# Introductory Algebra

### 1

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# CHAPTER 1 Whole Numbers, Integers, and Introduction to Algebra

Algebra has a language of its own. The picture shows just some of the words you may see and use in your study of algebra.

You may not realize it, but you already use algebra every day. Perhaps you figure out how much to tip a server in a restaurant. Maybe you calculate the amount of change you should get when you pay for something. It could even be when you compare batting averages of your favorite players. You can describe the algebra you use in specific words, and follow an orderly process. In this chapter, you will explore the words used to describe algebra and start on your path to solving algebraic problems easily, both in class and in your everyday life.

## 1.1 Whole Numbers

Learning Objectives

By the end of this section, you will be able to:

• Use place value with whole numbers
• Identify multiples and and apply divisibility tests
• Find prime factorization and least common multiples

As we begin our study of intermediate algebra, we need to refresh some of our skills and vocabulary. This chapter and the next will focus on whole numbers, integers, fractions, decimals, and real numbers. We will also begin our use of algebraic notation and vocabulary.

# Use Place Value with Whole Numbers

The most basic numbers used in algebra are the numbers we use to count objects in our world: 1, 2, 3, 4, and so on. These are called the counting numbers. Counting numbers are also called natural numbers. If we add zero to the counting numbers, we get the set of whole numbers.

Counting Numbers: 1, 2, 3, …

Whole Numbers: 0, 1, 2, 3, …

The notation “…” is called ellipsis and means “and so on,” or that the pattern continues endlessly.

We can visualize counting numbers and whole numbers on a number line .See Figure 1.

The numbers on the number line get larger as they go from left to right, and smaller as they go from right to left. While this number line shows only the whole numbers 0 through 6, the numbers keep going without end.
Figure 1

Our number system is called a place value system, because the value of a digit depends on its position in a number. Figure 2 shows the place values. The place values are separated into groups of three, which are called periods. The periods are ones, thousands, millions, billions, trillions, and so on. In a written number, commas separate the periods.

The number 5,278,194 is shown in the chart. The digit 5 is in the millions place. The digit 2 is in the hundred-thousands place. The digit 7 is in the ten-thousands place. The digit 8 is in the thousands place. The digit 1 is in the hundreds place. The digit 9 is in the tens place. The digit 4 is in the ones place.

Figure 2

EXAMPLE 1

In the number 63,407,218, find the place value of each digit:

1. 7
2. 0
3. 1
4. 6
5. 3
Solution

Place the number in the place value chart:

a) The 7 is in the thousands place.
b) The 0 is in the ten thousands place.
c) The 1 is in the tens place.
d) The 6 is in the ten-millions place.
e) The 3 is in the millions place.

TRY IT 1.1

For the number 27,493,615, find the place value of each digit:

a) 2 b) 1 c) 4 d) 7 e) 5

a) ten millions b) tens c) hundred thousands d) millions e) ones

TRY IT 1.2

For the number 519,711,641,328, find the place value of each digit:

a) 9 b) 4 c) 2 d) 6 e) 7

a) billions b) ten thousands c) tens d) hundred thousands e) hundred millions

When you write a check, you write out the number in words as well as in digits. To write a number in words, write the number in each period, followed by the name of the period, without the s at the end. Start at the left, where the periods have the largest value. The ones period is not named. The commas separate the periods, so wherever there is a comma in the number, put a comma between the words (see Figure 3). The number 74,218,369 is written as seventy-four million, two hundred eighteen thousand, three hundred sixty-nine.

Figure 3

HOW TO: Name a Whole Number in Words.

1. Start at the left and name the number in each period, followed by the period name.
2. Put commas in the number to separate the periods.
3. Do not name the ones period.

EXAMPLE 2

Name the number 8,165,432,098,710 using words.

Solution

Name the number in each period, followed by the period name.

Put the commas in to separate the periods.

So, is named as eight trillion, one hundred sixty-five billion, four hundred thirty-two million, ninety-eight thousand, seven hundred ten.

TRY IT 2.1

Name the number using words.

nine trillion, two hundred fifty-eight billion, one hundred thirty-seven million, nine hundred four thousand, sixty-one

TRY IT 2.2

Name the number using words.

seventeen trillion, eight hundred sixty-four billion, three hundred twenty-five million, six hundred nineteen thousand four

We are now going to reverse the process by writing the digits from the name of the number. To write the number in digits, we first look for the clue words that indicate the periods. It is helpful to draw three blanks for the needed periods and then fill in the blanks with the numbers, separating the periods with commas.

HOW TO: Write a Whole Number Using Digits.

1. Identify the words that indicate periods. (Remember, the ones period is never named.)
2. Draw three blanks to indicate the number of places needed in each period. Separate the periods by commas.
3. Name the number in each period and place the digits in the correct place value position.

EXAMPLE 3

Write nine billion, two hundred forty-six million, seventy-three thousand, one hundred eighty-nine as a whole number using digits.

Solution

Identify the words that indicate periods.
Except for the first period, all other periods must have three places. Draw three blanks to indicate the number of places needed in each period. Separate the periods by commas.
Then write the digits in each period.

The number is 9,246,073,189.

TRY IT 3.1

Write the number two billion, four hundred sixty-six million, seven hundred fourteen thousand, fifty-one as a whole number using digits.

TRY IT 3.2

Write the number eleven billion, nine hundred twenty-one million, eight hundred thirty thousand, one hundred six as a whole number using digits.

In 2016, Statistics Canada estimated the population of Toronto as 13,448,494. We could say the population of Toronto was approximately 13.4 million. In many cases, you don’t need the exact value; an approximate number is good enough.

The process of approximating a number is called rounding. Numbers are rounded to a specific place value, depending on how much accuracy is needed. Saying that the population of Toronto is approximately 13.4 million means that we rounded to the hundred thousands place.

EXAMPLE 4

Round 23,658 to the nearest hundred.

Solution

TRY IT 4.1

Round to the nearest hundred: .

TRY IT 4.2

Round to the nearest hundred: .

HOW TO: Round Whole Numbers.

1. Locate the given place value and mark it with an arrow. All digits to the left of the arrow do not change.
2. Underline the digit to the right of the given place value.
3. Is this digit greater than or equal to 5?
• Yes–add to the digit in the given place value.
• No–do not change the digit in the given place value.
4. Replace all digits to the right of the given place value with zeros.

EXAMPLE 5

Round to the nearest:

1. hundred
2. thousand
3. ten thousand
Solution

a)

 Locate the hundreds place in 103,978. Underline the digit to the right of the hundreds place. Since 7 is greater than or equal to 5, add 1 to the 9. Replace all digits to the right of the hundreds place with zeros. So, 104,000 is 103,978 rounded to the nearest hundred.

b)

 Locate the thousands place and underline the digit to the right of the thousands place. Since 9 is greater than or equal to 5, add 1 to the 3. Replace all digits to the right of the hundreds place with zeros. So, 104,000 is 103,978 rounded to the nearest thousand.

c)

 Locate the ten thousands place and underline the digit to the right of the ten thousands place. Since 3 is less than 5, we leave the 0 as is, and then replace the digits to the right with zeros. So, 100,000 is 103,978 rounded to the nearest ten thousand.

TRY IT 5.1

Round 206,981 to the nearest: a) hundred b) thousand c) ten thousand.

a) 207,000 b) 207,000 c) 210,000

TRY IT 5.2

Round 784,951 to the nearest: a) hundred b) thousand c) ten thousand.

a) 785,000 b) 785,000 c) 780,000

# Identify Multiples and Apply Divisibility Tests

The numbers 2, 4, 6, 8, 10, and 12 are called multiples of 2. A multiple of 2 can be written as the product of a counting number and 2

Similarly, a multiple of 3 would be the product of a counting number and 3

We could find the multiples of any number by continuing this process.

The Table 1 below shows the multiples of 2 through 9 for the first 12 counting numbers.

Table 1
Counting Number 1 2 3 4 5 6 7 8 9 10 11 12
Multiples of 2 2 4 6 8 10 12 14 16 18 20 22 24
Multiples of 3 3 6 9 12 15 18 21 24 27 30 33 36
Multiples of 4 4 8 12 16 20 24 28 32 36 40 44 48
Multiples of 5 5 10 15 20 25 30 35 40 45 50 55 60
Multiples of 6 6 12 18 24 30 36 42 48 54 60 66 72
Multiples of 7 7 14 21 28 35 42 49 56 63 70 77 84
Multiples of 8 8 16 24 32 40 48 56 64 72 80 88 96
Multiples of 9 9 18 27 36 45 54 63 72 81 90 99 108
Multiples of 10 10 20 30 40 50 60 70 80 90 100 110 120

Multiple of a Number

A number is a multiple of n if it is the product of a counting number and n.

Another way to say that 15 is a multiple of 3 is to say that 15 is divisible by 3. That means that when we divide 3 into 15, we get a counting number. In fact, is 5, so 15 is .

Divisible by a Number

If a number m is a multiple of n, then m is divisible by n.

Look at the multiples of 5 in Table 1. They all end in 5 or 0. Numbers with last digit of 5 or 0 are divisible by 5. Looking for other patterns in Table 1 that shows multiples of the numbers 2 through 9, we can discover the following divisibility tests:

Divisibility Tests

A number is divisible by:

• 2 if the last digit is 0, 2, 4, 6, or 8.
• 3 if the sum of the digits is divisible by 3.
• 5 if the last digit is 5 or 0.
• 6 if it is divisible by both 2 and 3.
• 10 if it ends with 0.

EXAMPLE 6

Is 5,625 divisible by 2? By 3? By 5? By 6? By 10?

Solution
 Is 5,625 divisible by 2? Does it end in 0,2,4,6, or 8? No. 5,625 is not divisible by 2. Is 5,625 divisible by 3? What is the sum of the digits? Is the sum divisible by 3? Yes. 5,625 is divisble by 3. Is 5,625 divisible by 5 or 10? What is the last digit? It is 5. 5,625 is divisble by 5 but not by 10. Is 5,625 divisible by 6? Is it divisible by both 2 or 3? No, 5,625 is not divisible by 2, so 5,625 is not divisible by 6.

TRY IT 6.1

Determine whether 4,962 is divisible by 2, by 3, by 5, by 6, and by 10

by 2, 3, and 6

TRY IT 6.2

Determine whether 3,765 is divisible by 2, by 3, by 5, by 6, and by 10

by 3 and 5

# Find Prime Factorization and Least Common Multiples

In mathematics, there are often several ways to talk about the same ideas. So far, we’ve seen that if m is a multiple of n, we can say that m is divisible by n. For example, since 72 is a multiple of 8, we say 72 is divisible by 8. Since 72 is a multiple of 9, we say 72 is divisible by 9. We can express this still another way.

Since , we say that 8 and 9 are factors of 72. When we write , we say we have factored 72

Other ways to factor 72 are . Seventy-two has many factors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 36, and 72

Factors

If , then a and b are factors of m.

Some numbers, like 72, have many factors. Other numbers have only two factors.

Prime Number and Composite Number

A prime number is a counting number greater than 1, whose only factors are 1 and itself.

A composite number is a counting number that is not prime. A composite number has factors other than 1 and itself.

The counting numbers from 2 to 19 are listed in Figure 4, with their factors. Make sure to agree with the “prime” or “composite” label for each!

Figure 4

The prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, and 19. Notice that the only even prime number is 2

A composite number can be written as a unique product of primes. This is called the prime factorization of the number. Finding the prime factorization of a composite number will be useful later in this course.

Prime Factorization

The prime factorization of a number is the product of prime numbers that equals the number.

To find the prime factorization of a composite number, find any two factors of the number and use them to create two branches. If a factor is prime, that branch is complete. Circle that prime!

If the factor is not prime, find two factors of the number and continue the process. Once all the branches have circled primes at the end, the factorization is complete. The composite number can now be written as a product of prime numbers.

EXAMPLE 7

Factor 48.
Solution

We say is the prime factorization of 48. We generally write the primes in ascending order. Be sure to multiply the factors to verify your answer!

If we first factored 48 in a different way, for example as , the result would still be the same. Finish the prime factorization and verify this for yourself.

TRY IT 7.1

Find the prime factorization of 80.

TRY IT 7.2

Find the prime factorization of 60.

HOW TO: Find the Prime Factorization of a Composite Number.

1. Find two factors whose product is the given number, and use these numbers to create two branches.
2. If a factor is prime, that branch is complete. Circle the prime, like a bud on the tree.
3. If a factor is not prime, write it as the product of two factors and continue the process.
4. Write the composite number as the product of all the circled primes.

EXAMPLE 8

Find the prime factorization of 252

Solution
 Step 1. Find two factors whose product is 252. 12 and 21 are not prime. Break 12 and 21 into two more factors. Continue until all primes are factored. Step 2. Write 252 as the product of all the circled primes.

TRY IT 8.1

Find the prime factorization of 126

TRY IT 8.2

Find the prime factorization of 294

One of the reasons we look at multiples and primes is to use these techniques to find the least common multiple of two numbers. This will be useful when we add and subtract fractions with different denominators. Two methods are used most often to find the least common multiple and we will look at both of them.

The first method is the Listing Multiples Method. To find the least common multiple of 12 and 18, we list the first few multiples of 12 and 18:

Notice that some numbers appear in both lists. They are the common multiples of 12 and 18

We see that the first few common multiples of 12 and 18 are 36, 72, and 108. Since 36 is the smallest of the common multiples, we call it the least common multiple. We often use the abbreviation LCM.

Least Common Multiple

The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers.

The procedure box lists the steps to take to find the LCM using the prime factors method we used above for 12 and 18

HOW TO: Find the Least Common Multiple by Listing Multiples.

1. List several multiples of each number.
2. Look for the smallest number that appears on both lists.
3. This number is the LCM.

EXAMPLE 9

Find the least common multiple of 15 and 20 by listing multiples.

Solution
 Make lists of the first few multiples of 15 and of 20, and use them to find the least common multiple. Look for the smallest number that appears in both lists. The first number to appear on both lists is 60, so 60 is the least common multiple of 15 and 20.

Notice that 120 is in both lists, too. It is a common multiple, but it is not the least common multiple.

TRY IT 9.1

Find the least common multiple by listing multiples: 9 and 12

36

TRY IT 9.2

Find the least common multiple by listing multiples: 18 and 24

72

Our second method to find the least common multiple of two numbers is to use The Prime Factors Method. Let’s find the LCM of 12 and 18 again, this time using their prime factors.

EXAMPLE 10

Find the Least Common Multiple (LCM) of 12 and 18 using the prime factors method.
Solution

Notice that the prime factors of 12 and the prime factors of 18 are included in the LCM . So 36 is the least common multiple of 12 and 18

By matching up the common primes, each common prime factor is used only once. This way you are sure that 36 is the least common multiple.

TRY IT 10.1

Find the LCM using the prime factors method: 9 and 12

36

TRY IT 10.2

Find the LCM using the prime factors method: 18 and 24

72

HOW TO: Find the Least Common Multiple Using the Prime Factors Method.

1. Write each number as a product of primes.
2. List the primes of each number. Match primes vertically when possible.
3. Bring down the columns.
4. Multiply the factors.

EXAMPLE 11

Find the Least Common Multiple (LCM) of 24 and 36 using the prime factors method.

Solution
 Find the primes of 24 and 36. Match primes vertically when possible.Bring down all columns. Multiply the factors. The LCM of 24 and 36 is 72.

TRY IT 11.1

Find the LCM using the prime factors method: 21 and 28

84

TRY IT 11.2

Find the LCM using the prime factors method: 24 and 32

96

# Key Concepts

• Place Value as in Figure 2.
• Name a Whole Number in Words
1. Start at the left and name the number in each period, followed by the period name.
2. Put commas in the number to separate the periods.
3. Do not name the ones period.
• Write a Whole Number Using Digits
1. Identify the words that indicate periods. (Remember the ones period is never named.)
2. Draw 3 blanks to indicate the number of places needed in each period. Separate the periods by commas.
3. Name the number in each period and place the digits in the correct place value position.
• Round Whole Numbers
1. Locate the given place value and mark it with an arrow. All digits to the left of the arrow do not change.
2. Underline the digit to the right of the given place value.
3. Is this digit greater than or equal to 5?
• Yes—add 1 to the digit in the given place value.
• No—do not change the digit in the given place value.
4. Replace all digits to the right of the given place value with zeros.
• Divisibility Tests: A number is divisible by:
• 2 if the last digit is 0, 2, 4, 6, or 8.
• 3 if the sum of the digits is divisible by 3.
• 5 if the last digit is 5 or 0.
• 6 if it is divisible by both 2 and 3.
• 10 if it ends with 0.
• Find the Prime Factorization of a Composite Number
1. Find two factors whose product is the given number, and use these numbers to create two branches.
2. If a factor is prime, that branch is complete. Circle the prime, like a bud on the tree.
3. If a factor is not prime, write it as the product of two factors and continue the process.
4. Write the composite number as the product of all the circled primes.
• Find the Least Common Multiple by Listing Multiples
1. List several multiples of each number.
2. Look for the smallest number that appears on both lists.
3. This number is the LCM.
• Find the Least Common Multiple Using the Prime Factors Method
1. Write each number as a product of primes.
2. List the primes of each number. Match primes vertically when possible.
3. Bring down the columns.
4. Multiply the factors.

# Glossary

composite number
A composite number is a counting number that is not prime. A composite number has factors other than 1 and itself.
counting numbers
The counting numbers are the numbers 1, 2, 3, …
divisible by a number
If a number is a multiple of , then is divisible by . (If 6 is a multiple of 3, then 6 is divisible by 3.)
factors
If , then are factors of . Since 3 · 4 = 12, then 3 and 4 are factors of 12.
least common multiple
The least common multiple of two numbers is the smallest number that is a multiple of both numbers.
multiple of a number
A number is a multiple of n if it is the product of a counting number and n.
number line
A number line is used to visualize numbers. The numbers on the number line get larger as they go from left to right, and smaller as they go from right to left.
origin
The origin is the point labeled 0 on a number line.
prime factorization
The prime factorization of a number is the product of prime numbers that equals the number.
prime number
A prime number is a counting number greater than 1, whose only factors are 1 and itself.
whole numbers
The whole numbers are the numbers 0, 1, 2, 3, ….

# Practice Makes Perfect

## Use Place Value with Whole Numbers

In the following exercises, find the place value of each digit in the given numbers.

 1. 51,493 a) 1 b) 4 c) 9 d) 5 e) 3 2. 87,210 a) 2 b) 8 c) 0 d) 7 e) 1 3. 164,285 a) 5 b) 6 c) 1 d) 8 e) 2 4. 395,076 a) 5 b) 3 c) 7 d) 0 e) 9 5. 93,285,170 a) 9 b) 8 c) 7 d) 5 e) 3 6. 36,084,215 a) 8 b) 6 c) 5 d) 4 e) 3 7. 7,284,915,860,132 a) 7 b) 4 c) 5 d) 3 e) 0 8. 2,850,361,159,433 a) 9 b) 8 c) 6 d) 4 e) 2

In the following exercises, name each number using words.

 9. 1,078 10. 5,902 11. 364,510 12. 146,023 13. 5,846,103 14. 1,458,398 15. 37,889,005 16. 62,008,465

In the following exercises, write each number as a whole number using digits.

 17. four hundred twelve 18. two hundred fifty-three 19. thirty-five thousand, nine hundred seventy-five 20. sixty-one thousand, four hundred fifteen 21. eleven million, forty-four thousand, one hundred sixty-seven 22. eighteen million, one hundred two thousand, seven hundred eighty-three 23. three billion, two hundred twenty-six million, five hundred twelve thousand, seventeen 24. eleven billion, four hundred seventy-one million, thirty-six thousand, one hundred six

In the following, round to the indicated place value.

 25. Round to the nearest ten. a) 386 b) 2,931 26. Round to the nearest ten. a) 792 b) 5,647 27. Round to the nearest hundred. a) 13,748 b) 391,794 28. Round to the nearest hundred. a) 28,166 b) 481,628 29. Round to the nearest ten. a) 1,492 b) 1,497 30. Round to the nearest ten. a) 2,791 b) 2,795 31. Round to the nearest hundred. a) 63,994 b) 63,940 32. Round to the nearest hundred. a) 49,584 b) 49,548

In the following exercises, round each number to the nearest a) hundred, b) thousand, c) ten thousand.

 33. 392,546 34. 619,348 35. 2,586,991 36. 4,287,965

## Identify Multiples and Factors

In the following exercises, use the divisibility tests to determine whether each number is divisible by 2, 3, 5, 6, and 10

 37. 84 38. 9,696 39. 75 40. 78 41. 900 42. 800 43. 986 44. 942 45. 350 46. 550 47. 22,335 48. 39,075

## Find Prime Factorizations and Least Common Multiples

In the following exercises, find the prime factorization.

 49. 86 50. 78 51. 132 52. 455 53. 693 54. 400 55. 432 56. 627 57. 2,160 58. 2,520

In the following exercises, find the least common multiple of the each pair of numbers using the multiples method.

 59. 8, 12 60. 4, 3 61. 12, 16 62. 30, 40 63. 20, 30 64. 44, 55

In the following exercises, find the least common multiple of each pair of numbers using the prime factors method.

 65. 8, 12 66. 12, 16 67. 28, 40 68. 84, 90 69. 55, 88 70. 60, 72

## Everyday Math

 71. Writing a Check Jorge bought a car for $24,493. He paid for the car with a check. Write the purchase price in words. 72. Writing a Check Marissa’s kitchen remodeling cost$18,549. She wrote a check to the contractor. Write the amount paid in words. 73. Buying a Car Jorge bought a car for $24,493. Round the price to the nearest a) ten b) hundred c) thousand; and d) ten-thousand. 74. Remodeling a Kitchen Marissa’s kitchen remodeling cost$18,549, Round the cost to the nearest a) ten b) hundred c) thousand and d) ten-thousand. 75. Population The population of China was 1,339,724,852 on November 1, 2010. Round the population to the nearest a) billion b) hundred-million; and c) million. 76. Astronomy The average distance between Earth and the sun is 149,597,888 kilometres. Round the distance to the nearest a) hundred-million b) ten-million; and c) million. 77. Grocery Shopping Hot dogs are sold in packages of 10, but hot dog buns come in packs of eight. What is the smallest number that makes the hot dogs and buns come out even? 78. Grocery Shopping Paper plates are sold in packages of 12 and party cups come in packs of eight. What is the smallest number that makes the plates and cups come out even?

## Writing Exercises

 79. What is the difference between prime numbers and composite numbers? 80. Give an everyday example where it helps to round numbers. 81. Explain in your own words how to find the prime factorization of a composite number, using any method you prefer.

 1. a) thousands b) hundreds c) tens d) ten thousands e) ones 3. a) ones b) ten thousands c) hundred thousands d) tens e) hundreds 5. a) ten millions b) ten thousands c) tens d) thousands e) millions 7. a) trillions b) billions c) millions d) tens e) thousands 9. one thousand, seventy-eight 11. three hundred sixty-four thousand, five hundred ten 13. five million, eight hundred forty-six thousand, one hundred three 15. thirty-seven million, eight hundred eighty-nine thousand, five 17. 412 19. 35,975 21. 11,044,167 23. 3,226,512,017 25. a) 390 b) 2,930 27. a) 13,700 b) 391,800 29. a) 1,490 b) 1,500 31. a) 64,000 b) 63,900 33. a) 392,500 b) 393,000 c) 390,000 35. a) 2,587,000 b) 2,587,000 c) 2,590,000 37. divisible by 2, 3, and 6 39. divisible by 3 and 5 41. divisible by 2, 3, 5, 6, and 10 43. divisible by 2 45. divisible by 2, 5, and 10 47. divisible by 3 and 5 49. 51. 53. 55. 57. 59. 24 61. 48 63. 60 65. 24 67. 420 69. 440 71. twenty-four thousand, four hundred ninety-three dollars 73. a) $24,490 b)$24,500 c) $24,000 d)$20,000 75. a) 1,000,000,000 b) 1,300,000,000 c) 1,340,000,000 77. 40 79. Answers may vary. 81. Answers may vary.

This chapter has been adapted from “Introduction to Whole Numbers” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

## 1.2 Use the Language of Algebra

Learning Objectives

By the end of this section, you will be able to:

• Use variables and algebraic symbols
• Identify expressions and equations
• Simplify expressions with exponents
• Simplify expressions using the order of operations

# Use Variables and Algebraic Symbols

Greg and Alex have the same birthday, but they were born in different years. This year Greg is years old and Alex is , so Alex is years older than Greg. When Greg was , Alex was . When Greg is , Alex will be . No matter what Greg’s age is, Alex’s age will always be years more, right?

In the language of algebra, we say that Greg’s age and Alex’s age are variable and the three is a constant. The ages change, or vary, so age is a variable. The years between them always stays the same, so the age difference is the constant.

In algebra, letters of the alphabet are used to represent variables. Suppose we call Greg’s age . Then we could use to represent Alex’s age. See the table below.

Greg’s age Alex’s age

Letters are used to represent variables. Letters often used for variables are .

Variables and Constants

A variable is a letter that represents a number or quantity whose value may change.

A constant is a number whose value always stays the same.

To write algebraically, we need some symbols as well as numbers and variables. There are several types of symbols we will be using. In 1.1 Whole Numbers, we introduced the symbols for the four basic arithmetic operations: addition, subtraction, multiplication, and division. We will summarize them here, along with words we use for the operations and the result.

Operation Notation Say: The result is…
Subtraction the difference of and
Multiplication The product of and
Division divided by The quotient of and

In algebra, the cross symbol, ×, is not used to show multiplication because that symbol may cause confusion. Does mean (three times ) or (three times )? To make it clear, use • or parentheses for multiplication.

We perform these operations on two numbers. When translating from symbolic form to words, or from words to symbolic form, pay attention to the words of or and to help you find the numbers.

• The sum of and means add plus , which we write as .
• The difference of and means subtract minus , which we write as .
• The product of and means multiply times , which we can write as .
• The quotient of and means divide by , which we can write as .

EXAMPLE 1

Translate from algebra to words:

Solution
 a. 12 plus 14 the sum of twelve and fourteen
 b. 30 times 5 the product of thirty and five
 c. 64 divided by 8 the quotient of sixty-four and eight
 d. minus the difference of and

TRY IT 1.1

Translate from algebra to words.

1. 18 plus 11; the sum of eighteen and eleven
2. 27 times 9; the product of twenty-seven and nine
3. 84 divided by 7; the quotient of eighty-four and seven
4. p minus q; the difference of p and q

TRY IT 1.2

Translate from algebra to words.

1. 47 minus 19; the difference of forty-seven and nineteen
2. 72 divided by 9; the quotient of seventy-two and nine
3. m plus n; the sum of m and n
4. 13 times 7; the product of thirteen and seven

When two quantities have the same value, we say they are equal and connect them with an equal sign.

Equality Symbol

The symbol is called the equal sign.

An inequality is used in algebra to compare two quantities that may have different values. The number line can help you understand inequalities. Remember that on the number line the numbers get larger as they go from left to right. So if we know that is greater than , it means that is to the right of on the number line. We use the symbols < and > for inequalities.

Inequality

< is read is less than

is to the left of on the number line

> is read is greater than

is to the right of on the number line

The expressions < > can be read from left-to-right or right-to-left, though in English we usually read from left-to-right. In general,

When we write an inequality symbol with a line under it, such as , it means or . We read this is less than or equal to . Also, if we put a slash through an equal sign, it means not equal.

We summarize the symbols of equality and inequality in the table below.

Algebraic Notation Say
is equal to
is not equal to
< is less than
> is greater than
is less than or equal to
is greater than or equal to

Symbols < and >

The symbols < and > each have a smaller side and a larger side.

smaller side < larger side
larger side > smaller side

The smaller side of the symbol faces the smaller number and the larger faces the larger number.

EXAMPLE 2

Translate from algebra to words:

1. >
2. <
Solution
 a. 20 is less than or equal to 35
 b. 11 is not equal to 15 minus 3
 c. > 9 is greater than 10 divided by 2
 d. < plus 2 is less than 10

TRY IT 2.1

Translate from algebra to words.

1. >
2. <
1. fourteen is less than or equal to twenty-seven
2. nineteen minus two is not equal to eight
3. twelve is greater than four divided by two
4. x minus seven is less than one

TRY IT 2.2

Translate from algebra to words.

1. <
2. >
1. nineteen is greater than or equal to fifteen
2. seven is equal to twelve minus five
3. fifteen divided by three is less than eight
4. y minus three is greater than six

EXAMPLE 3

The information in (Figure 1) compares the fuel economy in miles-per-gallon (mpg) of several cars. Write the appropriate symbol < >. in each expression to compare the fuel economy of the cars.

(credit: modification of work by Bernard Goldbach, Wikimedia Commons)
Figure 1
1. MPG of Prius_____ MPG of Mini Cooper
2. MPG of Versa_____ MPG of Fit
3. MPG of Mini Cooper_____ MPG of Fit
4. MPG of Corolla_____ MPG of Versa
5. MPG of Corolla_____ MPG of Prius
Solution
 a. MPG of Prius____MPG of Mini Cooper Find the values in the chart. 48____27 Compare. 48 > 27 MPG of Prius > MPG of Mini Cooper
 b. MPG of Versa____MPG of Fit Find the values in the chart. 26____27 Compare. 26 < 27 MPG of Versa < MPG of Fit
 c. MPG of Mini Cooper____MPG of Fit Find the values in the chart. 27____27 Compare. 27 = 27 MPG of Mini Cooper = MPG of Fit
 d. MPG of Corolla____MPG of Versa Find the values in the chart. 28____26 Compare. 28 > 26 MPG of Corolla > MPG of Versa
 e. MPG of Corolla____MPG of Prius Find the values in the chart. 28____48 Compare. 28 < 48 MPG of Corolla < MPG of Prius

TRY IT 3.1

Use Figure 1 to fill in the appropriate < >.

1. MPG of Prius_____MPG of Versa
2. MPG of Mini Cooper_____ MPG of Corolla
1. >
2. <

TRY IT 3.2

Use Figure 1 to fill in the appropriate < >.

1. MPG of Fit_____ MPG of Prius
2. MPG of Corolla _____ MPG of Fit
1. <
2. <

Grouping symbols in algebra are much like the commas, colons, and other punctuation marks in written language. They indicate which expressions are to be kept together and separate from other expressions. The table below lists three of the most commonly used grouping symbols in algebra.

 parentheses brackets braces

Here are some examples of expressions that include grouping symbols. We will simplify expressions like these later in this section.

# Identify Expressions and Equations

What is the difference in English between a phrase and a sentence? A phrase expresses a single thought that is incomplete by itself, but a sentence makes a complete statement. “Running very fast” is a phrase, but “The football player was running very fast” is a sentence. A sentence has a subject and a verb.

In algebra, we have expressions and equations. An expression is like a phrase. Here are some examples of expressions and how they relate to word phrases:

Expression Words Phrase
the sum of three and five
minus one the difference of and one
the product of six and seven
divided by the quotient of and

Notice that the phrases do not form a complete sentence because the phrase does not have a verb. An equation is two expressions linked with an equal sign. When you read the words the symbols represent in an equation, you have a complete sentence in English. The equal sign gives the verb. Here are some examples of equations:

Equation Sentence
The sum of three and five is equal to eight.
minus one equals fourteen.
The product of six and seven is equal to forty-two.
is equal to fifty-three.
plus nine is equal to two minus three.

Expressions and Equations

An expression is a number, a variable, or a combination of numbers and variables and operation symbols.

An equation is made up of two expressions connected by an equal sign.

EXAMPLE 4

Determine if each is an expression or an equation:

Solution
 a. This is an equation—two expressions are connected with an equal sign. b. This is an expression—no equal sign. c. This is an expression—no equal sign. d. This is an equation—two expressions are connected with an equal sign.

TRY IT 4.1

Determine if each is an expression or an equation:

1. equation
2. expression

TRY IT 4.2

Determine if each is an expression or an equation:

1. expression
2. equation

# Simplify Expressions with Exponents

To simplify a numerical expression means to do all the math possible. For example, to simplify we’d first multiply to get and then add the to get . A good habit to develop is to work down the page, writing each step of the process below the previous step. The example just described would look like this:

Suppose we have the expression . We could write this more compactly using exponential notation. Exponential notation is used in algebra to represent a quantity multiplied by itself several times. We write as and as . In expressions such as , the is called the base and the is called the exponent. The exponent tells us how many factors of the base we have to multiply.

We say is in exponential notation and is in expanded notation.

Exponential Notation

For any expression is a factor multiplied by itself times if is a positive integer.

The expression is read to the power.

For powers of and , we have special names.

The table below lists some examples of expressions written in exponential notation.

Exponential Notation In Words
to the second power, or squared
to the third power, or cubed
to the fourth power
to the fifth power

EXAMPLE 5

Write each expression in exponential form:

Solution
 a. The base 16 is a factor 7 times. b. The base 9 is a factor 5 times. c. The base is a factor 4 times. d. The base is a factor 8 times.

TRY IT 5.1

Write each expression in exponential form:

415

TRY IT 5.2

Write each expression in exponential form:

79

EXAMPLE 6

Write each exponential expression in expanded form:

Solution

a. The base is and the exponent is , so means

b. The base is and the exponent is , so means

TRY IT 6.1

Write each exponential expression in expanded form:

TRY IT 6.2

Write each exponential expression in expanded form:

To simplify an exponential expression without using a calculator, we write it in expanded form and then multiply the factors.

EXAMPLE 7

Simplify: .

Solution
 Expand the expression. Multiply left to right. Multiply.

TRY IT 7.1

Simplify:

1. 125
2. 1

TRY IT 7.2

Simplify:

1. 49
2. 0

# Simplify Expressions Using the Order of Operations

We’ve introduced most of the symbols and notation used in algebra, but now we need to clarify the order of operations. Otherwise, expressions may have different meanings, and they may result in different values.

For example, consider the expression:

Imagine the confusion that could result if every problem had several different correct answers. The same expression should give the same result. So mathematicians established some guidelines called the order of operations, which outlines the order in which parts of an expression must be simplified.

Order of Operations

When simplifying mathematical expressions perform the operations in the following order:

1. Parentheses and other Grouping Symbols

• Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.

2. Exponents

• Simplify all expressions with exponents.

3. Multiplication and Division

• Perform all multiplication and division in order from left to right. These operations have equal priority.

• Perform all addition and subtraction in order from left to right. These operations have equal priority.

Students often ask, “How will I remember the order?” Here is a way to help you remember: Take the first letter of each key word and substitute the silly phrase.

Please Excuse My Dear Aunt Sally.

 Please Parentheses Excuse Exponents My Dear Multiplication and Division Aunt Sally Addition and Subtraction

It’s good that ‘My Dear’ goes together, as this reminds us that multiplication and division have equal priority. We do not always do multiplication before division or always do division before multiplication. We do them in order from left to right.

Similarly, ‘Aunt Sally’ goes together and so reminds us that addition and subtraction also have equal priority and we do them in order from left to right.

EXAMPLE 8

Simplify the expressions:

Solution
 a. Are there any parentheses? No. Are there any exponents? No. Is there any multiplication or division? Yes. Multiply first. Add.
 b. Are there any parentheses? Yes. Simplify inside the parentheses. Are there any exponents? No. Is there any multiplication or division? Yes. Multiply.

TRY IT 8.1

Simplify the expressions:

1. 2
2. 14

TRY IT 8.2

Simplify the expressions:

1. 35
2. 99

EXAMPLE 9

Simplify:

Solution
 a. Are there any parentheses? No. Are there any exponents? No. Is there any multiplication or division? Yes. Multiply and divide from left to right. Divide. Multiply.
 b. Are there any parentheses? No. Are there any exponents? No. Is there any multiplication or division? Yes. Multiply and divide from left to right. Multiply. Divide.

TRY IT 9.1

Simplify:

18

TRY IT 9.2

Simplify:

9

EXAMPLE 10

Simplify: .

Solution
 Parentheses? Yes, subtract first. Exponents? No. Multiplication or division? Yes. Divide first because we multiply and divide left to right. Any other multiplication or division? Yes. Multiply. Any other multiplication or division? No. Any addition or subtraction? Yes.

TRY IT 10.1

Simplify:

16

TRY IT 10.2

Simplify:

23

When there are multiple grouping symbols, we simplify the innermost parentheses first and work outward.

EXAMPLE 11

.

Solution
 Are there any parentheses (or other grouping symbol)? Yes. Focus on the parentheses that are inside the brackets. Subtract. Continue inside the brackets and multiply. Continue inside the brackets and subtract. The expression inside the brackets requires no further simplification. Are there any exponents? Yes. Simplify exponents. Is there any multiplication or division? Yes. Multiply. Is there any addition or subtraction? Yes. Add. Add.

TRY IT 11.1

Simplify:

86

TRY IT 11.2

Simplify:

1

EXAMPLE 12

Simplify: .

Solution
 If an expression has several exponents, they may be simplified in the same step. Simplify exponents. Divide. Add. Subtract.

TRY IT 12.1

Simplify:

81

TRY IT 12.2

Simplify:

75

# Key Concepts

Operation Notation Say: The result is…
Multiplication The product of and
Subtraction the difference of and
Division divided by The quotient of and
• Equality Symbol
• is read as is equal to
• The symbol is called the equal sign.
• Inequality
• < is read is less than
• is to the left of on the number line
• > is read is greater than
• is to the right of on the number line
Algebraic Notation Say
is equal to
is not equal to
< is less than
> is greater than
is less than or equal to
is greater than or equal to
• Exponential Notation
• For any expression is a factor multiplied by itself times, if is a positive integer.
• means multiply factors of
• The expression of is read to the power.

Order of Operations When simplifying mathematical expressions perform the operations in the following order:

• Parentheses and other Grouping Symbols: Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.
• Exponents: Simplify all expressions with exponents.
• Multiplication and Division: Perform all multiplication and division in order from left to right. These operations have equal priority.
• Addition and Subtraction: Perform all addition and subtraction in order from left to right. These operations have equal priority.

# Glossary

expressions
An expression is a number, a variable, or a combination of numbers and variables and operation symbols.
equation
An equation is made up of two expressions connected by an equal sign.

# Practice Makes Perfect

## Use Variables and Algebraic Symbols

In the following exercises, translate from algebraic notation to words.

 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. < 12. < 13. 14. 15. 16. 17. > 18. > 19. 20. 21. 22.

## Identify Expressions and Equations

In the following exercises, determine if each is an expression or an equation.

 23 24 25 26 27 28 29 30

## Simplify Expressions with Exponents

In the following exercises, write in exponential form.

 31 32 33 34

In the following exercises, write in expanded form.

 35 36 37 38

## Simplify Expressions Using the Order of Operations

In the following exercises, simplify.

 39. a. b. 40. a. b. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64.

## Everyday Math

65. Basketball In the 2014 NBA playoffs, the San Antonio Spurs beat the Miami Heat. The table below shows the heights of the starters on each team. Use this table to fill in the appropriate symbol ( = ,<,  >).

Spurs Height Heat Height
Tim Duncan Rashard Lewis
Boris Diaw LeBron James
Kawhi Leonard Chris Bosh
Danny Green Ray Allen
1. Height of Tim Duncan____Height of Rashard Lewis
2. Height of Boris Diaw____Height of LeBron James
3. Height of Kawhi Leonard____Height of Chris Bosh
4. Height of Tony Parker____Height of Dwyane Wade
5. Height of Danny Green____Height of Ray Allen

66. Elevation In Colorado there are more than mountains with an elevation of over The table shows the ten tallest. Use this table to fill in the appropriate inequality symbol.

Mountain Elevation
Mt. Elbert
Mt. Massive
Mt. Harvard
Blanca Peak
La Plata Peak
Uncompahgre Peak
Crestone Peak
Mt. Lincoln
Grays Peak
Mt. Antero
1. Elevation of La Plata Peak____Elevation of Mt. Antero
2. Elevation of Blanca Peak____Elevation of Mt. Elbert
3. Elevation of Gray’s Peak____Elevation of Mt. Lincoln
4. Elevation of Mt. Massive____Elevation of Crestone Peak
5. Elevation of Mt. Harvard____Elevation of Uncompahgre Peak

## Writing Exercises

 67.Explain the difference between an expression and an equation. 68. Why is it important to use the order of operations to simplify an expression?

 1. 16 minus 9, the difference of sixteen and nine 3. 5 times 6, the product of five and six 5. 28 divided by 4, the quotient of twenty-eight and four 7. x plus 8, the sum of x and eight 9. 2 times 7, the product of two and seven 11. fourteen is less than twenty-one 13. thirty-six is greater than or equal to nineteen 15. 3 times n equals 24, the product of three and n equals twenty-four 17. y minus 1 is greater than 6, the difference of y and one is greater than six 19. 2 is less than or equal to 18 divided by 6; 2 is less than or equal to the quotient of eighteen and six 21. a is not equal to 7 times 4, a is not equal to the product of seven and four 23. equation 25. expression 27. expression 29. equation 31. 37 33. x5 35. 125 37. 256 39. a. 43 b. 55 41. 5 43. 34 45. 58 47. 6 49. 13 51. 4 53. 35 55. 10 57. 41 59. 81 61. 149 63. 50 65. a. > b. c. < d. < e. > 67. Answer may vary.

This chapter has been adapted from “Use the Language of Algebra” in Prealgebra (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

## 1.3 Evaluate, Simplify, and Translate Expressions

Learning Objectives

By the end of this section, you will be able to:

• Evaluate algebraic expressions
• Identify terms, coefficients, and like terms
• Simplify expressions by combining like terms
• Translate word phrases to algebraic expressions

# Evaluate Algebraic Expressions

In the last section, we simplified expressions using the order of operations. In this section, we’ll evaluate expressions—again following the order of operations.

To evaluate an algebraic expression means to find the value of the expression when the variable is replaced by a given number. To evaluate an expression, we substitute the given number for the variable in the expression and then simplify the expression using the order of operations.

EXAMPLE 1

Evaluate when

1.
2.
Solution

a. To evaluate, substitute for in the expression, and then simplify.

When , the expression has a value of .

b. To evaluate, substitute for in the expression, and then simplify.

When , the expression has a value of .

Notice that we got different results for parts a) and b) even though we started with the same expression. This is because the values used for were different. When we evaluate an expression, the value varies depending on the value used for the variable.

TRY IT 1.1

Evaluate:

1.
2.
1.  10
2.  19

TRY IT 1.2

Evaluate:

1.
2.
1.  4
2.  12

EXAMPLE 2

Evaluate

1.
2.
Solution

Remember means times , so means times .

a. To evaluate the expression when , we substitute for , and then simplify.

 Multiply. Subtract.

b. To evaluate the expression when , we substitute for , and then simplify.

 Multiply. Subtract.

Notice that in part a) that we wrote and in part b) we wrote . Both the dot and the parentheses tell us to multiply.

TRY IT 2.1

Evaluate:

1.  13
2.  5

TRY IT 2.2

Evaluate:

1.  8
2.  16

EXAMPLE 3

Evaluate when .

Solution

We substitute for , and then simplify the expression.

 Use the definition of exponent. Multiply.

When , the expression has a value of .

TRY IT 3.1

Evaluate:

.

64

TRY IT 3.2

Evaluate:

.

216

EXAMPLE 4

.

Solution

In this expression, the variable is an exponent.

 Use the definition of exponent. Multiply.

When , the expression has a value of .

TRY IT 4.1

Evaluate:

.

64

TRY IT 4.2

Evaluate:

.

81

EXAMPLE 5

.

Solution

This expression contains two variables, so we must make two substitutions.

 Multiply. Add and subtract left to right.

When and , the expression has a value of .

TRY IT 5.1

Evaluate:

33

TRY IT 5.2

Evaluate:

10

EXAMPLE 6

.

Solution

We need to be careful when an expression has a variable with an exponent. In this expression, means and is different from the expression , which means .

TRY IT 6.1

Evaluate:

.

40

TRY IT 6.2

Evaluate:

.

9

# Identify Terms, Coefficients, and Like Terms

Algebraic expressions are made up of terms. A term is a constant or the product of a constant and one or more variables. Some examples of terms are .

The constant that multiplies the variable(s) in a term is called the coefficient. We can think of the coefficient as the number in front of the variable. The coefficient of the term is . When we write , the coefficient is , since . The table below gives the coefficients for each of the terms in the left column.

Term Coefficient

An algebraic expression may consist of one or more terms added or subtracted. In this chapter, we will only work with terms that are added together. The table below gives some examples of algebraic expressions with various numbers of terms. Notice that we include the operation before a term with it.

Expression Terms

EXAMPLE 7

Identify each term in the expression . Then identify the coefficient of each term.

Solution

The expression has four terms. They are , and .

The coefficient of is .

The coefficient of is .

Remember that if no number is written before a variable, the coefficient is . So the coefficient of is .

The coefficient of a constant is the constant, so the coefficient of is .

TRY IT 7.1

Identify all terms in the given expression, and their coefficients:

The terms are 4x, 3b, and 2. The coefficients are 4, 3, and 2

TRY IT 7.2

Identify all terms in the given expression, and their coefficients:

The terms are 9a, 13a2, and a3, The coefficients are 9, 13, and 1

Some terms share common traits. Look at the following terms. Which ones seem to have traits in common?

Which of these terms are like terms?

• The terms and are both constant terms.
• The terms and are both terms with .
• The terms and both have .

Terms are called like terms if they have the same variables and exponents. All constant terms are also like terms. So among the terms ,

Like Terms

Terms that are either constants or have the same variables with the same exponents are like terms.

EXAMPLE 8

Identify the like terms:

Solution

a.

Look at the variables and exponents. The expression contains , and constants.

The terms and are like terms because they both have .

The terms and are like terms because they both have .

The terms and are like terms because they are both constants.

The term does not have any like terms in this list since no other terms have the variable raised to the power of .

b.

Look at the variables and exponents. The expression contains the terms

The terms and are like terms because they both have .

The terms are like terms because they all have .

The term has no like terms in the given expression because no other terms contain the two variables .

TRY IT 8.1

Identify the like terms in the list or the expression:

9, 15; 2x3 and 8x3, y2, and 11y2

TRY IT 8.2

Identify the like terms in the list or the expression:

4x3 and 6x3; 8x2 and 3x2; 19 and 24

# Simplify Expressions by Combining Like Terms

We can simplify an expression by combining the like terms. What do you think would simplify to? If you thought , you would be right!

We can see why this works by writing both terms as addition problems.

Add the coefficients and keep the same variable. It doesn’t matter what is. If you have of something and add more of the same thing, the result is of them. For example, oranges plus oranges is oranges. We will discuss the mathematical properties behind this later.

The expression has only two terms. When an expression contains more terms, it may be helpful to rearrange the terms so that like terms are together. The Commutative Property of Addition says that we can change the order of addends without changing the sum. So we could rearrange the following expression before combining like terms.

Now it is easier to see the like terms to be combined.

HOW TO: Combine like terms

1. Identify like terms.
2. Rearrange the expression so like terms are together.
3. Add the coefficients of the like terms.

EXAMPLE 9

Simplify the expression: .

Solution
 Identify the like terms. Rearrange the expression, so the like terms are together. Add the coefficients of the like terms. The original expression is simplified to…

TRY IT 9.1

Simplify:

16x + 17

TRY IT 9.2

Simplify:

17y + 7

EXAMPLE 10

Simplify the expression: .

Solution
 Identify the like terms. Rearrange the expression so like terms are together. Add the coefficients of the like terms.

These are not like terms and cannot be combined. So is in simplest form.

TRY IT 10.1

Simplify:

4x2 + 14x

TRY IT 10.2

Simplify:

12y2 + 15y

# Translate Words to Algebraic Expressions

In the previous section, we listed many operation symbols that are used in algebra, and then we translated expressions and equations into word phrases and sentences. Now we’ll reverse the process and translate word phrases into algebraic expressions. The symbols and variables we’ve talked about will help us do that. They are summarized in the table below.

Operation Phrase Expression
the sum of and
increased by
more than
the total of and
Subtraction minus
the difference of and
subtracted from
decreased by
less than
Multiplication times
the product of and
, , ,
Division divided by
the quotient of and
the ratio of and
divided into
, , ,

Look closely at these phrases using the four operations:

• the sum of and
• the difference of and
• the product of and
• the quotient of and

Each phrase tells you to operate on two numbers. Look for the words of and and to find the numbers.

EXAMPLE 11

Translate each word phrase into an algebraic expression:

1.  the difference of and
2.  the quotient of and
Solution

a. The key word is difference, which tells us the operation is subtraction. Look for the words of and and to find the numbers to subtract.

b. The key word is quotient, which tells us the operation is division.

This can also be written as

TRY IT 11.1

Translate the given word phrase into an algebraic expression:

1.  the difference of and
2.  the quotient of and
1.  47 − 41
2.  5x ÷ 2

TRY IT 11.2

Translate the given word phrase into an algebraic expression:

1.  the sum of and
2.  the product of and
1.  17 + 19
2.  7x

How old will you be in eight years? What age is eight more years than your age now? Did you add to your present age? Eight more than means eight added to your present age.

How old were you seven years ago? This is seven years less than your age now. You subtract from your present age. Seven less than means seven subtracted from your present age.

EXAMPLE 12

Translate each word phrase into an algebraic expression:

1.  Eight more than
2.  Seven less than
Solution

a. The key words are more than. They tell us the operation is addition. More than means “added to”.

b. The key words are less than. They tell us the operation is subtraction. Less than means “subtracted from”.

TRY IT 12.1

Translate each word phrase into an algebraic expression:

1.  Eleven more than
2.  Fourteen less than
1. x + 11
2. 11a − 14

TRY IT 12.2

Translate each word phrase into an algebraic expression:

1. more than
2. less than
1. j + 19
2. 2x − 21

EXAMPLE 13

Translate each word phrase into an algebraic expression:

1. five times the sum of and
2. the sum of five times and
Solution

a. There are two operation words: times tells us to multiply and sum tells us to add. Because we are multiplying times the sum, we need parentheses around the sum of and .

five times the sum of and

b. To take a sum, we look for the words of and and to see what is being added. Here we are taking the sum of five times and .

the sum of five times and

Notice how the use of parentheses changes the result. In part a), we add first and in part b), we multiply first.

TRY IT 13.1

Translate the word phrase into an algebraic expression:

1. four times the sum of and
2. the sum of four times and
1. 4(p + q)
2. 4p + q

TRY IT 13.2

Translate the word phrase into an algebraic expression:

1. the difference of two times
2. two times the difference of
1. 2x − 8
2. 2(x − 8)

Later in this course, we’ll apply our skills in algebra to solving equations. We’ll usually start by translating a word phrase to an algebraic expression. We’ll need to be clear about what the expression will represent. We’ll see how to do this in the next two examples.

EXAMPLE 14

The height of a rectangular window is inches less than the width. Let represent the width of the window. Write an expression for the height of the window.

Solution
 Write a phrase about the height. less than the width Substitute for the width. less than Rewrite ‘less than’ as ‘subtracted from’. subtracted from Translate the phrase into algebra.

TRY IT 14.1

The length of a rectangle is inches less than the width. Let represent the width of the rectangle. Write an expression for the length of the rectangle.

w − 5

TRY IT 14.2

The width of a rectangle is metres greater than the length. Let represent the length of the rectangle. Write an expression for the width of the rectangle.

l + 2

EXAMPLE 15

Blanca has dimes and quarters in her purse. The number of dimes is less than times the number of quarters. Let represent the number of quarters. Write an expression for the number of dimes.

Solution
 Write a phrase about the number of dimes. two less than five times the number of quarters Substitute for the number of quarters. less than five times Translate times . less than Translate the phrase into algebra.

TRY IT 15.1

Geoffrey has dimes and quarters in his pocket. The number of dimes is seven less than six times the number of quarters. Let represent the number of quarters. Write an expression for the number of dimes.

6q − 7

TRY IT 15.2

Lauren has dimes and nickels in her purse. The number of dimes is eight more than four times the number of nickels. Let represent the number of nickels. Write an expression for the number of dimes.

4n + 8

# Key Concepts

• Combine like terms.
1. Identify like terms.
2. Rearrange the expression so like terms are together.
3. Add the coefficients of the like terms

# Glossary

term
A term is a constant or the product of a constant and one or more variables.
coefficient
The constant that multiplies the variable(s) in a term is called the coefficient.
like terms
Terms that are either constants or have the same variables with the same exponents are like terms.
evaluate
To evaluate an algebraic expression means to find the value of the expression when the variable is replaced by a given number.

# Practice Makes Perfect

## Evaluate Algebraic Expressions

In the following exercises, evaluate the expression for the given value.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

## Identify Terms, Coefficients, and Like Terms

In the following exercises, list the terms in the given expression.

 21 22 23 24

In the following exercises, identify the coefficient of the given term.

 25 26 27 28

In the following exercises, identify all sets of like terms.

 29 30 31 32

## Simplify Expressions by Combining Like Terms

In the following exercises, simplify the given expression by combining like terms.

 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

## Translate English Phrases into Algebraic Expressions

In the following exercises, translate the given word phrase into an algebraic expression.

 49. The sum of 8 and 12 50. The sum of 9 and 1 51. The difference of 14 and 9 52. 8 less than 19 53. The product of 9 and 7 54. The product of 8 and 7 55. The quotient of 36 and 9 56. The quotient of 42 and 7 57. The difference of and 58. less than 59. The product of and 60. The product of and 61. The sum of and 62. The sum of and 63. The quotient of and 64. The quotient of and 65. Eight times the difference of and nine 66. Seven times the difference of and one 67. Five times the sum of and 68.  times five less than twice

In the following exercises, write an algebraic expression.

 69. Adele bought a skirt and a blouse. The skirt cost $15 more than the blouse. Let represent the cost of the blouse. Write an expression for the cost of the skirt. 70. Eric has rock and classical CDs in his car. The number of rock CDs is more than the number of classical CDs. Let represent the number of classical CDs. Write an expression for the number of rock CDs. 71. The number of girls in a second-grade class is less than the number of boys. Let represent the number of boys. Write an expression for the number of girls. 72. Marcella has fewer male cousins than female cousins. Let represent the number of female cousins. Write an expression for the number of boy cousins. 73. Greg has nickels and pennies in his pocket. The number of pennies is seven less than twice the number of nickels. Let represent the number of nickels. Write an expression for the number of pennies. 74. Jeannette has$5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let represent the number of tens. Write an expression for the number of fives. ## Everyday Math In the following exercises, use algebraic expressions to solve the problem.  75. Car insurance Justin’s car insurance has a$750 deductible per incident. This means that he pays $750 and his insurance company will pay all costs beyond$750. If Justin files a claim for $2,100, how much will he pay, and how much will his insurance company pay? 76. Home insurance Pam and Armando’s home insurance has a$2,500 deductible per incident. This means that they pay $2,500 and their insurance company will pay all costs beyond$2,500. If Pam and Armando file a claim for $19,400, how much will they pay, and how much will their insurance company pay? ## Writing Exercises  77. Explain why “the sum of x and y” is the same as “the sum of y and x,” but “the difference of x and y” is not the same as “the difference of y and x.” Try substituting two random numbers for and to help you explain. 78. Explain the difference between times the sum of and and “the sum of times and # Answers  1. 22 3. 26 5. 144 7. 32 9. 27 11. 21 13. 41 15. 9 17. 73 19. 54 21. 15x2, 6x, 2 23. 10y3, y, 2 25. 8 27. 5 29. x3, 8x3 and 14, 5 31. 16ab and 4ab; 16b2 and 9b2 33. 13x 35. 26a 37. 7c 39. 12x + 8 41. 10u + 3 43. 12p + 10 45. 22a + 1 47. 17x2 + 20x + 16 49. 8 + 12 51. 14 − 9 53. 9 ⋅ 7 55. 36 ÷ 9 57. x − 4 59. 6y 61. 8x + 3x 63. 65. 8 (y − 9) 67. 5 (x + y) 69. b + 15 71. b − 4 73. 2n − 7 75. He will pay$750. His insurance company will pay $1350. 77. Answers will vary. # Attributions This chapter has been adapted from “Evaluate, Simplify, and Translate Expressions” in Prealgebra (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information. ### 4 ## 1.4 Add and Subtract Integers Learning Objectives By the end of this section, you will be able to: • Use negatives and opposites • Simplify: expressions with absolute value • Add integers • Subtract integers # Use Negatives and Opposites Our work so far has only included the counting numbers and the whole numbers. But if you have ever experienced a temperature below zero or accidentally overdrawn your checking account, you are already familiar with negative numbers. Negative numbers are numbers less than . The negative numbers are to the left of zero on the number line. See Figure 1. Figure 1 The number line shows the location of positive and negative numbers. The arrows on the ends of the number line indicate that the numbers keep going forever. There is no biggest positive number, and there is no smallest negative number. Is zero a positive or a negative number? Numbers larger than zero are positive, and numbers smaller than zero are negative. Zero is neither positive nor negative. Consider how numbers are ordered on the number line. Going from left to right, the numbers increase in value. Going from right to left, the numbers decrease in value. See Figure 2. Figure 2 The numbers on a number line increase in value going from left to right and decrease in value going from right to left. Remember that we use the notation: a < b (read “a is less than b”) when a is to the left of b on the number line. a > b (read “a is greater than b”) when a is to the right of b on the number line. Now we need to extend the number line which showed the whole numbers to include negative numbers, too. The numbers marked by points in Figure 3 are called the integers. The integers are the numbers Figure 3 All the marked numbers are called integers. EXAMPLE 1 Order each of the following pairs of numbers, using < or >: a) ___ b) ___ c) ___ d) ___. Solution It may be helpful to refer to the number line shown.  a) 14 is to the right of 6 on the number line. ___ > b) −1 is to the left of 9 on the number line. ___ c) −1 is to the right of −4 on the number line. ___ d) 2 is to the right of −20 on the number line. ___ > TRY IT 1.1 Order each of the following pairs of numbers, using < or > a) ___ b) ___ c) ___ d) ___. Show answer a) > b) < c) > d) > TRY IT 1.2 Order each of the following pairs of numbers, using < or > a) ___ b) ___ c) ___ d) ___. Show answer a) < b) > c) < d) > You may have noticed that, on the number line, the negative numbers are a mirror image of the positive numbers, with zero in the middle. Because the numbers 2 and are the same distance from zero, they are called opposites. The opposite of 2 is , and the opposite of is 2 Opposite The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero. (Figure 4) illustrates the definition. The opposite of 3 is . Figure 4 Sometimes in algebra the same symbol has different meanings. Just like some words in English, the specific meaning becomes clear by looking at how it is used. You have seen the symbol “−” used in three different ways.  Between two numbers, it indicates the operation of subtraction. We read as “10 minus 4.” In front of a number, it indicates a negative number. We read −8 as “negative eight.” In front of a variable, it indicates the opposite. We read as “the opposite of .” Here there are two “−” signs. The one in the parentheses tells us the number is negative 2. The one outside the parentheses tells us to take the opposite of −2. We read as “the opposite of negative two.” Opposite Notation means the opposite of the number a. The notation is read as “the opposite of a.” EXAMPLE 2 Find: a) the opposite of 7 b) the opposite of c) . Solution  a) −7 is the same distance from 0 as 7, but on the opposite side of 0. The opposite of 7 is −7. b) 10 is the same distance from 0 as −10, but on the opposite side of 0. The opposite of −10 is 10. c) −(−6) The opposite of −(−6) is −6. TRY IT 2.1 Find: a) the opposite of 4 b) the opposite of c) . Show answer a) b) 3 c) 1 TRY IT 2.2 Find: a) the opposite of 8 b) the opposite of c) . Show answer a) b) 5 c) 5 Our work with opposites gives us a way to define the integers.The whole numbers and their opposites are called the integers. The integers are the numbers Integers The whole numbers and their opposites are called the integers. The integers are the numbers When evaluating the opposite of a variable, we must be very careful. Without knowing whether the variable represents a positive or negative number, we don’t know whether is positive or negative. We can see this in Example 3. EXAMPLE 3 Evaluate a) , when b) , when . Solution 1.  −x Write the opposite of 8. 2.  −x Write the opposite of −8. 8 TRY IT 3.1 Evaluate , when a) b) . Show answer a) b) 4 TRY IT 3.2 Evaluate , when a) b) . Show answer a) b) 11 # Simplify: Expressions with Absolute Value We saw that numbers such as are opposites because they are the same distance from 0 on the number line. They are both two units from 0. The distance between 0 and any number on the number line is called the absolute value of that number. Absolute Value The absolute value of a number is its distance from 0 on the number line. The absolute value of a number n is written as . For example, • units away from , so . • units away from , so . Figure 5 illustrates this idea. The integers units away from . Figure 5 The absolute value of a number is never negative (because distance cannot be negative). The only number with absolute value equal to zero is the number zero itself, because the distance from on the number line is zero units. Property of Absolute Value for all numbers Absolute values are always greater than or equal to zero! Mathematicians say it more precisely, “absolute values are always non-negative.” Non-negative means greater than or equal to zero. EXAMPLE 4 Simplify: a) b) c) . Solution The absolute value of a number is the distance between the number and zero. Distance is never negative, so the absolute value is never negative. a) b) c) TRY IT 4.1 Simplify: a) b) c) . Show answer a) 4 b) 28 c) 0 TRY IT 4.2 Simplify: a) b) c) . Show answer a) 13 b) 47 c) 0 In the next example, we’ll order expressions with absolute values. Remember, positive numbers are always greater than negative numbers! EXAMPLE 5 Fill in <, >, for each of the following pairs of numbers: a) ___ b) ___c) ___ d) –___ Solution  ___ – a) Simplify. Order. 5 ___ -5 5 > -5 > – b) Simplify. Order. ___ – 8 ___ -8 8 > -8 8 > – c) Simplify. Order. 9 ___ – -9 ___ -9 -9 = -9 -9 = – d) Simplify. Order. – ___ – 16 ____ -16 16 > -16 – > – TRY IT 5.1 Fill in <, >, or for each of the following pairs of numbers: a) ___- b) ___- c) ___ d) ___. Show answer a) > b) > c) < d) > TRY IT 5.2 Fill in <, >, or for each of the following pairs of numbers: a) ___- b) ___- c) ___ d) ___. Show answer a) > b) > c) > d) < We now add absolute value bars to our list of grouping symbols. When we use the order of operations, first we simplify inside the absolute value bars as much as possible, then we take the absolute value of the resulting number.  Parentheses ( ) Brackets [ ] Braces { } Absolute value | | In the next example, we simplify the expressions inside absolute value bars first, just like we do with parentheses. EXAMPLE 6 Simplify: . Solution  Work inside parentheses first: subtract 2 from 6. Multiply 3(4). Subtract inside the absolute value bars. Take the absolute value. Subtract. TRY IT 6.1 Simplify: . Show answer 16 TRY IT 6.2 Simplify: . Show answer 9 EXAMPLE 7 Evaluate: a) b) c) d) . Solution a)  Take the absolute value. 35 b)  Simplify. Take the absolute value. 20 c)  Take the absolute value. d)  Take the absolute value. TRY IT 7.1 Evaluate: a) b) c) d) . Show answer a) b) c) d) TRY IT 7.2 Evaluate: a) b) c) d) . Show answer a) b) c) d) # Add Integers Most students are comfortable with the addition and subtraction facts for positive numbers. But doing addition or subtraction with both positive and negative numbers may be more challenging. We will use two colour counters to model addition and subtraction of negatives so that you can visualize the procedures instead of memorizing the rules. We let one colour (blue) represent positive. The other colour (red) will represent the negatives. If we have one positive counter and one negative counter, the value of the pair is zero. They form a neutral pair. The value of this neutral pair is zero. We will use the counters to show how to add the four addition facts using the numbers and . To add , we realize that means the sum of 5 and 3  We start with 5 positives. And then we add 3 positives. We now have 8 positives. The sum of 5 and 3 is 8. Now we will add . Watch for similarities to the last example . To add , we realize this means the sum of .  We start with 5 negatives. And then we add 3 negatives. We now have 8 negatives. The sum of −5 and −3 is −8. In what ways were these first two examples similar? • The first example adds 5 positives and 3 positives—both positives. • The second example adds 5 negatives and 3 negatives—both negatives. In each case we got 8—either 8 positives or 8 negatives. When the signs were the same, the counters were all the same color, and so we added them. EXAMPLE 8 Add: a) b) . Solution a) 1 positive plus 4 positives is 5 positives. b) 1 negative plus 4 negatives is 5 negatives. TRY IT 8.1 Add: a) b) . Show answer a) 6 b) TRY IT 8.2 Add: a) b) . Show answer a) 7 b) So what happens when the signs are different? Let’s add . We realize this means the sum of and 3. When the counters were the same color, we put them in a row. When the counters are a different color, we line them up under each other.  −5 + 3 means the sum of −5 and 3. We start with 5 negatives. And then we add 3 positives. We remove any neutral pairs. We have 2 negatives left. The sum of −5 and 3 is −2. −5 + 3 = −2 Notice that there were more negatives than positives, so the result was negative. Let’s now add the last combination, .  5 + (−3) means the sum of 5 and −3. We start with 5 positives. And then we add 3 negatives. We remove any neutral pairs. We have 2 positives left. The sum of 5 and −3 is 2. 5 + (−3) = 2 When we use counters to model addition of positive and negative integers, it is easy to see whether there are more positive or more negative counters. So we know whether the sum will be positive or negative. EXAMPLE 9 Add: a) b) . Solution a)  −1 + 5 There are more positives, so the sum is positive. 4 b)  1 + (−5) There are more negatives, so the sum is negative. −4 TRY IT 9.1 Add: a) b) . Show answer a) 2 b) TRY IT 9.2 Add: a) b) . Show answer a) 3 b) Now that we have added small positive and negative integers with a model, we can visualize the model in our minds to simplify problems with any numbers. When you need to add numbers such as , you really don’t want to have to count out 37 blue counters and 53 red counters. With the model in your mind, can you visualize what you would do to solve the problem? Picture 37 blue counters with 53 red counters lined up underneath. Since there would be more red (negative) counters than blue (positive) counters, the sum would be negative. How many more red counters would there be? Because , there are 16 more red counters. Therefore, the sum of is . Let’s try another one. We’ll add . Again, imagine 74 red counters and 27 more red counters, so we’d have 101 red counters. This means the sum is . Let’s look again at the results of adding the different combinations of and . Addition of Positive and Negative Integers When the signs are the same, the counters would be all the same color, so add them. When the signs are different, some of the counters would make neutral pairs, so subtract to see how many are left. Visualize the model as you simplify the expressions in the following examples. EXAMPLE 10 Simplify: a) b) . Solution 1. Since the signs are different, we subtract The answer will be negative because there are more negatives than positives. 2. Since the signs are the same, we add. The answer will be negative because there are only negatives. TRY IT 10.1 Simplify: a) b) . Show answer a) b) TRY IT 10.2 Simplify: a) b) . Show answer a) b) The techniques used up to now extend to more complicated problems, like the ones we’ve seen before. Remember to follow the order of operations! EXAMPLE 11 Simplify: . Solution  Simplify inside the parentheses. Multiply. Add left to right. TRY IT 11.1 Simplify: . Show answer 13 TRY IT 11.2 Simplify: . Show answer 0 # Subtract Integers We will continue to use counters to model the subtraction. Remember, the blue counters represent positive numbers and the red counters represent negative numbers. Perhaps when you were younger, you read as take away When you use counters, you can think of subtraction the same way! We will model the four subtraction facts using the numbers and . To subtract , we restate the problem as take away  We start with 5 positives. We ‘take away’ 3 positives. We have 2 positives left. The difference of 5 and 3 is 2. 2 Now we will subtract . Watch for similarities to the last example . To subtract , we restate this as take away  We start with 5 negatives. We ‘take away’ 3 negatives. We have 2 negatives left. The difference of −5 and −3 is −2. −2 Notice that these two examples are much alike: The first example, we subtract 3 positives from 5 positives and end up with 2 positives. In the second example, we subtract 3 negatives from 5 negatives and end up with 2 negatives. Each example used counters of only one color, and the “take away” model of subtraction was easy to apply. EXAMPLE 12 Subtract: a) b) . Solution  a) Take 5 positive from 7 positives and get 2 positives. b) Take 5 negatives from 7 negatives and get 2 negatives. TRY IT 12.1 Subtract: a) b) . Show answer a) 2 b) TRY IT 12.2 Subtract: a) b) . Show answer a) 3 b) What happens when we have to subtract one positive and one negative number? We’ll need to use both white and red counters as well as some neutral pairs. Adding a neutral pair does not change the value. It is like changing quarters to nickels—the value is the same, but it looks different. • To subtract , we restate it as take away 3. We start with 5 negatives. We need to take away 3 positives, but we do not have any positives to take away. Remember, a neutral pair has value zero. If we add 0 to 5 its value is still 5. We add neutral pairs to the 5 negatives until we get 3 positives to take away.  −5 − 3 means −5 take away 3. We start with 5 negatives. We now add the neutrals needed to get 3 positives. We remove the 3 positives. We are left with 8 negatives. The difference of −5 and 3 is −8. −5 − 3 = −8 And now, the fourth case, . We start with 5 positives. We need to take away 3 negatives, but there are no negatives to take away. So we add neutral pairs until we have 3 negatives to take away.  5 − (−3) means 5 take away −3. We start with 5 positives. We now add the needed neutrals pairs. We remove the 3 negatives. We are left with 8 positives. The difference of 5 and −3 is 8. 5 − (−3) = 8 EXAMPLE 13 Subtract: a) b) . Solution a)  Take 1 positive from the one added neutral pair. −3 − 1 −4 b)  Take 1 negative from the one added neutral pair. 3 − (−1) 4 TRY IT 13.1 Subtract: a) b) . Show answer a) b) 10 TRY IT 13.2 Subtract: a) b) . Show answer a) b) 11 Have you noticed that subtraction of signed numbers can be done by adding the opposite? In Example 13, is the same as and is the same as . You will often see this idea, the subtraction property, written as follows: Subtraction Property Subtracting a number is the same as adding its opposite. Look at these two examples. . Of course, when you have a subtraction problem that has only positive numbers, like , you just do the subtraction. You already knew how to subtract long ago. But knowing that gives the same answer as helps when you are subtracting negative numbers. Make sure that you understand how and give the same results! EXAMPLE 14 Simplify: a) and b) and . Solution  a) Subtract. b) Subtract. TRY IT 14.1 Simplify: a) and b) and . Show answer a) b) TRY IT 14.2 Simplify: a) and b) and . Show answer a) b) Look at what happens when we subtract a negative. Subtracting a negative number is like adding a positive! You will often see this written as . Does that work for other numbers, too? Let’s do the following example and see. EXAMPLE 15 Simplify: a) and b) and . Solution a) b)  a) Subtract. b) Subtract. TRY IT 15.1 Simplify: a) and b) and . Show answer a) b) TRY IT 15.2 Simplify: a) and b) and . Show answer a) 23 b) 3 Let’s look again at the results of subtracting the different combinations of and . Subtraction of Integers When there would be enough counters of the colour to take away, subtract. When there would be not enough counters of the colour to take away, add. What happens when there are more than three integers? We just use the order of operations as usual. EXAMPLE 16 Simplify: . Solution  Simplify inside the parentheses first. Subtract left to right. Subtract. TRY IT 16.1 Simplify: . Show answer 3 TRY IT 16.2 Simplify: . Show answer 13 Access these online resources for additional instruction and practice with adding and subtracting integers. You will need to enable Java in your web browser to use the applications. # Key Concepts • Addition of Positive and Negative Integers • Property of Absolute Value: for all numbers. Absolute values are always greater than or equal to zero! • Subtraction of Integers • Subtraction Property: Subtracting a number is the same as adding its opposite. # Glossary absolute value The absolute value of a number is its distance from 0 on the number line. The absolute value of a number is written as . integers The whole numbers and their opposites are called the integers: …−3, −2, −1, 0, 1, 2, 3… opposite The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero: means the opposite of the number. The notation is read “the opposite of .” # Practice Makes Perfect ## Use Negatives and Opposites of Integers In the following exercises, order each of the following pairs of numbers, using < or >.  1. a) ___ b) ___ c) ___ d) ___ 2. a) ___ b) ___ c) ___ d) ___ In the following exercises, find the opposite of each number.  3. a) 2 b) 4. a) 9 b) In the following exercises, simplify.  5 6 7 8 In the following exercises, evaluate.  9. when a) b) 10. when a) b) ## Simplify Expressions with Absolute Value In the following exercises, simplify.  11. a) b) c) 12. a) b) c) In the following exercises, fill in <, >, or for each of the following pairs of numbers.  13. a) ___ b) ___ 14. a) ___ b) ___ In the following exercises, simplify.  15 16 17 18 19 20 21 22 In the following exercises, evaluate.  23. a) b) 24. a) b) ## Add Integers In the following exercises, simplify each expression.  25 26. 27 28. 29 30. 31 32. 6 33 34. ## Subtract Integers In the following exercises, simplify.  35. 36. 37. 38. 39. 40. 41. a) b) 42. a) b) 43. a) b) 44. a) b) In the following exercises, simplify each expression.  45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 ## Everyday Math 71. Elevation The highest elevation in North America is Mount McKinley, Alaska, at 20,320 feet above sea level. The lowest elevation is Death Valley, California, at 282 feet below sea level. Use integers to write the elevation of: a) Mount McKinley. b) Death Valley. 72. Extreme temperatures The highest recorded temperature on Earth was ° Celsius, recorded in the Sahara Desert in 1922. The lowest recorded temperature was ° below ° Celsius, recorded in Antarctica in 1983 Use integers to write the: a) highest recorded temperature. b) lowest recorded temperature. 73. Provincial budgets For 2019 the province of Quebec estimated it would have a budget surplus of$5.6 million. That same year, Alberta estimated it would have a budget deficit of $7.5 million. Use integers to write the budget of: a) Quebec. b) Alberta. 74. University enrolmentsThe number of international students enrolled in Canadian postsecondary institutions has been on the rise for two decades, with their numbers increasing at a higher rate than that of Canadian students. Enrolments of international students rose by 24,315 from 2015 to 2017. Meanwhile, there was a slight decline in the number of Canadian students, by 912 for the same fiscal years. Use integers to write the change: a) in International Student enrolment from Fall 2015 to Fall 2017. b) in Canadian student enrolment from Fall 2015 to Fall 2017. 75. Stock Market The week of September 15, 2008 was one of the most volatile weeks ever for the US stock market. The closing numbers of the Dow Jones Industrial Average each day were:  Monday Tuesday Wednesday Thursday Friday What was the overall change for the week? Was it positive or negative? 76. Stock Market During the week of June 22, 2009, the closing numbers of the Dow Jones Industrial Average each day were:  Monday Tuesday Wednesday Thursday Friday What was the overall change for the week? Was it positive or negative? ## Writing Exercises  77. Give an example of a negative number from your life experience. 78. What are the three uses of the sign in algebra? Explain how they differ. 79. Explain why the sum of and 2 is negative, but the sum of 8 and is positive. 80. Give an example from your life experience of adding two negative numbers. # Answers  1. a) > b) < c) < d) > 3. a)b) 6 5. 4 7. 15 9. a)b) 12 11. a) 32 b) 0 c) 16 13. a) < b) 15. 17. 56 19. 0 21. 8 23. a) b) 25. 27. 32 29. 31. 108 33. 29 35. 6 37. 39. 12 41. a) 16 b) 16 43. a) 45 b) 45 45. 27 47. 49. 51. 53. -99 55. 57. 59. 22 61. 63. 0 65. 6 67. 69. 71. a) 20,320 b) 73. a)$5.6 million b) million 75. 77. Answers may vary 79. Answers may vary

This chapter has been adapted from “Add and Subtract Integers” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

## 1.5 Multiply and Divide Integers

Learning Objectives

By the end of this section, you will be able to:

• Multiply integers
• Divide integers
• Simplify expressions with integers
• Evaluate variable expressions with integers
• Translate English phrases to algebraic expressions
• Use integers in applications

# Multiply Integers

Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we will use the model just to help us discover the pattern.

We remember that means add a, b times. Here, we are using the model just to help us discover the pattern.

The next two examples are more interesting.

What does it mean to multiply 5 by It means subtract 5, 3 times. Looking at subtraction as “taking away,” it means to take away 5, 3 times. But there is nothing to take away, so we start by adding neutral pairs on the workspace. Then we take away 5 three times.

In summary:

Notice that for multiplication of two signed numbers, when the:

• signs are the same, the product is positive.
• signs are different, the product is negative.

We’ll put this all together in the chart below

Multiplication of Signed Numbers

For multiplication of two signed numbers:

 Same signs Product Example Two positives Two negatives Positive Positive
 Different signs Product Example Positive \cdot negative Negative \cdot positive Negative Negative

EXAMPLE 1

Multiply: a) b) c) d) .

Solution
 a) Multiply, noting that the signs are different so the product is negative. b) Multiply, noting that the signs are the same so the product is positive. c) Multiply, with different signs. d) Multiply, with same signs.

TRY IT 1.1

Multiply: a) b) c) d) .

a) b) 28 c) d) 60

TRY IT 1.2

Multiply: a) b) c) d) .

a) b) 54 c) d) 39

When we multiply a number by 1, the result is the same number. What happens when we multiply a number by ? Let’s multiply a positive number and then a negative number by to see what we get.

Each time we multiply a number by , we get its opposite!

Multiplication by

Multiplying a number by gives its opposite.

EXAMPLE 2

Multiply: a) b) .

Solution
 a) Multiply, noting that the signs are different so the product is negative. b) Multiply, noting that the signs are the same so the product is positive.

TRY IT 2.1

Multiply: a) b) .

a) b) 17

TRY IT 2.2

Multiply: a) b) .

a) b) 16

# Divide Integers

What about division? Division is the inverse operation of multiplication. So, because . In words, this expression says that 15 can be divided into three groups of five each because adding five three times gives 15. Look at some examples of multiplying integers, to figure out the rules for dividing integers.

Division follows the same rules as multiplication!

For division of two signed numbers, when the:

• signs are the same, the quotient is positive.
• signs are different, the quotient is negative.

And remember that we can always check the answer of a division problem by multiplying.

Multiplication and Division of Signed Numbers

For multiplication and division of two signed numbers:

• If the signs are the same, the result is positive.
• If the signs are different, the result is negative.
Same signs Result
Two positives Positive
Two negatives Positive

If the signs are the same, the result is positive.

Different signs Result
Positive and negative Negative
Negative and positive Negative

If the signs are different, the result is negative.

EXAMPLE 3

Divide: a) b) .

Solution
 a) Divide. With different signs, the quotient is negative. b) Divide. With signs that are the same, the quotient is positive.

TRY IT 3.1

Divide: a) b) .

a) b) 39

TRY IT 3.2

Divide: a) b) .

a) b) 23

# Simplify Expressions with Integers

What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember My Dear Aunt Sally?

Let’s try some examples. We’ll simplify expressions that use all four operations with integers—addition, subtraction, multiplication, and division. Remember to follow the order of operations.

EXAMPLE 4

Simplify: .

Solution

TRY IT 4.1

Simplify: .

TRY IT 4.2

Simplify: .

EXAMPLE 5

Simplify: a) b) .

Solution
 a) Write in expanded form. Multiply. Multiply. Multiply. b) Write in expanded form. We are asked to find the opposite of. Multiply. Multiply. Multiply.

Notice the difference in parts a) and b). In part a), the exponent means to raise what is in the parentheses, the to the power. In part b), the exponent means to raise just the 2 to the power and then take the opposite.

TRY IT 5.1

Simplify: a) b) .

a) 81 b)

TRY IT 5.2

Simplify: a) b) .

a) 49 b)

The next example reminds us to simplify inside parentheses first.

EXAMPLE 6

Simplify: .

Solution
 Subtract in parentheses first. Multiply. Subtract.

TRY IT 6.1

Simplify: .

29

TRY IT 6.2

Simplify: .

52

EXAMPLE 7

Simplify: .

Solution
 Exponents first. Multiply. Divide.

TRY IT 7.1

Simplify: .

4

TRY IT 7.2

Simplify: .

9

EXAMPLE 8

Simplify: .

Solution
 Multiply and divide left to right, so divide first. Multiply. Add.

TRY IT 8.1

Simplify: .

21

TRY IT 8.2

Simplify: .

6

# Evaluate Variable Expressions with Integers

Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.

EXAMPLE 9

When , evaluate: a) b) .

Solution

a)

 Simplify. −4

b)

TRY IT 9.1

When , evaluate a) b) .

a) b) 10

TRY IT 9.2

When , evaluate a) b) .

a) b) 17

EXAMPLE 10

Evaluate when and .

Solution
 Add inside parenthesis. (6)2 Simplify. 36

TRY IT 10.1

Evaluate when and .

196

TRY IT 10.2

Evaluate when and .

8

EXAMPLE 11

Evaluate when a) and b) .

Solution

a)

 Subtract. 8

b)

 Subtract. 32

TRY IT 11.1

Evaluate: when a) and b) .

a) b) 36

TRY IT 11.2

Evaluate: when a) and b) .

a) b) 9

EXAMPLE 12

Evaluate: when .

Solution

Substitute . Use parentheses to show multiplication.

 Substitute. Evaluate exponents. Multiply. Add. 52

TRY IT 12.1

Evaluate: when .

39

TRY IT 12.2

Evaluate: when .

13

# Translate Phrases to Expressions with Integers

Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.

EXAMPLE 13

Translate and simplify: the sum of 8 and , increased by 3

Solution
 the sum of 8 and , increased by 3. Translate. Simplify. Be careful not to confuse the brackets with an absolute value sign. Add.

TRY IT 13.1

Translate and simplify the sum of 9 and , increased by 4

TRY IT 13.2

Translate and simplify the sum of and , increased by 7

When we first introduced the operation symbols, we saw that the expression may be read in several ways. They are listed in the chart below.

minus
the difference of and
subtracted from
less than

Be careful to get a and b in the right order!

EXAMPLE 14

Translate and then simplify a) the difference of 13 and b) subtract 24 from .

Solution
 a) Translate. Simplify. b) Translate. Remember, “subtract from means . Simplify.

TRY IT 14.1

Translate and simplify a) the difference of 14 and b) subtract 21 from .

a) b)

TRY IT 14.2

Translate and simplify a) the difference of 11 and b) subtract 18 from .

a) b)

Once again, our prior work translating English to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is “product” and for division is “quotient.”

EXAMPLE 15

Translate to an algebraic expression and simplify if possible: the product of and 14

Solution
 Translate. Simplify.

TRY IT 15.1

Translate to an algebraic expression and simplify if possible: the product of and 12

TRY IT 15.2

Translate to an algebraic expression and simplify if possible: the product of 8 and .

EXAMPLE 16

Translate to an algebraic expression and simplify if possible: the quotient of and .

Solution
 Translate. Simplify.

TRY IT 16.1

Translate to an algebraic expression and simplify if possible: the quotient of and .

TRY IT 16.2

Translate to an algebraic expression and simplify if possible: the quotient of and .

# Use Integers in Applications

We’ll outline a plan to solve applications. It’s hard to find something if we don’t know what we’re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we’ll write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.

EXAMPLE 17

The temperature in Sparwood, British Columbia, one morning was 11 degrees. By mid-afternoon, the temperature had dropped to degrees. What was the difference of the morning and afternoon temperatures?

Solution

TRY IT 17.1

The temperature in Whitehorse, Yukon, one morning was 15 degrees. By mid-afternoon the temperature had dropped to 30 degrees below zero. What was the difference in the morning and afternoon temperatures?

The difference in temperatures was 45 degrees.

TRY IT 17.2

The temperature in Quesnel, BC, was degrees at lunchtime. By sunset the temperature had dropped to degrees. What was the difference in the lunchtime and sunset temperatures?

The difference in temperatures was 9 degrees.

HOW TO: Apply a Strategy to Solve Applications with Integers

1. Read the problem. Make sure all the words and ideas are understood
2. Identify what we are asked to find.
3. Write a phrase that gives the information to find it.
4. Translate the phrase to an expression.
5. Simplify the expression.
6. Answer the question with a complete sentence.

EXAMPLE 18

The Mustangs football team received three penalties in the third quarter. Each penalty gave them a loss of fifteen yards. What is the number of yards lost?

Solution
 Step 1. Read the problem. Make sure all the words and ideas are understood. Step 2. Identify what we are asked to find. the number of yards lost Step 3. Write a phrase that gives the information to find it. three times a 15-yard penalty Step 4. Translate the phrase to an expression. Step 5. Simplify the expression. Step 6. Answer the question with a complete sentence. The team lost 45 yards.

TRY IT 18.1

The Bears played poorly and had seven penalties in the game. Each penalty resulted in a loss of 15 yards. What is the number of yards lost due to penalties?

The Bears lost 105 yards.

TRY IT 18.2

Bill uses the ATM on campus because it is convenient. However, each time he uses it he is charged a $2 fee. Last month he used the ATM eight times. How much was his total fee for using the ATM? Show answer A$16 fee was deducted from his checking account.

# Key Concepts

• Multiplication and Division of Two Signed Numbers
• Same signs—Product is positive
• Different signs—Product is negative
• Strategy for Applications
1. Identify what you are asked to find.
2. Write a phrase that gives the information to find it.
3. Translate the phrase to an expression.
4. Simplify the expression.
5. Answer the question with a complete sentence.

# Practice Makes Perfect

## Multiply Integers

In the following exercises, multiply.

 1 2 3 4 5 6 7 8

## Divide Integers

In the following exercises, divide.

 9 10 11 12 13 14

## Simplify Expressions with Integers

In the following exercises, simplify each expression.

 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

## Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

 33. when a) b) 34. when a) b) 35. a) when b) when 36. a) when b) when 37.  when 38. when 39. when 40. when 41. when 42. when 43. when a) b) 44. when a) b) 45. when a) b) 46. when a) b) 47. when 48. when 49. when 50. when

## Translate English Phrases to Algebraic Expressions

In the following exercises, translate to an algebraic expression and simplify if possible.

 51. the sum of 3 and , increased by 7 52. the sum of and , increased by 23 53. the difference of 10 and 54. subtract 11 from 55. the difference of and 56. subtract from 57. the product of and 58. the product of and 59. the quotient of and 60. the quotient of and 61. the quotient of and the sum of a and b 62. the quotient of and the sum of m and n 63. the product of and the difference of 64. the product of and the difference of

## Use Integers in Applications

In the following exercises, solve.

 65. Temperature On January , the high temperature in Lytton, British Columbia, was ° . That same day, the high temperature in Fort Nelson, British Columbia was °. What was the difference between the temperature in Lytton and the temperature in Embarrass? 66. Temperature On January , the high temperature in Palm Springs, California, was °, and the high temperature in Whitefield, New Hampshire was °. What was the difference between the temperature in Palm Springs and the temperature in Whitefield? 67. Football At the first down, the Chargers had the ball on their 25 yard line. On the next three downs, they lost 6 yards, gained 10 yards, and lost 8 yards. What was the yard line at the end of the fourth down? 68. Football At the first down, the Steelers had the ball on their 30 yard line. On the next three downs, they gained 9 yards, lost 14 yards, and lost 2 yards. What was the yard line at the end of the fourth down? 69. Checking Account Ester has $124 in her checking account. She writes a check for$152. What is the new balance in her checking account? 70. Checking Account Selina has $165 in her checking account. She writes a check for$207. What is the new balance in her checking account? 71. Checking Account Kevin has a balance of in his checking account. He deposits $225 to the account. What is the new balance? 72. Checking Account Reymonte has a balance of in his checking account. He deposits$281 to the account. What is the new balance?

 73. Stock market Javier owns 300 shares of stock in one company. On Tuesday, the stock price dropped $12 per share. What was the total effect on Javier’s portfolio? 74. Weight loss In the first week of a diet program, eight women lost an average of 3 pounds each. What was the total weight change for the eight women? ## Writing Exercises  75. In your own words, state the rules for multiplying integers. 76. In your own words, state the rules for dividing integers. 77. Why is 78. Why is # Answers  1. 3. 5. 7. 14 9. 11. 13 13. 15. 17. 64 19. 21. 90 23. 9 25. 41 27. 29. 31. 5 33. a) b) 16 35. a) b) 10 37. 39. 41. 121 43. a) 1 b) 33 45. a)b) 25 47. 21 49. 51. 53. 55. 57. 59. 61. 63. 65. ° 67. 21 69. 71.$187 73. 75. Answers may vary 77. Answers may vary

This chapter has been adapted from “Multiply and Divide Integers” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

# Review Exercises

## Use Place Value with Whole Number

In the following exercises find the place value of each digit.

 1. 26,915 a) 1 b) 2 c) 9 d) 5 e) 6 2. 359,417 a) 9 b) 3 c) 4 d) 7 e) 1 3. 58,129,304 a) 5 b) 0 c) 1 d) 8 e) 2 4. 9,430,286,157 a) 6 b) 4 c) 9 d) 0 e) 5

In the following exercises, name each number.

 5. 6,104 6. 493,068 7. 3,975,284 8. 85,620,435

In the following exercises, write each number as a whole number using digits.

 9. three hundred fifteen 10. sixty-five thousand, nine hundred twelve 11. ninety million, four hundred twenty-five thousand, sixteen 12. one billion, forty-three million, nine hundred twenty-two thousand, three hundred eleven

In the following exercises, round to the indicated place value.

 Round to the nearest ten. 13. a) 407 b) 8,564 Round to the nearest hundred. 14. a) 25,846 b) 25,864

In the following exercises, round each number to the nearest a) hundred b) thousand c) ten thousand.

 15. 864,951 16. 3,972,849

## Identify Multiples and Factors

In the following exercises, use the divisibility tests to determine whether each number is divisible by 2, by 3, by 5, by 6, and by 10

 17. 168 18. 264 19. 375 20. 750 21. 1430 22. 1080

## Find Prime Factorizations and Least Common Multiples

In the following exercises, find the prime factorization.

 23. 420 24. 115 25. 225 26. 2475 27. 1560 28. 56 29. 72 30. 168 31. 252 32. 391

In the following exercises, find the least common multiple of the following numbers using the multiples method.

 33. 6,15 34. 60, 75

In the following exercises, find the least common multiple of the following numbers using the prime factors method.

 35. 24, 30 36. 70, 84

## Use Variables and Algebraic Symbols

In the following exercises, translate the following from algebra to English.

 37. 25 – 7 38. 5 · 6 39. 45 ÷ 5 40. x + 8 41. 42 ≥ 27 42. 3n = 24 43. 3 ≤ 20 ÷ 4 44. a ≠ 7 · 4

In the following exercises, determine if each is an expression or an equation.

 45. 6 · 3 + 5 46. y – 8 = 32

## Simplify Expressions Using the Order of Operations

In the following exercises, simplify each expression.

 47. 35 48. 108

In the following exercises, simplify

 49. 6 + 10/2 + 2 50. 9 + 12/3 + 4 51. 20 ÷ (4 + 6) · 5 52. 33 · (3 + 8) · 2 53. 42 +52 54. (4 + 5)2

## Evaluate an Expression

In the following exercises, evaluate the following expressions.

 55. 9x + 7 when x = 3 56. 5x – 4 when x = 6 57. x4 when x = 3 58. 3x when x = 3 59. x2 + 5x – 8 when x = 6 60. 2x + 4y – 5 when x = 7, y = 8

## Simplify Expressions by Combining Like Terms

In the following exercises, identify the coefficient of each term.

 61. 12n 62. 9x2

In the following exercises, identify the like terms.

 63. 3n, n2, 12, 12p2, 3, 3n2 64. 5, 18r2, 9s, 9r, 5r2, 5s

In the following exercises, identify the terms in each expression.

 65. 11x2 + 3x + 6 66. 22y3 + y + 15

In the following exercises, simplify the following expressions by combining like terms.

 67. 17a + 9a 68. 18z + 9z 69. 9x + 3x + 8 70. 8a + 5a + 9 71. 7p + 6 + 5p – 4 72. 8x + 7 + 4x – 5

## Translate an English Phrase to an Algebraic Expression

In the following exercises, translate the following phrases into algebraic expressions.

# CHAPTER 2 Operations with Rational Numbers and Introduction to Real Numbers

All the numbers we use in the intermediate algebra course are real numbers. The chart below shows us how the number sets we use in algebra fit together. In this chapter we will work with rational numbers, but you will be also introduced to irrational numbers. The set of rational numbers together with the set of irrational numbers make up the set of real numbers.

## 2.1 Visualize Fractions

Learning Objectives

By the end of this section, you will be able to:

• Find equivalent fractions
• Simplify fractions
• Multiply fractions
• Divide fractions
• Simplify expressions written with a fraction bar
• Translate phrases to expressions with fractions

# Find Equivalent Fractions

Fractions are a way to represent parts of a whole. The fraction means that one whole has been divided into 3 equal parts and each part is one of the three equal parts. See (Figure 1). The fraction represents two of three equal parts. In the fraction , the 2 is called the numerator and the 3 is called the denominator.

Figure 1.

The circle on the left has been divided into 3 equal parts. Each part is of the 3 equal parts. In the circle on the right, of the circle is shaded (2 of the 3 equal parts).

Fraction

A fraction is written , where and

• a is the numerator and b is the denominator.

A fraction represents parts of a whole. The denominator b is the number of equal parts the whole has been divided into, and the numerator a indicates how many parts are included.

If a whole pie has been cut into 6 pieces and we eat all 6 pieces, we ate pieces, or, in other words, one whole pie.

So . This leads us to the property of one that tells us that any number, except zero, divided by itself is 1

Property of One

Any number, except zero, divided by itself is one.

If a pie was cut in pieces and we ate all 6, we ate pieces, or, in other words, one whole pie. If the pie was cut into 8 pieces and we ate all 8, we ate pieces, or one whole pie. We ate the same amount—one whole pie.

The fractions and have the same value, 1, and so they are called equivalent fractions. Equivalent fractions are fractions that have the same value.

Let’s think of pizzas this time. (Figure 2) shows two images: a single pizza on the left, cut into two equal pieces, and a second pizza of the same size, cut into eight pieces on the right. This is a way to show that is equivalent to . In other words, they are equivalent fractions.

Figure 2.

Since the same amount is of each pizza is shaded, we see that is equivalent to . They are equivalent fractions.

Equivalent Fractions

Equivalent fractions are fractions that have the same value.

How can we use mathematics to change into How could we take a pizza that is cut into 2 pieces and cut it into 8 pieces? We could cut each of the 2 larger pieces into 4 smaller pieces! The whole pizza would then be cut into pieces instead of just 2. Mathematically, what we’ve described could be written like this as . See (Figure 3).

Figure 3.

Cutting each half of the pizza into pieces, gives us pizza cut into 8 pieces: .

This model leads to the following property:

Equivalent Fractions Property

If are numbers where , then

If we had cut the pizza differently, we could get