since the graph of the data forms a line,
the rate of change is constant. Use the
graph to find the constant ratio.
80
−
1
distance traveled
−−
# of rotations
160
80
−
or −
2
1
y
Distance (in.)
140
120
100
80
60
40
20
240
80
−
or −
3
1
320
80
−
or −
4
1
x
0
1 2 3 4 5 6 7
Rotations
The bicycle travels 80 inches for each rotation of the tires.
Example 2
The number of trading cards varies directly as the number of
packages. If there are 84 cards in 7 packages, how many cards are
in 12 packages?
Let x = the number of packages and y = the total number of cards.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
y = kx
Direct variation
84 = k(7)
y
12 = k
Simplify.
y = 12x
= 84, x = 7
Substitute for k
= 12.
Use the equation to find y when x = 12.
y = 12x
y = 12(12)
x
y = 144
Multiply.
= 12
There are 144 cards in 12 packages.
Exercises
Write an expression and solve the given situation.
1. TICKETS Four friends bought movie tickets for $41. The next day seven
friends bought movie tickets for $71.75. What is the price of one ticket?
2. JOBS Barney earns $24.75 in three hours. If the amount that earns varies
directly with the number of hours, how much would he earn in 20 hours?
Chapter 3
25
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A
Slope-Intercept Form
Linear equations are often written in the form y = mx + b. This is called the slope-intercept form.
When an equation is written in this form, m is the slope and b is the y-intercept.
Example 1
State the slope and the y-intercept of the graph of y = x - 3.
y=x-3
y = 1x + (-3)
↑
↑
y = mx + b
Write the original equation.
Write the equation in the form y
= mx + b.
m = 1, b = -3
The slope of the graph is 1, and the y-intercept is -3.
You can use the slope-intercept form of an equation to graph the equation.
Example 2
Graph y = 2x + 1 using the slope and y-intercept.
Step 1 Find the slope and y-intercept.
y = 2x + 1
slope = 2, y-intercept = 1
right 1
y
Step 2 Graph the y-intercept 1.
up 2
y = 2x + 1
0
x
Step 4 Draw a line through the two points.
Exercises
State the slope and the y-intercept for the graph of each equation.
1. y = x + 1
1
3. y = −
x-1
2. y = 2x - 4
2
Graph each equation using the slope and the y-intercept.
4. y = 2x + 2
Chapter 3
2
y
y
0
1
6. y = −
x+2
5. y = x - 1
x
0
30
y
x
0
x
Course 3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
Step 3 Write the slope 2 as −
. Use
1
it to locate a second point on
the line.
2 ←
change in y : up 2 units
m=−
change in x : right 1 unit
1 ←
3-2
B
NAME ________________________________________ DATE _____________ PERIOD _____
Reteach
Graph Functions Using Intercepts
Standard form is when an equation is written in the form Ax + By = C.
Example
State the x- and y-intercepts of 3x + 2y = 6. Then graph the function.
STEP 1
Find the x-intercept.
To find the x-intercept, let y = 0.
3x + 2y = 6
Write the equation.
3x + 2(0) = 6
Replace y with 0.
3x + 0 = 6
Multiply.
3x = 6
Simplify.
x=2
Divide each side by 3.
The x-intercept is 2.
STEP 2
Find the y-intercept.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
To find the y-intercept, let x = 0.
3x + 2y = 6
Write the equation.
3(0) + 2y = 6
Replace x with 0.
0 + 2y = 6
Multiply.
2y = 6
Simplify.
y=3
Divide each side by 2.
The y-intercept is 3.
STEP 3
y
(0, 3)
Graph the points (2, 0) and (0, 3) on a coordinate
plane. Then connect the points.
(2, 0)
0
x
Exercises
State the x- and y-intercepts of each function. Then graph the
function.
1. 3x + 5y = –15
2. – 2x + y = 8
y
Chapter 3
y
y
1
0
3. – 4x – 3y = –12
1
2
1
x
0
2
35
x
0
1
x
Course 3
NAME ________________________________________ DATE _____________ PERIOD _____
3-3
Reteach
C
Solve Systems of Equations by Graphing
Example
Solve the system y = 2x + 3 and y = x – 1 by graphing.
Graph each equation on the same coordinate plane.
y
y = 2x + 3
y=x-1
1
0
1
x
(-4, -5)
The graphs appear to intersect at (–4, –5).
Check this estimate by replacing x with –4 and y with –5.
y = 2x + 3
Check
–5
2(–4) + 3
–5 = –5 !
y=x–1
–5
–4 – 1
–5 = –5 !
The solution of the system is (–4, –5).
Exercises
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Solve each system of equations by graphing.
1. y = 2x + 5
y = –x + 8
2. y = –x – 3
y=x+1
y
y
2
1
0
2
0
x
3. y = –3x + 9
y = −3x + 3
1
x
y
1
2
Chapter 3
x
4. y = –2x + 4
y = –x + 3
y
0
1
2
0
x
46
Course 3
3-3
A
D
NAME ________________________________________ DATE _____________ PERIOD _____
Reteach
Solve Systems of Equations by Substitution
Real-World Example
You own three times as many shares of ABC stock as you do of RST
stock. Altogether you have 380 shares of stock.
a. Write a system of equations to represent this situation.
Draw a bar diagram.
x:
RST
y:
ABC
ABC

#
#
#
#
 380
#
#
#
#

ABC
Use the diagram to write the system.
y = 3x
There are 3 times as many shares ABC stocks as RST stocks.
x + y = 380
The total number of stocks owned is 380.
b. Solve the system by substitution. Interpret the solution.
Since y is equal to 3x, you can replace y with 3x in the second equation.
Write the equation.
x + 3x = 380
Replace y with 3x.
4x = 380
Simplify.
4x
380
−
=−
4
4
Division Property of Equality
x = 95
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
x + y = 380
Simplify.
Since x = 95 and y = 3x, then y = 285 when x = 95. The solution of
this system of equations is (95, 285). This means that you own 95
shares of RST stock and 285 shares of ABC stock.
Exercises
Solve each system of equations by substitution.
1. y = x + 3
y = 4x
2. y = -x - 2
y = -2x
3. y = x + 14
y = 8x
4. y = x - 6
y = 2x
5. y = -x + 8
y = 3x
6. y = -x
y = -2x
Chapter 3
52
Course 3
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B
NAME ________________________________________ DATE _____________ PERIOD _____
Reteach
Write Equations
The table shows several verbal phrases for each algebraic expression.
Phrases
8 more than a number
the sum of 8 and a number
x plus 8
x increased by 8
Phrases
4 multiplied by n
4 times a number
the product of 4 and n
Expression
Phrases
Expression
the difference of r and 6
6 subtracted from a
number
r-6
x+8
6 less than a number
r minus 6
Expression
Phrases
Expression
a number divided by 3
z
4n
the quotient of z and 3
−
3
the ratio of z and 3
The table shows several verbal sentences that represent the same equation.
Sentences
9 less than a number is equal to 45.
The difference of a number and 9 is 45.
A number decreased by 9 is 45.
45 is equal to a number minus 9.
Equation
n - 9 = 45
Write each verbal phrase as an algebraic expression.
2. the quotient of g and 15
3. the product of 5 and b
4. p increased by 10
5. 14 less than f
6. the difference of 32 and x
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. the sum of 8 and t
Define a variable. Then write an equation to model each situation.
7. 5 more than a number is 6.
8. The product of 7 and b is equal to 63.
9. The sum of r and 45 is 79.
10. The quotient of x and 7 is equal to 13.
11. The original price decreased by $5 is $34.
12. 5 shirts at $d each is $105.65.
Chapter 4
16
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C
Solve Addition and Subtraction Equations
You can use the following properties to solve addition and subtraction equations.
• Addition Property of Equality — If you add the same number to each side of an equation, the two
sides remain equal.
• Subtraction Property of Equality — If you subtract the same number from each side of an equation,
the two sides remain equal.
Example 1
w + 19 =
Solve w + 19 = 45. Check your solution.
45
Write the equation.
- 19 = -19
Subtraction Property of Equality
w = 26
Simplify.
Check w + 19 = 45
26 + 19 ! 45
45 = 45 "
Example 2
Replace w with 26. Is this sentence true?
The sentence is true.
Solve h - 25 = -76. Check your solution.
h - 25 = -76
+ 25 = +25
h = -51
Check h - 25 = -76
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Write the original equation.
-51 - 25 ! -76
-76 = -76 "
Write the equation.
Addition Property of Equality
Simplify.
Write the original equation.
Replace h with -51. Is this sentence true?
The sentence is true.
Exercises
Solve each equation. Check your solution.
1. s - 4 = 12
2. d + 2 = 21
3. h + 6 = 15
4. x + 5 = -8
5. b - 10 = -34
6. f - 22 = -6
7. 17 + c = 41
8. v - 36 = 25
9. y - 29 = -51
10. 19 = z - 32
11. 13 + t = -29
12. 55 = 39 + k
13. 62 + b = 45
14. x - 39 = -65
15. -56 = -47 + n
Chapter 4
21
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D
Solve Multiplication and Division Equations
You can use the following properties to solve multiplication and division equations.
• Multiplication Property of Equality — If you multiply each side of an equation by the same number,
the two sides remain equal.
• Division Property of Equality — If you divide each side of an equation by the same nonzero number,
the two sides remain equal.
Example 1
Solve 19w = 114. Check your solution.
19w = 114
Write the equation.
19w
114
−
=−
19
19
Division Property of Equality
w=6
Check
Simplify.
19w = 114
Write the original equation.
19(6) " 114
Replace w with 6.
114 = 114 #
Example 2
d
Solve −
= -9. Check your solution.
15
d
−
= -9
15
d
−
(15) = -9(15)
15
d = -135
d
−
= -9
15
-135
−
" -9
15
-9 = -9 #
Write the equation.
Multiplication Property of Equality
Simplify.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Check
This sentence is true.
Write the original equation.
Replace d with
-135.
This sentence is true.
Exercises
Solve each equation. Check your solution.
r
=6
1. −
2. 2d = 12
4. -8x = 40
5. − = -6
x
6. −
= -7
7. 17c = -68
h
8. −
= 12
9. 29t = -145
11. 13t = -182
12. 117 = -39k
5
10. 125 = 5z
Chapter 4
3. 7h = -21
f
8
-10
-11
26
Course 3
NAME ________________________________________ DATE _____________ PERIOD _____
4-2
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B
Solve Two-Step Equations
A two-step equation contains two operations. To solve a two-step equation, undo each operation in
reverse order.
Solve -2a + 6 = 14. Check your solution.
Example 1
-2a + 6 = 14
Write the equation.
-6 = -6
-2a
Subtraction Property of Equality
= 8
Simplify.
-2a
8
−
=−
-2
-2
Division Property of Equality
a = -4
Simplify.
-2a + 6 = 14
Check
-2(-4) + 6 " 14
Write the equation.
Replace a with
-4 to see if the sentence is true.
14 = 14 # The sentence is true.
The solution is -4.
Sometimes it is necessary to combine like terms before solving an equation.
Example 2
Solve 5 = 8x - 2x - 7. Check your solution.
Write the equation.
5 = 6x - 7
Combine like terms.
5 + 7 = 6x - 7 + 7
Addition Property of Equality
12 = 6x
Simplify.
6x
12
−
=−
Division Property of Equality
6
6
2=x
The solution is 2.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5 = 8x - 2x - 7
Simplify.
Check this solution.
Exercises
Solve each equation. Check your solution.
1. 2d + 7 = 9
2. 11 = 3z + 5
3. 2s - 4 = 6
4. -12 = 5r + 8
5. -6p - 3 = 9
6. -14 = 3x + x - 2
7. 5c + 2 - 3c = 10
8. 3 + 7n + 2n = 21
9. 21 = 6r + 5 - 7r
10. 8 - 5b = -7
11. -10 = 6 - 4m
12. -3t + 4 = 19
a
=5
13. 2 + −
1
14. - −
q - 7 = -3
v
15. 4 - −
=0
6
Chapter 4
3
32
5
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4-2
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C
Write Two-Step Equations
Some verbal sentences translate into two-step equations.
Example 1
Translate each sentence into an equation.
Sentence
Equation
Four more than three times a number is 19.
3n + 4 = 19
Five is seven less than twice a number.
5 = 2n - 7
Seven more than the quotient of a number and 3 is 10.
n
= 10
7+−
3
After a sentence has been translated into a two-step equation, you can solve the equation.
Example 2
Translate the sentence into an equation. Then find the number.
Thirteen more than five times a number is 28.
Words
Thirteen more than five times a number is 28.
Variable
Let n = the number.
Equation
5n + 13 =
28
-13 = -13
Write the equation.
Subtraction Property of Equality
5n = 15
Simplify.
15
5n
−
=−
Division Property of Equality
3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
5
n=3
Therefore, the number is 3.
Exercises
Translate each sentence into an equation. Then find each number.
1. Five more than twice a number is 7.
2. Fourteen more than three times a number is 2.
3. Seven less than twice a number is 5.
4. Two more than four times a number is -10.
5. Eight less than three times a number is -14.
6. Three more than the quotient of a number and 2 is 7.
Chapter 4
38
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A
Powers and Exponents
The product of repeated factors can be expressed as a power. A power consists of a base and an
exponent. The exponent tells how many times the base is used as a factor.
Write each expression using exponents.
Example 1
a. 7  7  7  7
7  7  7  7 = 74
The number 7 is a factor 4 times. So, 7 is the base and 4 is the exponent.
b. y  y  x  y  x
yyxyx=yyyxx
Commutative Property
= (y  y  y) · (x · x)
Associative Property
=y x
Definition of exponents
3
2
To evaluate a power, perform the repeated multiplication to find the product.
Example 2
Evaluate (-6)4.
(-6)4 = (-6)  (-6)  (-6)  (-6)
= 1,296
Write the power as a product.
Multiply.
Example 3
Evaluate m2 + (n - m)3 if m = -3 and n = 2.
m2 + (n - m)3 = (-3)2 + (2 - (-3))3
Replace m with -3 and n with 2.
= (-3)2 + (5)3
Perform operations inside parentheses.
= (-3  -3) + (5  5  5)
Write the powers as products.
= 9 + 125 or 134
Add.
Exercises
Write each expression using exponents.
1. 8  8  8  8  8
2. a  a  a  a  a  a
3. 5  5  9  9  5  9  5  5
5. (-3)5
3
6. −
Evaluate each expression.
4. 24
(4)
3
ALGEBRA Evaluate each expression if a = 5 and b = -4.
7. a2 + b2
Chapter 5
8. (a + b)2
9. a + b2
12
Course 3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
The order of operations states that exponents are evaluated before multiplication, division, addition,
and subtraction.
5-1
B
NAME ________________________________________ DATE _____________ PERIOD _____
Reteach
Multiply and Divide Monomials
The Product of Powers rule states that to multiply powers with the same base, add their exponents.
Example 1
Simplify. Express using exponents.
a. 23  22
23  22 = 23 + 2
The common base is 2.
= 25
Add the exponents.
b. 2s6(7s7)
2s6(7s7) = (2  7)(s6  s7)
Commutative and Associative Properties
= 14(s6 + 7)
The common base is s.
= 14s13
Add the exponents.
The Quotient of Powers rule states that to divide powers with the same base, subtract their
exponents.
Example 2
k8
Simplify −
. Express using exponents.
k
8
k
−
= k8 -1 The common base is k.
1
k
= k7
Subtract the exponents.
Example 3
(-2)10  56  63
(-2)  5  6
(
) ( ) ( )
(-2)10  56  63
(-2)10
56
63
−
−
=
 −
 −
53
62
(-2)6  53  62
(-2)6
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
.
Simplify −
6
3
2
Group by common base.
= (–2)4  53  61
Subtract the exponents.
= 16  125  6 or 12,000
Simplify.
Exercises
Simplify. Express using exponents.
1. 52  55
2. e2  e7
3. 2a5  6a
4. 4x2(–5x6)
79
5. −
3
v14
6. −
6
15w7
7. −
2
10m8
8. −
7
5
 37  43
9. 2−
1
5
2 3 4
Chapter 5
v
5w
2m
415  (-5)6
4  (-5)
10. −
12
4
18
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C
Powers of Monomials
Power of a Power: To find the power of a power, multiply the exponents.
Power of a Product: To find the power of a product, find the power of each factor and multiply.
Simplify (53)6.
Example 1
(53)6 = 53 · 6
= 518
Power of a power
Simplify.
Example 2
Simplify (-3m2n4)3.
(-3m2n4)3 = (-3)3 · m2 · 3 · n4 · 3
= -27m6n12
Power of a product
Simplify.
Exercises
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Simplify.
1. (43)5
2. (42)7
3. (92)4
4. (k4)2
5. [(63)2]2
6. [(32)2]3
7. (5q4r2)5
8. (3y2z2)6
9. (7a4b3c7)2
10. (-4d3e5)2
11. (-5g4h9)7
Chapter 5
12. (0.2k8)2
23
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A
Negative Exponents
Any nonzero number to the zero power is 1. Any nonzero number to the negative n power is the
multiplicative inverse of the number to the nth power.
Example 1
Write each expression using a positive exponent.
b. a−4
a. 7−3
1
7−3 = −
3
1
a−4 = −
4
Definition of negative exponent
7
Definition of negative exponent
a
Evaluate each expression.
Example 2
a. 5−4
b. (−3)−5
1
5−4 = −
4
1
(−3)−5 = −
5
Definition of negative exponent
5
Definition of negative exponent
(−3)
1
=−
−243
1
=−
54 = 5 · 5 · 5 · 5
625
(−3)5 = (−3)
(−3)
· (−3) · (−3) ·
· (−3)
Example 3
1
Write −
as an expression using a negative exponent.
5
6
1
−
= 6−5
5
Definition of negative exponent
6
Simplify. Express using positive exponents.
w−5
b. −
−7
a. x−3 · x5
x−3· x5 = x(−3) + 5
= x2
w
w−5
−
= w−5 − (−7)
w−7
Product of Powers
Add the exponents.
Quotient of Powers
= w2
Subtract the exponents.
Exercises
Write each expression using a positive exponent.
1. a−8
2. 6−3
3. n−4
5. 9−3
6. (−2)−5
Evaluate each expression.
4. 7−2
Write each fraction as an expression using a negative exponent.
1
7. −
7
5
1
8. −
6
1
9. −
8
3
x
Simplify. Express using positive exponents.
10. 4−2 · 4−4
Chapter 5
h−2
12. −
4
11. r−3 · r5
h
32
Course 3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Example 4
NAME ________________________________________ DATE _____________ PERIOD _____
5-2
Reteach
B
Scientific Notation
A number in scientific notation is written as the product of a factor that is at least one but less than ten
and a power of ten.
Write 8.65 × 107 in standard form.
Example 1
8.65 × 107 = 8.65 × 10,000,000
= 86,500,000
"
!!!!!!
Example 2
107 = 10 ∙ 10 ∙ 10 ∙ 10 ∙ 10 ∙ 10 ∙ 10 or 10,000,000
The decimal point moves 7 places to the right.
Write 9.2 × 10–3 in standard form.
9.2 × 10-3 = 9.2 × 0.001
The decimal point moves 3 places to the left.
= 0.0092
"
!!
Example 3
Write 76,250 in scientific notation.
76,250
= 7.625 × 10,000
"
!!!
= 7.625 × 104
Example 4
Since 76,250 is >1, the exponent is positive.
Write 0.00157 in scientific notation.
0.00157 = 1.57 × 0.001
!!
"
= 1.57 × 10–3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
The decimal point moves 4 places.
The decimal point moves 3 places.
Since 0.00157 is <1, the exponent is negative.
Exercises
Write each number in standard form.
1. 5.3 × 101
2. 9.4 × 103
3. 7.07 × 105
4. 2.6 × 10-3
5. 8.651 × 10-2
6. 6.7 × 10-6
Write each number in scientific notation.
7. 561
9. 56,400,000
8. 14
10. 0.752
11. 0.0064
12. 0.000581
Chapter 5
37
Course 3
NAME ________________________________________ DATE _____________ PERIOD _____
5-2
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C
Compute with Scientific Notation
You can use the Product of Powers and Quotient of Powers properties to multiply and divide numbers
written in scientific notation.
Example 1
Evaluate (3.4 × 105)(2.3 × 103). Express the result in scientific notation.
(3.4 × 105)(2.3 × 103) = (3.4 × 2.3)(105 × 103)
Commutative and Associative Properties
= (7.82)(105 × 103)
Multiply 3.4 by 2.3.
= 7.82 × 105 + 3
Product of Powers
8
= 7.82 × 10
Add the exponents.
Example 2
2.325 × 104
Evaluate −
. Express the result in scientific notation.
2
3.1 × 10
2.325
2.325 × 10
= −
−
2
3.1
3.1 × 10
10
) (−
10 )
Associative Property
104
= (0.75) −
2
( 10 )
Divide 2.325 by 3.1.
= 0.75 × 104 – 2
Quotient of Powers
(
4
4
2
2
Subtract the exponents.
= 0.75
" × 102
Write 0.75 × 102 in scientific notation.
= 7.5 × 10
Since the decimal point moved 1 place to the right,
subtract 1 from the exponent.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
= 0.75 × 10
Example 3
Evaluate (5.24 × 105) + (8.65 × 106). Express the result in
scientific notation.
(5.24 × 105) + (8.65 × 106) = (5.24 × 105) + (86.5 × 105)
= (5.24 + 86.5) × 105
5
Write 8.65 × 106 as 86.5 × 105.
Distributive Property
= 91.74 × 10
Add 5.24 and 86.5.
= 9.174 × 106
Write 91.74 × 105 in scientific
notation.
Exercises
Evaluate each expression. Express the result in scientific notation.
1. (6.7 × 104)(2.9 × 105)
2. (4.3 × 104) + (5.21 × 105)
5.46 × 105
3. −
3
4. (9.6 × 105) – (3.7 × 103)
8.4 × 10
Chapter 5
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Course 3
5-3
A
NAME ________________________________________ DATE _____________ PERIOD _____
Reteach
Square Roots
A square root of a number is one of its two equal factors. A radical sign, √€ , is used to indicate a
positive square root. Every positive number has both a negative and positive square root.
Examples
Find each square root.
√1
€
Find the positive square root of 1; 12 = 1.
- √€€
16
Find the negative square root of 16; (-4)2 = 16.
± √€€
0.25
Find both square roots of 0.25; 0.52 = 0.25.
√€€
-49
There is no real square root because no number times itself is equal to -49.
Example 5
4
a =−
9
2
4
Solve a2 = −
. Check your solution(s).
9
Write the equation.
√9
4
−
a=± €
Definition of square root
2
2
a=−
or - −
3
3
2
Check −
3
· −23 = −94 and (-−23 ) (-−32 ) = −49 .
2
2
and - −
.
The equation has two solutions, −
3
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3
Exercises
Find each square root.
€
1. √4
2. √€
9
3. - √€€
49
4. - √€€
25
5. ± √€€
0.01
6. - √€€
0.64
7.
9
−
√€€
16
8.
-1
−
√€€
25
ALGEBRA Solve each equation. Check your solution(s).
9. x2 = 121
10. a2 = 3,600
81
11. p2 = −
121
12. t2 = −
Chapter 5
48
100
196
Course 3
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NAME ________________________________________ DATE _____________ PERIOD _____
Reteach
Estimate Square Roots
Most numbers are not perfect squares. You can estimate square roots for these numbers.
Example 1
•
•
Estimate √€€
204 to the nearest whole number.
The largest perfect square less than 204 is 196.
The smallest perfect square less than 204 is 225.
196 < 204 < 225
142 < 204
< 152
√""
142 < √""
204 < √""
152
14 < √""
204 < 15
Write an inequality.
196 = 142 and 225 = 152.
Find the square root of each number.
Simplify.
So, √""
204 is between 14 and 15. Since 204 is closer to 196 than 225,
the best whole number estimate for √""
204 is 14.
Example 2
•
•
Estimate √€€
79.3 to the nearest whole number.
The largest perfect square less than 79.3 is 64.
The smallest perfect square less than 79.3 is 81.
82 < 79.3 < 92
√"
82 < √""
79.3 < √"
92
8 < √""
79.3 < 9
Write an inequality.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
64 < 79.3 < 81
64 = 82 and 81 = 92.
Find the square root of each number.
Simplify.
So, √""
79.3 is between 8 and 9. Since 79.3 is closer to 81 than 64,
the best whole number estimate for √""
79.3 is 9.
Exercises
Estimate to the nearest whole number.
"
1. √8
2. √""
37
3. √""
14
4. √""
26
5. √""
62
6. √""
48
103
7. √""
8. √""
141
9. √""
14.3
""
10. √51.2
11. √""
82.7
12. √"""
175.2
Chapter 5
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Course 3
NAME ________________________________________ DATE _____________ PERIOD _____
5-3
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D
Compare Real Numbers
Numbers may be classified by identifying to which of the following sets they belong.
Integers …, -2, -1, 0, 1, 2, …
Whole Numbers
0, 1, 2, 3, 4, …
Rational Numbers
a
numbers that can be expressed in the form −
, where a and b are
b
integers and b ≠ 0
Irrational Numbers
a
, where a and b
numbers that cannot be expressed in the form −
b
are integers and b ≠ 0
Examples
Name all sets of numbers to which each real number belongs.
5
whole number, integer, rational number
0.666…
Decimals that terminate or repeat are rational numbers, since they can
2
be expressed as fractions.
0.666… = −
- √$$
25
Since - √$$
25 = -5, it is an integer and a rational number.
√$$
11
√$$
11 ≈ 3.31662479… Since the decimal does not terminate or repeat, it
is an irrational number.
3
To compare real numbers, write each number as a decimal and then compare the
decimal values.
Example 5
Replace
1
with <, >, or = to make 2 −
√#
5 a true statement.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
4
Write each number as a decimal.
1
= 2.25
2−
4
√5
$ ≈ 2.236067…
1
Since 2.25 is greater than 2.236067…, 2 −
> √$
5.
4
Exercises
Name all sets of numbers to which each real number belongs.
1. 30
2. -11
4
3. 5 −
4. √$$
21
5. 0
6. - √$
9
6
7. −
8. - √$$
101
7
3
Replace each
9. 2.7
Chapter 5
√$
7
with <, >, or = to make a true statement.
10. √$$
11
1
3−
2
1
11. 4 −
6
60
√$$
17
−
12. 3.8
√$$
15
Course 3
9-1
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NAME ________________________________________ DATE _____________ PERIOD _____
Reteach
Literal Equations
Example 1
The equation V = Bh can be used to find the volume of a prism. Solve
the equation for B.
V = Bh
Write the equation.
Bh
V
−
=−
Division Property of Equality
h
h
V
−=B
h
Simplify.
V
So, B = −
.
h
Example 2
Sanchez is in charge of packaging. The company he works for
requires that the volume of each box he uses be 108 cubic inches.
Find the height of the box that will meet this requirement. Use the
equation V = bwh.
Step 1
V = bwh
bwh
V
−
=−
h in.
Write the equation.
Division Property of Equality
4 in.
9 in.
Simplify.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
bw
bw
V
−=h
bw
Solve the equation for h.
Step 2
V
−
=h
bw
108
−=h
9(4)
3=h
Write the equation.
Replace V with 108,
b with 9, and w with 4.
Divide.
The height of the box needs to be 3 inches.
Exercises
Solve each equation for the indicated variable.
1. A = bh, for b
2. L = πrℓ, for ℓ
3. a2 + b2 = c2, for c
1
4. A = −
bh, for b
2
5. SANDBOX Lauren is building a square sandbox for her daughter. She
wants the area to be 27 square feet.
a. Solve the area formula for a square, A = s2, for s.
b. Find the length s of the sandbox to the nearest tenth.
Chapter 9
12
Course 3