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Chapter 1Introduction to Algebra: Integers
1.1 Place Value
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1.1 Place Value
Objectives
1. Identify whole numbers.
3. Write a whole number in words or digits.
2. Identify the place value of a digit throughhundred-trillions.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.1 - 4
Objective 1: Identify whole numbers.
Our number system is a place value system. Each location in which a number is placed
gives it a different value.
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Objective 1: Identify whole numbers.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.1 - 6
Objective 1: Identify whole numbers.
Whole Numbers: created from the
digits 0, 1, 2, 3, …, 9
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Objective 2: Identify the place value of a digit through hundred-trillions.
Example Identify the place value of
each 8 in 6,598,274,806.
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Objective 3: Write a whole number in words or digits.
Example Write 6,058,120 in words.
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Objective 3: Write a whole number in words or digits.
Example Write the number using digits.
Seventy-seven billion, thirty thousand, five hundred
77,000,030,500
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Chapter 1Introduction to Algebra: Integers
1.2 Introduction to Signed Numbers
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1.2 Introduction to Signed Numbers
Objectives
1. Write positive and negative numbers used in everyday situations.
4. Find the absolute value of integers.
3. Use the < and > symbols to compare integers.
2. Graph signed numbers on a number line.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.2 - 12
Objective 1: Write positive and negative numbers used in everyday situations.
Numbers greater than zero are called positive numbers.
Numbers less than zero are called negative numbers.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.2 - 13
Objective 1: Write positive and negative numbers used in everyday situations.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.2 - 14
Objective 1: Write positive and negative numbers used in everyday situations.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.2 - 15
Objective 1: Write positive and negative numbers used in everyday situations.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.2 - 16
Objective 1: Write positive and negative numbers used in everyday situations.
Example Write “a loss of $500” as a
number with its appropriate sign.
– $500
Raised negative sign
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Objective 2: Graph signed numbers on a number line.
A number line is like a thermometer turned sideways.
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Objective 2: Graph signed numbers on a number line.
Graph each number on the
number line.
(a) –5 (b) 3 (c) 1 ½ (d) 0 (e) –1
Example
(a) (c) (b)(e)(d)
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.2 - 19
Objective 3: Use the < and > symbols to compare integers.
Integers are the numbers
…, –5, –4, –3, –2, –1, 0, 1, 2, 3 ,4, 5,… “>” means greater than “<“ means less than
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.2 - 20
Objective 3: Use the < and > symbols to compare integers.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.2 - 21
Objective 3: Use the < and > symbols to compare integers.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.2 - 22
Objective 3: Use the < and > symbols to compare integers.
Write < or > between each
pair of numbers to make a true
statement.
(a) 0 ____ 2
(b) 1 ____–4
(c) –4 ____–2
0 < 2
1 > –4
Example
–4 < –2
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.2 - 23
Objective 4: Find the absolute value of integers.
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Objective 4: Find the absolute value of integers.
Find each absolute value.
(a) |4|
(b) |–4|
4 spaces, so |4| = 4
4 spaces, so |–4| = 4
Example
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Chapter 1Introduction to Algebra:
Integers
1.3 Adding Integers
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1.3 Adding Integers
Objectives
1. Add integers.
2. Identify properties of addition.
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Objective 1: Add integers.
Numbers that you add together are called addends, and the result is called a sum.
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Objective 1: Add integers.
Use a number line to find
–5 + –4 (use a football analogy).
Example
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Objective 1: Add integers.
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Objective 1: Add integers.
Add the integers –8 + –7.
Step 1 Add the absolute values.
|–8| = 8 and |–7| = 7
Add 8 + 7 to get 15.
Both negative Sum is negative
Step 2 Use the common sign as the sign of the sum.
–8 + –7= –15
Example
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Objective 1: Add integers.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 32
Objective 1: Add integers.
Add the integers 3 + –8.
Step 1 |3| = 3 and |–8| = 8
Subtract 8 – 3 to get 5.
Step 2 –8 has the larger absolute value and is
negative, so the sum is also
negative.
3 + –8 = –5
Example
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 33
Objective 1: Add integers.
A football team has to gain at least
10 yards during four plays in order to keep the ball.
Suppose on four plays a team lost 6 yards, gained 8
yards, lost 2 yards, and gained 7 yards. Did the
team gain enough to keep the ball?
Example
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Objective 1: Add integers.
Example (continued)
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 35
Objective 2: Identify properties of addition.
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Objective 2: Identify properties of addition.
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Objective 2: Identify properties of addition.
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Objective 2: Identify properties of addition.
Example Add 6 + 9 + –9 using the associative property.
Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 1.4 - 39
Chapter 1Introduction to Algebra:
Integers
1.4 Subtracting Integers
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1.4 Subtracting Integers
Objectives
1. Find the opposite of a signed number.
2. Subtract integers.
3. Combine adding and subtracting of integers.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.4 - 41
Objective 1: Find the opposite of a signed number.
Opposite integers are the same distance from 0 on the number line but are on opposite sides of 0.
When you add opposites, the sum is always 0. The opposites are also called additive inverses.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.4 - 42
Objective 1: Find the opposite of a signed number.
Example Find the opposite of each number and show that the sum of the numbers is 0.
(a) 6
(b) –8
(c) 0
The opposite of 6 is –6. 6 + –6 = 0
The opposite of –8 is 8. –8 + 8 = 0
The opposite of 0 is 0. 0 + 0 = 0
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.4 - 43
Objective 2: Subtract integers.
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Objective 2: Subtract integers.
Example Subtract the integers.
4 – 10
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Objective 2: Subtract integers.
Example Subtract the integers.
–9 – –6
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Objective 3: Combine adding and subtracting of integers.
Example Simplify by completing all the calculations.
–5 – 10 – 12 +1
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Chapter 1Introduction to Algebra:
Integers
1.5 Problem Solving: Rounding and Estimating
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1.5 Problem Solving: Rounding and Estimating
Objectives
1. Locate the place to which a number is to be rounded.
2. Round integers.
3. Use front end rounding to estimate answers in addition and subtraction.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.5 - 49
Objective 1: Locate the place to which a number is to be rounded.
Rounding a number means finding a number that is close to the original number, but easier to work with.
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Objective 1: Locate the place to which a number is to be rounded.
Draw a line under the place to which the number is to be rounded.Example
Round –23 to the nearest ten.
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Objective 1: Locate the place to which a number is to be rounded.
Draw a line under the place to which the number is to be rounded.
Round –54,702 to the nearest thousand.
Example
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.5 - 52
Objective 2: Round integers.
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Objective 2: Round integers.
Round 349 to the nearest hundred.
Step 1
Underline the place to which the number is being rounded.
Example
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.5 - 54
Objective 2: Round integers.
Round 349 to the nearest hundred.
Step 2
Because the next digit is 4 or less, do not change the digit in the underlined place.
Example
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.5 - 55
Objective 2: Round integers.
Round 349 to the nearest hundred.
Step 3
Change all digits to the right of the underlined place to zeros.
Example
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.5 - 56
Objective 2: Round integers.
Round 36,833 to the nearest thousand.
Step 1
Underline the place to which the number is being rounded.
Example
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.5 - 57
Objective 2: Round integers.
Round 36,833 to the nearest thousand.
Step 2
Because the next digit is 5 or more, add 1 to the underlined digit.
Example
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Objective 2: Round integers.
Round 36,833 to the nearest thousand.
Step 3
Change all digits to the right of the underlined place to zeros.
Example
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.5 - 59
Objective 2: Round integers.
Round 13,961 to the nearest hundred.
Step 1
Example
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.5 - 60
Objective 2: Round integers.
Round 13,961 to the nearest hundred.
Step 2
Example
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Objective 2: Round integers.
Round 13,961 to the nearest hundred.
Step 3
Example
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.5 - 62
Objective 3: Use front end rounding to estimate answers in addition and subtraction.
If you purchase a sofa for $988 and a chair for $209, you may estimate the total cost.
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Objective 3: Use front end rounding to estimate answers in addition and subtraction.
In front end rounding, each number is rounded to the highest possible place, so all the digits become 0 except the first digit.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.5 - 64
Meisha’s paycheck showed gross pay of $823. It also listed deductions of $291. Estimate her net pay.
Example
Objective 3: Use front end rounding to estimate answers in addition and subtraction.
Front end rounding:
Net pay estimate: $800 – $300 = $500.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.5 - 65
Meisha’s paycheck showed gross pay of $823. It also listed deductions of $291. Find her exact net pay.
Example
Objective 3: Use front end rounding to estimate answers in addition and subtraction.
Exact pay:
$823 – $291 = $532
Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 1.6 - 66
Chapter 1Introduction to Algebra:
Integers
1.6 Multiplying Integers
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.6 - 67
1.6 Multiplying Integers
Objectives
1. Use a raised dot or parentheses to express multiplication.
2. Multiply integers.
3. Identify properties of multiplication.
4. Estimate answers to application problems involving multiplication.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.6 - 68
Objective 1: Use a raised dot or parentheses to express multiplication.
Numbers being multiplied are called factors and the answer is called the product.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.6 - 69
Objective 1: Use a raised dot or parentheses to express multiplication.
Rewrite the multiplication in three different ways.
10 × 7
Example
The factors are 10 and 7 and the product is 70.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.6 - 70
Objective 1: Use a raised dot or parentheses to express multiplication.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.6 - 71
Objective 2: Multiply integers.
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Objective 2: Multiply integers.
Multiply the integers.
–2 • 8
Example
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Objective 2: Multiply integers.
Multiply the integers.
–10 (–6 )
Example
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Objective 2: Multiply integers.
Multiply the integers.
–3 • (4 • –5)
Example
Multiply numbers in parentheses first.
Then multiply the resulting pair of numbers.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.6 - 75
Objective 2: Multiply integers.
Multiply the integers.
–2 • –2 • –2
Example
Multiply the first pair of numbers.Then multiply the resulting pair of numbers.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.6 - 76
Objective 3: Identify properties of multiplication.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.6 - 77
Objective 3: Identify properties of multiplication.
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Objective 3: Identify properties of multiplication.
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Objective 3: Identify properties of multiplication.
Illustrate the commutative property for the product below.
–7 • –4
Example
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Objective 3: Identify properties of multiplication.
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Objective 3: Identify properties of multiplication.
Illustrate the associative property for the product below.
5 • 10 • 2
Example
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Objective 3: Identify properties of multiplication.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.6 - 83
Objective 3: Identify properties of multiplication.
Illustrate the distributive property for the product below.
–2(–5 + 1)
Example
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.6 - 84
Objective 4: Estimate answers to application problems involving multiplication.
Last year a video store had to replace 392 defective videos at a cost of $19 each. Estimate the amount of money the store lost on the videos.
Example
Front end rounding: 392 rounds to 400 and –$19 rounds to –$20.
Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 1.7 - 85
Chapter 1Introduction to Algebra:
Integers
1.7 Dividing Integers
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.7 - 86
1.7 Dividing Integers
Objectives
1. Divide integers.
2. Identify properties of division.
3. Combine multiplying and dividing of integers.
4. Estimate answers to application problems involving division.
5. Interpret remainders in division application problems.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.7 - 87
Objective 1: Divide integers.
The answer to a division problem is called the quotient.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.7 - 88
Objective 1: Divide integers.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.7 - 89
Objective 1: Divide integers.
Divide the integers.
(a)
(b)
Example
Different signs,quotient is negative. 4
5
20
5
20
4
24
Same sign,quotient is positive. 6
4
24
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.7 - 90
Objective 2: Identify properties of division.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.7 - 91
Objective 3: Combine multiplying and dividing of integers.
Simplify.
6(–10) ÷ (–3 • 2)
Example
Do operations inside parentheses first.Start at the left and perform operations from left to right.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.7 - 92
Objective 3: Combine multiplying and dividing of integers.
Simplify.
–24 ÷ –2(4) ÷ –6
Example
No operations inside parentheses. Start at the left and perform operations from left to right.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.7 - 93
Objective 4: Estimate answers to application problems involving division.
During a 24-hour lab experiment, the temperature of a solution dropped 96 degrees. Estimate the average drop in temperature each hour.
Example
Front end rounding: –96 degrees rounds to –100
and 24 hours rounds to 20.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.7 - 94
Objective 5: Interpret remainders in division application problems.
The math department a a college has $360 in its budget to buy calculators. If the calculators cost $25 each, how many can be purchased? How much money will be left over?
Example
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.7 - 95
Objective 5: Interpret remainders in division application problems.
Luke needs to rent tents for 135 Scouts going on a camping trip. Each tent sleeps 6 people. How many tents should he rent?
Example
Luke will need to rent 23 tents.
Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 1.8 - 96
Chapter 1Introduction to Algebra:
Integers
1.8 Exponents and Order of Operations
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.8 - 97
1.8 Exponents and Order of Operations
Objectives
1. Use exponents to write repeated factors.
2. Simplify expressions containing exponents.
3. Use the order of operations.
4. Simplify expressions with fraction bars.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.8 - 98
Objective 1: Use exponents to write repeated factors.
An exponent can be used to represent repeated multiplication.
The base is the number being repeatedly multiplied.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.8 - 99
Objective 2: Simplifying expressions containing exponents.
Simplify each expression.
(a) (–5)2
(b) (–5)3
Example
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.8 - 100
Objective 2: Simplifying expressions containing exponents.
Simplify the expression.
23(–3)2
Example
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.8 - 101
Objective 3: Use the order of operations.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.8 - 102
Objective 3: Use the order of operations.
Simplify.
9 + 3(20 – 4) ÷ 8
Example
9 + 3(20 – 4) ÷ 8
9 + 3(16) ÷ 8
9 + 48 ÷ 8
9 + 6
15
Work inside parentheses first.
Work left to right performing multiplication and division.
Add last.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.8 - 103
Objective 3: Use the order of operations.
Simplify.
3 + 2(6 – 8) • (15 ÷ 3)
Example
3 + 2(6 – 8) • (15 ÷ 3)
3 + 2(–2) • (5)
3 + –4 • 5
3 + –20 –17
Work inside parentheses first.
Work left to right performing multiplications.
Add last.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.8 - 104
Objective 3: Use the order of operations.
Simplify.
(–4)3 – (4 – 6)2(–3)
Example
(–4)3 – (4 – 6)2(–3)
(–4)3 – (–2)2(–3)
–64 – 4(–3)
–52
Work inside parentheses first.
Simplify exponents.
Subtract last.–64 – –12
Multiply.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.8 - 105
Objective 3: Use the order of operations.