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LESSON
2.1
Date ————————————
Study Guide
For use with pages 64–70
GOAL
Graph and compare positive and negative numbers.
Whole numbers are the numbers 0, 1, 2, 3, . . ..
Integers are the numbers . . ., 23, 22, 21, 0, 1, 2, 3, . . ..
A rational number is a number that can be written as a quotient of
two integers.
Two numbers that are the same distance from 0 on a number line but
are on opposite sides of 0 are called opposites.
LESSON 2.1
The absolute value of a number a is the distance between a and 0 on
the number line. The symbol ⏐a⏐ represents the absolute value of a.
A conditional statement has a hypothesis and a conclusion.
EXAMPLE 1
Graph and compare integers
Graph 27 and 25 on a number line. Then tell which is greater.
Solution
25
29
28
27
26
25
24
23
22
21
0
On the number line, 25 is to the right of 27. So, 25 > 27.
Exercises for Example 1
Graph the numbers on a number line. Then tell which number is greater.
1. 24 and 22
EXAMPLE 2
2.
0 and 23
3. 1 and 21
Classify numbers
Tell whether each of the following numbers is a whole number,
}
an integer, or a rational number: 217, 0.3, 8, and Ï 2.
Number
Whole number?
Integer?
Rational number?
217
No
Yes
Yes
0.3
No
No
Yes
8
Yes
Yes
Yes
No
No
No
}
Ï2
8
Algebra 1
Chapter 2 Resource Book
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27
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LESSON
2.1
Study Guide
Date ————————————
continued
For use with pages 64–70
EXAMPLE 3
Order rational numbers
2
1
Order the rational numbers 2.2, 21.7, }
, 0, and 2}
from least
5
3
to greatest.
Solution
Begin by graphing the numbers on a number line.
2
21.7
22
1
3
2
5
0
21
0
2.2
1
2
1
3
2
From least to greatest, the numbers are 21.7, 2 }3 , 0, }5 , and 2.2.
Tell whether each number in the list is a whole number, an integer, or a
rational number. Then order the numbers from least to greatest.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
EXAMPLE 4
4. 2, 1.5, 23, 20.5
5. 0, 20.3, 20.6, 20.1
1
6. 23.2, 1, 2.5, 2 }
2
1
2 2
1
7. }, 2 }, }, 21}
5
3 5
2
LESSON 2.1
Exercises for Examples 2 and 3
Find the opposites and absolute values of numbers
For the given value of a, find 2a and ⏐a⏐.
2
b. a 5 }
7
a. a 5 21.3
Solution
a. If a 5 21.3, then 2a 5 2(21.3) 5 1.3.
If a 5 21.3, then ⏐a⏐ 5 ⏐21.3⏐ 5 2(21.3) 5 1.3.
1 2
2
2
2
b. If a 5 }, then 2a 5 2 } 5 2 } .
7
7
7
2
2
2
If a 5 }7 , then ⏐a⏐ 5 ⏐}7 ⏐5 }7.
Exercises for Example 4
For the given value of a, find 2a and ⏐a⏐.
8. a 5 5
9.
a 5 211
10. a 5 23.91
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Chapter 2 Resource Book
9
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LESSON
2.2
Date ————————————
Study Guide
For use with pages 73 –79
GOAL
Add positive and negative numbers.
Vocabulary
The number 0 is the additive identity.
The opposite of a is its additive inverse.
Rules of Addition
Same signs To add two numbers with the same sign, add their absolute
values. The sum has the same sign as the numbers added.
Different signs To add two numbers with different signs, subtract the
lesser absolute value from the greater absolute value. The sum has the
same sign as the number with the greater absolute value.
Properties of Addition
Commutative Property The order in which you add two numbers
does not change the sum.
Associative Property The way you group three numbers in a sum
does not change the sum.
Identity Property The sum of a number and 0 is the number.
EXAMPLE 1
Add real numbers
a. 17 1 (224) 5 ⏐224⏐ 2 ⏐17⏐
Rule of different signs
5 24 2 17
Take absolute values.
5 27
Subtract. Use the sign of the number
with the greater absolute value.
Rule of same sign
b. 21.3 1 (25.8) 5 ⏐21.3⏐ 1 ⏐25.8⏐
5 1.3 1 5.8
Take absolute values.
5 27.1
Add and use the sign of the numbers
added.
LESSON 2.2
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
Inverse Property The sum of a number and its opposite is 0.
Exercises for Example 1
Find the sum.
1. 23.6 1 7.1
1
1
1
4. 23 } 1 22}
5
2
2
2.
5.3 1 (29)
3. 20.2 1 (20.6)
5.
11 1 (215) 1 8
6. 25 1 (28) 1 6
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Chapter 2 Resource Book
19
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LESSON
2.2
Study Guide
Date ————————————
continued
For use with pages 73 –79
EXAMPLE 2
Identify the properties of addition
Identify the property being illustrated.
Statement
a. 215 1 0 5 215
b. 12 1 (217) 5 217 1 12
Property Illustrated
Identity property of addition
Commutative property of addition
Exercises for Example 2
Identify the property being illustrated.
7. 9 1 (2x) 5 2x 1 9
8. 5.1 1 (25.1) 5 0
9. [3 1 (22)] 1 1 5 3 1[(22) 1 1]
EXAMPLE 3
Solve a multi-step problem
Week
Price change for company A
Price change for company B
1
$.05
2$.06
2
2$.08
$.13
3
$.11
2$.04
Solution
STEP 1
Calculate the total change in gas prices for each company.
Company A:
Total change 5 0.05 1 (20.08) 1 0.11
Company B:
Total change 5 20.06 1 0.13 1 (20.04)
5 20.08 1 (0.05 1 0.11)
5 0.13 1 [2(0.06 1 0.04)]
5 20.08 1 0.16
5 0.13 1 (20.1)
5 0.08
5 0.03
STEP 2
Compare the total change in gas prices: 0.08 > 0.03.
Company A had the greater total change in gas prices.
Exercise for Example 3
10. In Example 3, suppose that the changes in gas prices for week 4 are 2$.06 for
company A and $.07 for company B and the changes in gas prices for week 5
are $.04 for company A and 2$.03 for company B. Which company has the
greater total change in gas prices for the five weeks?
20
Algebra 1
Chapter 2 Resource Book
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
LESSON 2.2
Gas Prices The table shows the changes in gas prices for two companies. Which
company had the greater total change in gas prices for the three weeks?
Name ———————————————————————
LESSON
2.3
Date ————————————
Study Guide
For use with pages 80–85
GOAL
Subtract real numbers.
Subtraction Rule
To subtract b from a, add the opposite of b to a.
EXAMPLE 1
Subtract real numbers
Find the difference.
a. 12 2 (28)
b. 215 2 11
Solution
a. 12 2 (28) 5 12 1 8
Add the opposite of 28.
5 20
b. 215 2 11 5 215 1 (211)
Add.
Add opposite of 11.
5 226
Add.
Exercises for Example 1
EXAMPLE 2
1. 29 2 (23)
2. 17 2 21
3. 23 2 (27)
4. 22.7 2 3.8
5. 11 2 (215) 2 7
6. 27 2 6 2 (29)
Evaluate a variable expression
Evaluate the expression x 2 5.1 2 y, when x 5 3.7 and y 5 22.3.
Solution
x 2 5.1 2 y 5 3.7 2 5.1 2 (22.3)
Substitute 3.7 for x and 22.3 for y.
5 3.7 1 (25.1) 1 2.3
Add the opposites of 5.1 and 22.3.
5 0.9
Add.
Exercises for Example 2
Evaluate the expression when x 5 25 and y 5 3.
7. x 1 y 2 7
8.
92y2x
9. x 2 (5 2 y)
Algebra 1
Chapter 2 Resource Book
LESSON 2.3
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
Find the difference.
31
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LESSON
2.3
Study Guide
Date ————————————
continued
For use with pages 80–85
EXAMPLE 3
Evaluate change
Stock A share of a stock on the New York Stock Exchange was valued at $31.26 at
the opening of trading. The value of the stock at closing was $27.97. What was the
change in stock value?
Solution
STEP 1
Write a verbal model of the situation.
Change in
value
STEP 2
5
Closing
value
2
Opening
value
Find the change in value.
Change in value 5 27.97 2 31.26
Substitute values.
5 27.97 1 (231.26)
Add the opposite of 31.26.
5 23.29
Add.
The change in value of the stock was 2$3.29.
Exercises for Example 3
11. From 288C to 218C
12. From 578F to 438F
13. From 2118C to 2158C
LESSON 2.3
10. From 128F to 228F
32
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Chapter 2 Resource Book
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Find the change in temperature.
Name ———————————————————————
LESSON
LESSON 2.4
2.4
Date ————————————
Study Guide
For use with pages 87–93
GOAL
Multiply real numbers.
Vocabulary
The number 1 is called the multiplicative identity.
The Sign of a Product
p The product of two real numbers with the same sign is positive.
p The product of two real numbers with different signs is negative.
Properties of Multiplication
Commutative Property The order in which two numbers are
multiplied does not change the product.
Associative Property The way you group three numbers when
multiplying does not change the product.
Identity Property The product of a number and 1 is that number.
Property of Zero The product of a number and 0 is 0.
Property of 21 The product of a number and 21 is the opposite of
the number.
Multiply real numbers
Find the product.
a. (27)(23)
b. 4(22)(26)
Solution
a. (27)(23) 5 21
b. 4(22)(26) 5 (28)(26)
5 48
Same signs; product is positive.
Multiply 4 and 22; product is negative.
Same signs; product is positive.
Exercises for Example 1
Find the product.
42
1. 25(4)
2. 9(28)
3. 212(26)
1
4. (24) 2}
2
5. 4(28)(5)
6. 28(22)(28)
Algebra 1
Chapter 2 Resource Book
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EXAMPLE 1
Name ———————————————————————
LESSON
2.4
Study Guide
Date ————————————
continued
For use with pages 87–93
Identify the properties of multiplication
Identify the property being illustrated.
Solution
Statement
a. x p 5 5 5 p x
b. 26 p (21) 5 6
c. (23 p x) p 2 5 23 p (x p 2)
Property Illustrated
Commutative property of multiplication
Multiplicative property of 21
Associative property of multiplication
LESSON 2.4
EXAMPLE 2
Exercises for Example 2
Identify the property being illustrated.
7. 25 p 1 5 25
8. [5 p (22)] p (23) 5 5 p [(22) p (23)]
9. 0 p 11 5 0
10. 23 p (212) 5 212 p (23)
EXAMPLE 3
Use properties of multiplication
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
Find the product (23x) p (22). Justify your steps.
Solution
(23x) p (22) 5 (22) p (23x)
Commutative property of multiplication
5 [22 p (23)]x
Associative Property of multiplication
5 6x
Product of 22 and 23 is 6.
Exercises for Example 3
Find the product. Justify your steps.
11. 29(2x)
12. w(23)(12)
13. (28)(5)(2z)
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Chapter 2 Resource Book
43
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LESSON
2.5
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Study Guide
For use with pages 96 –101
GOAL
Apply the distributive property.
Vocabulary
Two expressions that have the same value for all values of the variable
are called equivalent expressions.
The distributive property can be used to find the product of a number
and a sum or difference:
Words
Algebra
The product of a and (b 1 c):
a(b 1 c) 5 ab 1 ac
(b 1 c)a 5 ba 1 ca
a(b 2 c) 5 ab 2 ac
(b 2 c)a 5 ba 2 ca
The parts of an expression that are added together are called terms.
The number part of a term with a variable part is called the coefficient
of the term.
LESSON 2.5
The product of a and (b 2 c):
A constant term has a number part but no variable part.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
Like terms are terms that have the same variable parts raised to the
same power. Constant terms are also like terms.
EXAMPLE 1
Apply the distributive property
Use the distributive property to write an equivalent expression.
a. 7(x 1 3)
b. 22y(3y 1 8)
Solution
a. 7(x 1 3) 5 7(x) 1 7(3)
Distribute 7.
5 7x 1 21
b. 22y(3y 1 8) 5 22y(3y) 1 (22y)(8)
Simplify.
Distribute 22y.
5 26y 2 2 16y
Simplify.
Exercises for Example 1
Use the distributive property to write an equivalent expression.
1. 8(x 2 3)
2.
( y 1 6)(8)
3. 23(4z 2 5)
4. 2(7 2 3m)
5.
} (9n 1 12)
1
3
6. (22p 1 1)(23p)
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Chapter 2 Resource Book
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LESSON
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Study Guide
Date ————————————
continued
For use with pages 96 –101
EXAMPLE 2
Identify parts of an expression
Identify the terms, like terms, coefficients, and constant terms of the
expression 211 2 8y 1 6 1 3y.
Solution
Write the expression as a sum: 211 1 (28y) 1 6 1 3y
Terms: 211, 28y, 6, 3y
Like terms: 28y and 3y, 211 and 6
Coefficients: 28, 3
Constant terms: 211, 6
LESSON 2.5
Exercises for Example 2
Identify the terms, like terms, coefficients, and constant terms of the
expression.
7. 7p 2 12 23p 2 8
EXAMPLE 3
8. 5t 2 1 7q 2 11q 1 9t 2
Solve a multi-step problem
Beverages Every weekday after tennis practice, you either buy a bottle of milk for
$.75 or a bottle of juice for $.85. You practiced 20 days in the past month. Find the cost
of beverages if you buy 12 bottles of milk. Let b be the number of beverages bought.
STEP 1
Write a verbal model. Then write an equation.
Total
cost
5
Price of
milk
p
Number of
bottles
of milk
1
Price of
juice
p
Number of
bottles
of juice
C
5
0.75
p
b
1
0.85
p
(20 2 b)
C 5 0.75b 1 0.85(20 2 b)
5 0.75b 1 17 2 0.85b
5 17 2 0.1b
STEP 2
Find the value of C when b 5 12.
C 5 17 2 0.1b
Write equation.
5 17 2 0.1(12)
Substitute 12 for b.
5 15.80
Simplify.
You spend $15.80 on beverages after tennis practice.
Exercise for Example 3
9. In Example 3, suppose you practice 25 days the next month and buy 18 bottles
of milk. How much do you spend on drinks?
54
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Chapter 2 Resource Book
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Solution
Name ———————————————————————
LESSON
2.6
Date ————————————
Study Guide
For use with pages 103 –108
GOAL
Divide real numbers.
Vocabulary
1
The reciprocal of a nonzero number a, written }a, is called the
multiplicative inverse of a.
Inverse Property of Multiplication
The product of a nonzero number and its multiplicative inverse is 1.
1
Algebra: a p }a 5 1, a Þ 0
Division Rule
To divide a number a by a nonzero number b, multiply a by the
multiplicative inverse of b.
The Sign of a Quotient
• The quotient of two real numbers with the same sign is positive.
• The quotient of two real numbers with different signs is negative.
• The quotient of 0 and any nonzero real number is 0.
Find multiplicative inverses of numbers
Find the multiplicative inverse of the number.
7
a. }
8
b.
LESSON 2.6
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
EXAMPLE 1
22
Solution
7 8
7 8
a. The multiplicative inverse of } is } because } p } 5 1.
8 7
8 7
1
1
b. The multiplicative inverse of 22 is 2} because 22 p 2} 5 1.
2
2
Exercises for Example 1
Find the multiplicative inverse of the number.
5
1. }
7
2. 28
11
3. 2}
14
4. 1
Algebra 1
Chapter 2 Resource Book
65
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LESSON
2.6
Study Guide
Date ————————————
continued
For use with pages 103 –108
EXAMPLE 2
Divide real numbers
Find the quotient.
1 2
3
1
b. } 4 2}
7
2
2
a. 224 4 }
3
Solution
3
2
a. 224 4 } 5 224 p } 5 236
3
2
1 2
1
3
3
6
b. } 4 2} 5 } p (22) 5 2}
2
7
7
7
Exercises for Example 2
Find the quotient.
5. 27 4 (23)
EXAMPLE 3
6.
3
8
1 74 2
2
7. 2} 4 (23)
5
} 4 2}
Simplify an expression
248x 1 12
Simplify the expression }}
.
4
248x 1 12
4
} 5 (248x 1 12) 4 4
Rewrite fraction as division expression.
LESSON 2.6
1
5 (248x 1 12) p }4
1
Division rule
1
5 248x p }4 1 12 p }4
Distributive property
5 212x 1 3
Simplify.
Exercises for Example 3
Find the quotient.
7x 2 28
8. }
14
66
Algebra 1
Chapter 2 Resource Book
9.
220x 1 15
5
}
212x 2 21
10. }
26
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
Solution
Name ———————————————————————
LESSON
2.7
Date ————————————
Study Guide
For use with pages 109 –118
GOAL
Find square roots and compare real numbers.
Vocabulary
If b2 5 a then b is a square root of a. All positive real numbers have
two square roots, a positive square root (or principle square root) and
a negative square root.
}
A square root is written with the radical symbol Ï . The number or
expression inside the radical symbol is the radicand.
The square of an integer is called a perfect square.
An irrational number is a number that cannot be written as a quotient
of two integers.
The set of real numbers is the set of all rational and irrational
numbers.
EXAMPLE 1
Find square roots
Evaluate the expression.
}
a. Ï 400
b.
}
2Ï16
}
c. 6Ï 81
Solution
The positive square root of 400 is 20.
The negative square root of 16 is 24.
The positive and negative square roots of 81 are 9 and 29.
}
b. 2Ï 16 5 24
}
c. 6Ï 81 5 69
Exercises for Example 1
Evaluate the expression.
}
1. Ï 289
EXAMPLE 2
2.
}
2Ï100
}
3. 6Ï 441
Approximate a square root
}
Approximate Ï52 to the nearest integer.
Solution
The greatest perfect square less than 52 is 49. The least perfect square greater
than 52 is 64.
49 < 52 < 64
}
}
}
Ï49 < Ï52 < Ï64
}
7 < Ï52 < 8
Write a compound inequality that compares 52 to
both 49 and 64.
Take positive square root of each number.
Find square root of each perfect square.
}
LESSON 2.7
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
}
a. Ï 400 5 20
}
Because 52 is closer to 49 than to 64, Ï52 is closer to 7 than to 8. So, Ï52 is
about 7.
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Chapter 2 Resource Book
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LESSON
2.7
Study Guide
Date ————————————
continued
For use with pages 109 –118
Exercises for Example 2
Approximate the square root to the nearest integer.
}
4. Ï 75
EXAMPLE 3
5.
}
}
Ï240
6. 2Ï 20
Classify numbers
Tell whether each of the following numbers is a real number, a
rational number, an irrational number, an integer, or a whole
}
}
}
number: Ï 64, Ï 17, 2Ï 36.
Number
Real
number?
Rational
number?
Irrational
number?
Integer?
Whole
number?
Yes
Yes
No
Yes
Yes
Yes
No
Yes
No
No
Yes
Yes
No
Yes
No
}
Ï64
}
Ï17
}
2Ï 36
Graph and order real numbers
}
}
3 }
Order the numbers from least to greatest: }
, Ï 16, 22.2, 2Ï 12, Ï6.
5
Solution
Begin by graphing the numbers on a number line.
2
24
12
23
3
5
22.2
22
21
0
6
1
2
}
16
3
3
}
4
5
}
Read the numbers from left to right: 2Ï12 , 22.2, }5, Ï6 , Ï16 .
LESSON 2.7
Exercises for Examples 3 and 4
76
Tell whether each number in the list is a real number, a rational
number, an irrational number, an integer, or a whole number.
Then order the numbers from least to greatest.
}
}
1
7. Ï 10 , 2}, 2Ï 8 , 22, 1.3
2
Algebra 1
Chapter 2 Resource Book
}
}
1
8. 2Ï 3 , 2}, 2Ï 11 , 22.5, 4
3
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
EXAMPLE 4