Introductory Algebra

Introductory Algebra

Izabela Mazur

Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis, and OpenStax

BCcampus

Victoria, B.C.

Contents

1

About BCcampus Open Education

Introductory Algebra by Izabela Mazur was funded by BCcampus Open Education.

BCcampus Open Education began in 2012 as the B.C. Open Textbook Project with the goal of making post-secondary education in British Columbia more accessible by reducing students’ costs through the use of open textbooks and other OER. BCcampus supports the post-secondary institutions of British Columbia as they adapt and evolve their teaching and learning practices to enable powerful learning opportunities for the students of B.C. BCcampus Open Education is funded by the British Columbia Ministry of Advanced Education and Skills Training, and the Hewlett Foundation.

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For more information about open education in British Columbia, please visit the BCcampus Open Education website. If you are an instructor who is using this book for a course, please fill out our Adoption of an Open Textbook form.

2

For Students: How to Access and Use this Textbook

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Tips for Using This Textbook

I

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.

The image shows a collage of mathematical terms such as algebra, decimals, equations, numbers etcetera. The words are written horizontally, vertically, and in different colors.

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

1.1 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.

\phantom{\rule{1.5em}{0ex}}Counting Numbers: 1, 2, 3, …

\phantom{\rule{1.5em}{0ex}}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.
A horizontal number line with arrows on each end and values of zero to six runs along the bottom of the diagram. A second horizontal line with a left-facing arrow lies above the first and extend from zero to three. This line is labled “smaller”. A third horizontal line with a right-facing arrow lies above the first two, but runs from three to six and is labeled “larger”.
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.

This figure is a table illustrating the number 5,278,194 within the place value system. The table is shown with a header row, labeled “Place Value”, divided into a second header row labeled “Trillions”, “Billions”, “Millions”, “Thousands” and “Ones”. Under the header “Trillions” are three labeled columns, written from bottom to top, that read “Hundred trillions”, “Ten trillions” and “Trillions”. Under the header “Billions” are three labeled columns, written from bottom to top, that read “Hundred billions”, “Ten billions” and “Billions”. Under the header “Millions” are three labeled columns, written from bottom to top, that read “Hundred millions”, “Ten millions” and “Millions”. Under the header “Thousands” are three labeled columns, written from bottom to top, that read “Hundred thousands”, “Ten thousands” and “Thousands”. Under the header “Ones” are three labeled columns, written from bottom to top, that read “Hundreds”, “Tens” and “Ones”. From left to right, below the columns labeled “Millions”, “Hundred thousands”, “Ten thousands”, “Thousands”, “Hundreds”, “Tens”, and “Ones”, are the following values: 5, 2, 7, 8, 1, 9, 4. This means there are 5 millions, 2 hundred thousands, 7 ten thousands, 8 thousands, 1 hundreds, 9 tens, and 4 ones in the number five million two hundred seventy-nine thousand one hundred ninety-four.
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:

This figure is a table illustrating the number 63,407,218 within the place value system. The table is shown with a header row, labeled “Place Value”, divided into a second header row labeled “Trillions”, “Billions”, “Millions”, “Thousands” and “Ones”. Under the header “Trillions” are three labeled columns, written from bottom to top, that read “Hundred trillions”, “Ten trillions” and “Trillions”. Under the header “Billions” are three labeled columns, written from bottom to top, that read “Hundred billions”, “Ten billions” and “Billions”. Under the header “Millions” are three labeled columns, written from bottom to top, that read “Hundred millions”, “Ten millions” and “Millions”. Under the header “Thousands” are three labeled columns, written from bottom to top, that read “Hundred thousands”, “Ten thousands” and “Thousands”. Under the header “Ones” are three labeled columns, written from bottom to top, that read “Hundreds”, “Tens” and “Ones”. From left to right, below the columns labeled “Ten millions”, “Millions”, “Hundred thousands”, “Ten thousands”, “Thousands”, “Hundreds”, “Tens”, and “Ones”, are the following values: 6, 3, 4, 0, 7, 2, 1, 8. This means there are 6 ten millions, 3 millions, 4 hundred thousands, 0 ten thousands, 7 thousands, 2 hundreds, 1 ten, and 8 ones in the number sixty-three million, four hundred seven thousand, two hundred eighteen.

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

Show answer

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

Show answer

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.

In this figure, the numbers 74, 218 and 369 are listed in a row, separated by commas. Each number has a curly bracket beneath it with the word “millions” written below the number 74, “thousands” written below the number 218, and “ones” written below the number 369. A left-facing arrow points at these three words, labeling them “periods”. One row down is the number “74”, a right-facing arrow and the words “Seventy-four million” followed by a comma. The next row below is the number “218”, a right-facing arrow and the words “two hundred eighteen thousand” followed by a comma. On the bottom row is the number “369”, a right-facing arrow and the words “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.

In this figure, the numbers 8, 165, 432, 098 and 710 are listed in a row, separated by commas. Each number has a horizontal bracket beneath with the word “trillions” written below the number 8, “billions” written below the number 165, “millions” written below the number 432, “thousands” written below the number 098, and “ones” written below the number 710. One row down is the number 8, a right-facing arrow and the words “Eight trillion” followed by a comma. On the next row below is the number 165, a right-facing arrow and the words “One hundred sixty-five billion” followed by a comma. On the next row below is the number 432, a right-facing arrow and the words “Four hundred thirty-two million” followed by a comma. On the next row below is the number “098”, a right-facing arrow and the words “Ninety-eight thousand” followed by a comma. On the bottom row is the number 710, a right-facing arrow and the words “Seven hundred ten”.

Put the commas in to separate the periods.

So, 8,165,432,098,710 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 9,258,137,904,061 using words.

Show answer

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 17,864,325,619,004 using words.

Show answer

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.

An image has two lines of text. The upper lines read “nine billion”, followed by a comma, and “two hundred forty six million”, also followed by a comma. The words “billion” and “million” are underlined and each phrase has a curly bracket underneath. The lower lines read “seventy three thousand”, followed by a comma, and “one hundred eighty nine”. The word “thousand” is underlined and each phrase has a curly bracket underneath.

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.

Show answer

2,466,714,051

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.

Show answer

11,921,830,106

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

This figure is a table that has three columns and four rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains the numbers corresponding with the written steps and instructions. In the top row, the first cell says: “Step 1. Locate the given place value with an arrow. All digits to the left do not change.” In the the second cell, the instructions say: “Locate the hundreds place in 23,658.” In the third cell, there is the number 23,658 with an arrow pointing to the digit 6, labeling it “hundreds place.”One row down, the instructions in the first cell say: “Step 2. Underline the digit to the right of the given place value.” In the second cell, the instructions say: “Underline the 5, which is to the right of the hundreds place.” In the third cell, there is the number 23,658 again, the same arrow pointing to the digit 6, labeling it the hundreds place. The 5 is also underlined in this cell.One row down, the first cell says: “Step 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.” In the second cell, the instructions say: “Add 1 to the 6 in the hundreds place, since 5 is greater than or equal to 5.” The third cell contains the number 23,658 again, with an arrow pointing at the digit 6 and the text “add 1”. There is also a curly bracket under the digits 5 and 8, with an arrow pointing at them and the text “replace with 0s.”In the bottom row, the first cell says: “Step 4. Replace all digits to the right of the given place value with zeros. So, 23,700 is rounded to the nearest hundred.” In the second cell, the instructions say: “Replace all digits to the right of the hundreds place with zeros.” The third cell contains the number 23,700, which we have reached by rounding the number 23,658 to the nearest hundred.

TRY IT 4.1

Round to the nearest hundred: 17,852.

Show answer

17,900

TRY IT 4.2

Round to the nearest hundred: 468,751.

Show answer

468,800

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 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.

EXAMPLE 5

Round 103,978 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. \phantom{\rule{1.5em}{0ex}}.
Since 3 is less than 5, we leave the 0 as is, and then replace the digits to the right with zeros. \phantom{\rule{1.5em}{0ex}}.
\phantom{\rule{1.5em}{0ex}}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.

Show answer

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.

Show answer

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

A diagram made up of two rows of numbers. The top row reads “2, 4, 6, 8, 10, 12,” followed by an elipsis. Below 2 is 2 times 1, below 4 is 2 times 2, below 6 is 2 times 3, below 8 is 2 times 4, below 10 is 2 times 5, and below 12 is 2 times 6.

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

A diagram made up of two rows of numbers. The top row reads “3, 6, 9, 12, 15, 18,” followed by an elipsis. Below 3 is 3 times 1, below 6 is 3 times 2, below 9 is 3 times 3, below 12 is 3 times 4, below 15 is 3 times 5, and below 18 is 3 times 6.

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, 15\div 3 is 5, so 15 is 5\cdot 3.

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? 5+6+2+5=18
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

Show answer

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

Show answer

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

An image shows the equation 8 times 9 equals 72. Written below the expression 8 times 9 is a curly bracket and the word “factors” while written below 72 is a horizontal bracket and the word “product”.

Other ways to factor 72 are 1\cdot 72,2\cdot 36,3\cdot 24,4\cdot 18,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}6\cdot 12. Seventy-two has many factors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 36, and 72

Factors

If a\cdot b=m, 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!

 

A table is shown with eleven rows and seven columns. The first row is a header row, and each cell labels the contents of the column below it. In the header row, the first three cells read from left to right “Number”, “Factors”, and “Prime or Composite?” The entire fourth column is blank. The last three cells read from left to right “Number”, “Factor”, and “Prime or Composite?” again. In each subsequent row, the first cell contains a number, the second contains its factors, and the third indicates whether the number is prime or composite. The three columns to the left of the blank middle column contain this information for the number 2 through 10, and the three columns to the right of the blank middle column contain this information for the number 11 through 19. On the left side of the blank column, in the first row below the header row, the cells read from left to right: “2”, “1,2”, and “Prime”. In the next row, the cells read from left to right: “3”, “1,3”, and “Prime”. In the next row, the cells read from left to right: “4”, “1,2,4”, and “Composite”. In the next row, the cells read from left to right: “5”, “1,5”, and “Prime”. In the next row, the cells read from left to right: “6”, “1,2,3,6” and “Composite”. In the next row, the cells read from left to right: “7”, “1,7”, and “Prime”. In the next row, the cells read from left to right: “8”, “1,2,4,8”, and “Composite”. In the next row, the cells read from left to right: “9”, “1,3,9”, and “Composite”. In the bottom row, the cells read from left to right: “10”, “1,2,5,10”, and “Composite”. On the right side of the blank column, in the first row below the header row, the cells read from left to right: “11”, “1,11”, and “Prime”. In the next row, the cells read from left to right: “12”, “1,2,3,4,6,12”, and “Composite”. In the next row, the cells read from left to right: “13”, “1,13”, and “Prime”. In the next row, the cells read from left to right “14”, “1,2,7,14”, and “Composite”. In the next row, the cells read from left to right: “15”, “1,3,5,15”, and “Composite”. In the next row, the cells read from left to right: “16”, “1,2,4,8,16”, and “Composite”. In the next row, the cells read from left to right, “17”, “1,17”, and “Prime”. In the next row, the cells read from left to right, “18”, “1,2,3,6,9,18”, and “Composite”. In the bottom row, the cells read from left to right: “19”, “1,19”, and “Prime”.
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

This figure is a table that has three columns and four rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions and some math. The third column contains most of the math work corresponding with the written steps and instructions. In the top row, the first cell says: “Step 1. Find two factors whose product is the given number. Use these numbers to create two branches.” The second cell contains the algebraic equation 48 equals 2 times 24. In the third cell, there is a factor tree with 48 at the top. Two branches descend from 48 and terminate at 2 and 24 respectively.One row down, the instructions in the first cell say: “Step 2. If a factor is prime, that branch is complete. Circle the prime.” In the second cell, the instructions say: “2 is prime. Circle the prime.” In the third cell, the factor tree from step 1 is repeated, but the 2 at the bottom of the tree is now circled.One row down, the first cell says: “Step 3. If a factor is not prime, write it as the product of two factors and continue the process.” In the second cell, the instructions say: “24 is not prime. Break it into 2 more factors.” The third cell contains the original factor tree, with 48 at the top and two downward-pointing branches terminating at 2, which is underlined, and 24. Two more branches descend from 24 and terminate at 4 and 6 respectively. One line down, the instructions in the middle of the cell say “4 and 6 are not prime. Break them each into two factors.” In the cell on the right, the factor tree is repeated once more. Two branches descend from the 4 and terminate at 2 and 2. Both 2s are circled. Two more branches descend from 6 and terminate at a 2 and a 3, which are both circled. The instructions on the left say “2 and 3 are prime, so circle them.”In the bottom row, the first cell says: “Step 4. Write the composite number as the product of all the circled primes.” The second cell is left blank. The third cell contains the algebraic equation 48 equals 2 times 2 times 2 times 2 times 3.

We say 2\cdot 2\cdot 2\cdot 2\cdot 3 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 6\cdot 8, 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.

Show answer

2\cdot 2\cdot 2\cdot 2\cdot 5

TRY IT 7.2

Find the prime factorization of 60.

Show answer

2\cdot 2\cdot 3\cdot 5

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. 252=2\cdot 2\cdot 3\cdot 3\cdot 7

TRY IT 8.1

Find the prime factorization of 126

Show answer

2\cdot 3\cdot 3\cdot 7

TRY IT 8.2

Find the prime factorization of 294

Show answer

2\cdot 3\cdot 7\cdot 7

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:

Two rows of numbers are shown. The first row begins with 12, followed by a colon, then 12, 24, 36, 48, 60, 72, 84, 96, 108, and an elipsis. 36, 72, and 108 are bolded written in red. The second row begins with 18, followed by a colon, then 18, 36, 54, 72, 90, 108, and an elipsis. Again, the numbers 36, 72, and 108 are bolded written in red. On the line below is the phrase “Common Multiples”, a colon and the numbers 36, 72, and 108, written in red. One line below is the phrase “Least Common Multiple”, a colon and the number 36, written in blue.

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

Show answer

36

TRY IT 9.2

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

Show answer

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

This figure is a table that has three columns and four rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions and some math. The third column contains most of the math work corresponding with the written steps and instructions. In the top row, the first cell says: “Step 1. Write each number as a product of primes.” The second cell is left blank. In the third cell, there are two factor trees. In the first factor tree, two branches descend from 18 and terminate at 3 and 6 respectively. The 3 is prime and therefore circled. Two more branches descend from the 6 and terminate in 2 and 3, both of which are circled. In the second factor tree, two branches descend from 12 and terminate at 3 and 4. The 3 is circled. Two more branches descend from 4, terminating at 2 and 2, both of which are circled.One row down, the instructions in the first cell say: “Step 2. List the primes of each number. Match primes vertically when possible.” In the second cell, the instructions say: “List the primes of 12. List the primes of 18. Line up with the primes of 12 when possible. If not create a new column.” The third cell contains the prime factorization of 12 written as the equation 12 equals 2 times 2 times 3. Below this equation is another showing the prime factorization of 18 written as the equation 18 equals 2 times 3 times 3. The two equations line up vertically at the equal symbol. The first 2 in the prime factorization of 12 aligns with the 2 in the prime factorization of 18. Under the second 2 in the prime factorization of 12 is a gap in the prime factorization of 18. Under the 3 in the prime factorization of 12 is the first 3 in the prime factorization of 18. The second 3 in the prime factorization has no factors above it from the prime factorization of 12.One row down, the instructions in the first cell say: “Bring down the number from each column.” The second cell is blank. The third cell contains the prime factorizations of 12 and 18 again, illustrated as two equations aligned just as they were before. This time, a horizontal line is drawn under the prime factorization of 18. Below this line is the equation LCM equal to 2 times 2 times 3 times 3. Arrows are drawn down vertically from the prime factorization of 12 through the prime factorization of 18 ending at the LCM equation. The first arrow starts at the first 2 in the prime factorization of 12 and continues down through the 2 in the prime factorization of 18, ending with the first 2 in the LCM. The second arrow starts at the next 2 in the prime factorization of 12 and continues down through the gap in the prime factorization of 18, ending with the second 2 in the LCM. The third arrow starts at the 3 in the prime factorization of 12 and continues down through the first 3 in the prime factorization of 18, ending with the first 3 in the LCM. The last arrow starts at the second 3 in the prime factorization of 18 and points down to the second 3 in the LCM.In the bottom row of the table, the first cell says: “Step 4: Multiply the factors.” The second cell is bank. The third cell contains the equation LCM equals 36.

Notice that the prime factors of 12 \left(2\cdot 2\cdot 3\right) and the prime factors of 18 \left(2\cdot 3\cdot 3\right) are included in the LCM \left(2\cdot 2\cdot 3\cdot 3\right). 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

Show answer

36

TRY IT 10.2

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

Show answer

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

Show answer

84

TRY IT 11.2

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

Show answer

96

Key Concepts

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 m is a multiple of n, then m is divisible by n. (If 6 is a multiple of 3, then 6 is divisible by 3.)
factors
If a·b=m, then a\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b are factors of m. 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.

Answers

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. 2\cdot 43 51. 2\cdot 2\cdot 3\cdot 11 53. 3\cdot 3\cdot 7\cdot 11
55. 2\cdot 2\cdot 2\cdot 2\cdot 3\cdot 3\cdot 3 57. 2\cdot 2\cdot 2\cdot 2\cdot 3\cdot 3\cdot 3\cdot 5 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.

Attributions

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.

2

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 20 years old and Alex is 23, so Alex is 3 years older than Greg. When Greg was 12, Alex was 15. When Greg is 35, Alex will be 38. No matter what Greg’s age is, Alex’s age will always be 3 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 3 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 g. Then we could use g+3 to represent Alex’s age. See the table below.

Greg’s age Alex’s age
12 15
20 23
35 38
g g+3

Letters are used to represent variables. Letters often used for variables are x,y,a,b,\text{and}\phantom{\rule{0.2em}{0ex}}c.

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…
Addition a+b a\phantom{\rule{0.2em}{0ex}}\text{plus}\phantom{\rule{0.2em}{0ex}}b the sum of a and b
Subtraction a-b a\phantom{\rule{0.2em}{0ex}}\text{minus}\phantom{\rule{0.2em}{0ex}}b the difference of a and b
Multiplication a\cdot b,\left(a\right)\left(b\right),\left(a\right)b,a\left(b\right) a\phantom{\rule{0.2em}{0ex}}\text{times}\phantom{\rule{0.2em}{0ex}}b The product of a and b
Division a\div b,a/b,\phantom{\rule{0.2em}{0ex}}\frac{a}{b},b\overline{)a} a divided by b The quotient of a and b

In algebra, the cross symbol, ×, is not used to show multiplication because that symbol may cause confusion. Does 3xy mean 3 \phantom{\rule{0.2em}{0ex}} \times \phantom{\rule{0.2em}{0ex}}y (three times y) or 3 \cdot x \cdot y (three times x\phantom{\rule{0.2em}{0ex}}\text{times}\phantom{\rule{0.2em}{0ex}}y)? 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.

EXAMPLE 1

Translate from algebra to words:

  1. \phantom{\rule{0.2em}{0ex}}12+14\phantom{\rule{0.2em}{0ex}}
  2. \phantom{\rule{0.2em}{0ex}}\left(30\right)\left(5\right)\phantom{\rule{0.2em}{0ex}}
  3. \phantom{\rule{0.2em}{0ex}}64 \div 8\phantom{\rule{0.2em}{0ex}}
  4. \phantom{\rule{0.2em}{0ex}}x-y
Solution
a.
12+14
12 plus 14
the sum of twelve and fourteen
b.
\left(30\right)\left(5\right)
30 times 5
the product of thirty and five
c.
64\div 8
64 divided by 8
the quotient of sixty-four and eight
d.
x-y
x minus y
the difference of x and y

TRY IT 1.1

Translate from algebra to words.

  1. \phantom{\rule{0.2em}{0ex}}18+11\phantom{\rule{0.4em}{0ex}}
  2. \phantom{\rule{0.2em}{0ex}}\left(27\right)\left(9\right)\phantom{\rule{0.4em}{0ex}}
  3. \phantom{\rule{0.2em}{0ex}}84\div 7
  4. p-q
Show Answer
  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. \phantom{\rule{0.2em}{0ex}}47-19\phantom{\rule{0.4em}{0ex}}
  2. \phantom{\rule{0.2em}{0ex}}72\div 9\phantom{\rule{0.4em}{0ex}}
  3. \phantom{\rule{0.2em}{0ex}}m+n\phantom{\rule{0.4em}{0ex}}
  4. \phantom{\rule{0.2em}{0ex}}\left(13\right)\left(7\right)
Show Answer
  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

a=b\phantom{\rule{0.2em}{0ex}}\text{is read}\phantom{\rule{0.2em}{0ex}}a\phantom{\rule{0.2em}{0ex}}\text{is equal to}\phantom{\rule{0.2em}{0ex}}b

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 b is greater than a, it means that b is to the right of a on the number line. We use the symbols < and > for inequalities.

Inequality

a < b is read a is less than b

a is to the left of b on the number line

The figure shows a horizontal number line that begins with the letter a on the left then the letter b to its right.

a > b is read a is greater than b

a is to the right of b on the number line

The figure shows a horizontal number line that begins with the letter b on the left then the letter a to its right.

The expressions a < b\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}a > \phantom{\rule{0.2em}{0ex}}b can be read from left-to-right or right-to-left, though in English we usually read from left-to-right. In general,

"7 is less than 11" equivalent to "11 is greater than 7"

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

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

Algebraic Notation Say
a=b a is equal to b
a\ne b a is not equal to b
a < b a is less than b
a > b a is greater than b
a\le b a is less than or equal to b
a\ge b a is greater than or equal to b

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. \phantom{\rule{0.2em}{0ex}}20\le 35
  2. \phantom{\rule{0.2em}{0ex}}11\ne 15-3
  3. \phantom{\rule{0.2em}{0ex}}9 > \phantom{\rule{0.2em}{0ex}}10\div 2
  4. \phantom{\rule{0.2em}{0ex}}x+2 < \phantom{\rule{0.2em}{0ex}}10
Solution
a.
20\le 35
20 is less than or equal to 35
b.
11\ne 15-3
11 is not equal to 15 minus 3
c.
9 > 10\div 2
9 is greater than 10 divided by 2
d.
x+2 < 10
x plus 2 is less than 10

TRY IT 2.1

Translate from algebra to words.

  1. \phantom{\rule{0.2em}{0ex}}\text{14}\le 27\phantom{\rule{0.2em}{0ex}}
  2. \phantom{\rule{0.2em}{0ex}}19-2\ne 8\phantom{\rule{0.2em}{0ex}}
  3. \phantom{\rule{0.2em}{0ex}}12 > 4\phantom{\rule{0.2em}{0ex}}\div 2\phantom{\rule{0.2em}{0ex}}
  4. \phantom{\rule{0.2em}{0ex}}x-7 < \phantom{\rule{0.2em}{0ex}}1
Show Answer
  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. \phantom{\rule{0.2em}{0ex}}19\ge 15\phantom{\rule{0.2em}{0ex}}
  2. \phantom{\rule{0.2em}{0ex}}7=12-5\phantom{\rule{0.2em}{0ex}}
  3. \phantom{\rule{0.2em}{0ex}}15\div 3 < 8\phantom{\rule{0.2em}{0ex}}
  4. \phantom{\rule{0.2em}{0ex}}y-3 > 6\phantom{\rule{0.2em}{0ex}}
Show Answer
  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 \text{symbol},\text{=}, < ,\text{or} >. in each expression to compare the fuel economy of the cars.

(credit: modification of work by Bernard Goldbach, Wikimedia Commons)
This table has two rows and six columns. The first column is a header column and it labels each row The first row is labeled “Car” and the second “Fuel economy (mpg)”. To the right of the ‘Car’ row are the labels: “Prius”, “Mini Cooper”, “Toyota Corolla”, “Versa”, “Honda Fit”. Each of these columns contains an image of the labeled car model. To the right of the “Fuel economy (mpg)” row are the algebraic equations: the letter p, the equals symbol, the number forty-eight; the letter m, the equals symbol, the number twenty-seven; the letter c, the equals symbol, the number twenty-eight; the letter v, the equals symbol, the number twenty-six; and the letter f, the equals symbol, the number twenty-seven.
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 \text{symbol},\text{=}, < ,\text{or} >.

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

TRY IT 3.2

Use Figure 1 to fill in the appropriate \text{symbol},\text{=}, < ,\text{or} >.

  1. MPG of Fit_____ MPG of Prius
  2. MPG of Corolla _____ MPG of Fit
Show Answer
  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.

Common Grouping Symbols
parentheses \left(\phantom{\rule{0.5em}{0ex}}\right)
brackets \left[\phantom{\rule{0.5em}{0ex}}\right]
braces \left\{\phantom{\rule{0.5em}{0ex}}\right\}

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

8\left(14-8\right)\phantom{\rule{4em}{0ex}}21-3\left[2+4\left(9-8\right)\right]\phantom{\rule{4em}{0ex}}24\div \left\{13-2\left[1\left(6-5\right)+4\right]\right\}

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
3+5 3\phantom{\rule{0.2em}{0ex}}\text{plus}\phantom{\rule{0.2em}{0ex}}5 the sum of three and five
n-1 n minus one the difference of n and one
6\cdot 7 6\phantom{\rule{0.2em}{0ex}}\text{times}\phantom{\rule{0.2em}{0ex}}7 the product of six and seven
\frac{x}{y} x divided by y the quotient of x and y

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
3+5=8 The sum of three and five is equal to eight.
n-1=14 n minus one equals fourteen.
6\cdot 7=42 The product of six and seven is equal to forty-two.
x=53 x is equal to fifty-three.
y+9=2y-3 y plus nine is equal to two y 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:

  1. \phantom{\rule{0.2em}{0ex}}16-6=10
  2. \phantom{\rule{0.2em}{0ex}}4\cdot 2+1
  3. \phantom{\rule{0.2em}{0ex}}x\div 25
  4. \phantom{\rule{0.2em}{0ex}}y+8=40
Solution
a. \phantom{\rule{0.2em}{0ex}}16-6=10 This is an equation—two expressions are connected with an equal sign.
b. \phantom{\rule{0.2em}{0ex}}4\cdot 2+1 This is an expression—no equal sign.
c. \phantom{\rule{0.2em}{0ex}}x\div 25 This is an expression—no equal sign.
d. \phantom{\rule{0.2em}{0ex}}y+8=40 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. \phantom{\rule{0.2em}{0ex}}23+6=29\phantom{\rule{0.4em}{0ex}}
  2. \phantom{\rule{0.2em}{0ex}}7\cdot 3-7
Show Answer
  1. equation
  2. expression

TRY IT 4.2

Determine if each is an expression or an equation:

  1. \phantom{\rule{0.2em}{0ex}}y\div 14\phantom{\rule{0.4em}{0ex}}
  2. \phantom{\rule{0.2em}{0ex}}x-6=21
Show Answer
  1. expression
  2. equation

Simplify Expressions with Exponents

To simplify a numerical expression means to do all the math possible. For example, to simplify 4\cdot2+1 we’d first multiply 4\cdot2 to get 8 and then add the 1 to get 9. 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:

4\cdot2+1
8+1
9

Suppose we have the expression 2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2. 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 2\cdot2\cdot2 as {2}^{3} and 2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2 as {2}^{9}. In expressions such as {2}^{3}, the 2 is called the base and the 3 is called the exponent. The exponent tells us how many factors of the base we have to multiply.

The image shows the number two with the number three, in superscript, to the right of the two. The number two is labeled as “base” and the number three is labeled as “exponent”.

\text{means multiply three factors of 2}

We say {2}^{3} is in exponential notation and 2\cdot2\cdot2 is in expanded notation.

Exponential Notation

For any expression {a}^{n},a is a factor multiplied by itself n times if n is a positive integer.

{a}^{n}\text{means multiply}\phantom{\rule{0.2em}{0ex}}n\phantom{\rule{0.2em}{0ex}}\text{factors of}\phantom{\rule{0.2em}{0ex}}a

At the top of the image is the letter a with the letter n, in superscript, to the right of the a. The letter a is labeled as “base” and the letter n is labeled as “exponent”. Below this is the letter a with the letter n, in superscript, to the right of the a set equal to n factors of a.

The expression {a}^{n} is read a to the {n}^{th} power.

For powers of n=2 and n=3, we have special names.

\begin{array}{l}{a}^{2}\phantom{\rule{0.2em}{0ex}}\text{is read as}\phantom{\rule{0.2em}{0ex}}\text{"}a\phantom{\rule{0.2em}{0ex}}\text{squared"}\\ {a}^{3}\phantom{\rule{0.2em}{0ex}}\text{is read as}\phantom{\rule{0.2em}{0ex}}\text{"}a\phantom{\rule{0.2em}{0ex}}\text{cubed"}\end{array}

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

Exponential Notation In Words
{7}^{2} 7 to the second power, or 7 squared
{5}^{3} 5 to the third power, or 5 cubed
{9}^{4} 9 to the fourth power
{12}^{5} 12 to the fifth power

EXAMPLE 5

Write each expression in exponential form:

  1. \phantom{\rule{0.2em}{0ex}}16\cdot16\cdot16\cdot16\cdot16\cdot16\cdot16
  2. \phantom{\rule{0.2em}{0ex}}9\cdot9\cdot9\cdot9\cdot9
  3. \phantom{\rule{0.2em}{0ex}}x\cdot x\cdot x\cdot x
  4. \phantom{\rule{0.2em}{0ex}}a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a
Solution
a. The base 16 is a factor 7 times. {16}^{7}
b. The base 9 is a factor 5 times. {9}^{5}
c. The base x is a factor 4 times. {x}^{4}
d. The base a is a factor 8 times. {a}^{8}

TRY IT 5.1

Write each expression in exponential form:

41\cdot41\cdot41\cdot41\cdot41

Show Answer

415

TRY IT 5.2

Write each expression in exponential form:

7\cdot7\cdot7\cdot7\cdot7\cdot7\cdot7\cdot7\cdot7

Show Answer

79

EXAMPLE 6

Write each exponential expression in expanded form:

  1. \phantom{\rule{0.2em}{0ex}}{8}^{6}\phantom{\rule{0.2em}{0ex}}
  2. \phantom{\rule{0.2em}{0ex}}{x}^{5}
Solution

a. The base is 8 and the exponent is 6, so {8}^{6} means 8\cdot 8\cdot 8\cdot 8\cdot 8\cdot 8

b. The base is x and the exponent is 5, so {x}^{5} means x\cdot x\cdot x\cdot x\cdot x

TRY IT 6.1

Write each exponential expression in expanded form:

  1. \phantom{\rule{0.2em}{0ex}}{4}^{8}
  2. \phantom{\rule{0.2em}{0ex}}{a}^{7}
Show Answer
  1. 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4
  2. a \cdot a \cdot a \cdot a \cdot a \cdot a \cdot a

TRY IT 6.2

Write each exponential expression in expanded form:

  1. {\phantom{\rule{0.2em}{0ex}}8}^{8}\phantom{\rule{0.4em}{0ex}}
  2. \phantom{\rule{0.2em}{0ex}}{b}^{6}
Show Answer
  1. 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8
  2. b \cdot b \cdot b \cdot b \cdot b \cdot b

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

EXAMPLE 7

Simplify: {3}^{4}.

Solution
{3}^{4}
Expand the expression. 3\cdot 3\cdot 3\cdot 3
Multiply left to right. 9\cdot 3\cdot 3
27\cdot 3
Multiply. 81

TRY IT 7.1

Simplify:

  1. \phantom{\rule{0.2em}{0ex}}{5}^{3}\phantom{\rule{0.4em}{0ex}}
  2. \phantom{\rule{0.2em}{0ex}}{1}^{7}
Show Answer
  1. 125
  2. 1

TRY IT 7.2

Simplify:

  1. \phantom{\rule{0.2em}{0ex}}{7}^{2}\phantom{\rule{0.4em}{0ex}}
  2. \phantom{\rule{0.2em}{0ex}}{0}^{5}
Show Answer
  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:

4+3\cdot 7
\begin{array}{cccc}\hfill \text{Some students say it simplifies to 49.}\hfill & \phantom{\rule{2em}{0ex}}& & \hfill \text{Some students say it simplifies to 25.}\hfill \\ \begin{array}{ccc}& & \hfill 4+3\cdot 7\hfill \\ \text{Since}\phantom{\rule{0.2em}{0ex}}4+3\phantom{\rule{0.2em}{0ex}}\text{gives 7.}\hfill & \phantom{\rule{2em}{0ex}}& \hfill 7\cdot 7\hfill \\ \text{And}\phantom{\rule{0.2em}{0ex}}7\cdot 7\phantom{\rule{0.2em}{0ex}}\text{is 49.}\hfill & \phantom{\rule{2em}{0ex}}& \hfill 49\hfill \end{array}& & & \begin{array}{ccc}& & \hfill 4+3\cdot 7\hfill \\ \phantom{\rule{0.2em}{0ex}}\text{Since}\phantom{\rule{0.2em}{0ex}}3\cdot 7\phantom{\rule{0.2em}{0ex}}\text{is 21.}\hfill & & \hfill 4+21\hfill \\ \phantom{\rule{0.2em}{0ex}}\text{And}\phantom{\rule{0.2em}{0ex}}21+4\phantom{\rule{0.2em}{0ex}}\text{makes 25.}\hfill & & \hfill 25\hfill \end{array}\hfill \end{array}

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.

4. Addition and Subtraction

  • 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:

  1. \phantom{\rule{0.2em}{0ex}}4+3\cdot 7\phantom{\rule{0.2em}{0ex}}
  2. \phantom{\rule{0.2em}{0ex}}\left(4+3\right)\cdot 7
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. \phantom{\rule{0.2em}{0ex}}12-5\cdot 2\phantom{\rule{0.4em}{0ex}}
  2. \phantom{\rule{0.2em}{0ex}}\left(12-5\right)\cdot 2
Show Answer
  1. 2
  2. 14

TRY IT 8.2

Simplify the expressions:

  1. \phantom{\rule{0.2em}{0ex}}8+3\cdot 9\phantom{\rule{0.4em}{0ex}}
  2. \left(8+3\right)\cdot 9
Show Answer
  1. 35
  2. 99

EXAMPLE 9

Simplify:

  1. \phantom{\rule{0.2em}{0ex}}\text{18}\div \text{9}\cdot \text{2}\phantom{\rule{0.2em}{0ex}}
  2. \phantom{\rule{0.2em}{0ex}}\text{18}\cdot \text{9}\div \text{2}
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:

42\div 7\cdot 3

Show Answer

18

TRY IT 9.2

Simplify:

12\cdot 3\div 4

Show Answer

9

EXAMPLE 10

Simplify: 18\div 6+4\left(5-2\right).

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:

30\div 5+10\left(3-2\right)

Show Answer

16

TRY IT 10.2

Simplify:

70\div 10+4\left(6-2\right)

Show Answer

23

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

EXAMPLE 11

\text{Simplify:}\phantom{\rule{0.2em}{0ex}}5+{2}^{3}+3\left[6-3\left(4-2\right)\right].

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:

9+{5}^{3}-\left[4\left(9+3\right)\right]

Show Answer

86

TRY IT 11.2

Simplify:

{7}^{2}-2\left[4\left(5+1\right)\right]

Show Answer

1

EXAMPLE 12

Simplify: {2}^{3}+{3}^{4}\div 3-{5}^{2}.

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:

{3}^{2}+{2}^{4}\div 2+{4}^{3}

Show Answer

81

TRY IT 12.2

Simplify:

{6}^{2}-{5}^{3}\div 5+{8}^{2}

Show Answer

75

ACCESS ADDITIONAL ONLINE RESOURCES

Key Concepts

Operation Notation Say: The result is…
Addition a+b a\phantom{\rule{0.2em}{0ex}}\text{plus}\phantom{\rule{0.2em}{0ex}}b the sum of a and b
Multiplication a\cdot b,\left(a\right)\left(b\right),\left(a\right)b,a\left(b\right) a\phantom{\rule{0.2em}{0ex}}\text{times}\phantom{\rule{0.2em}{0ex}}b The product of a and b
Subtraction a-b a\phantom{\rule{0.2em}{0ex}}\text{minus}\phantom{\rule{0.2em}{0ex}}b the difference of a and b
Division a\div b,a/b,\phantom{\rule{0.2em}{0ex}}\frac{a}{b},b\overline{)a} a divided by b The quotient of a and b
Algebraic Notation Say
a=b a is equal to b
a\ne b a is not equal to b
a < b a is less than b
a > b a is greater than b
a\le b a is less than or equal to b
a\ge b a is greater than or equal to b

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

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. 16-9 2. 25-7
3. 5 \cdot 6 4. 3 \cdot 9
5. 28 \div 4 6. 45 \div 5
7. x+8 8. x+11
9. \left(2\right)\left(7\right) 10. \left(4\right)\left(8\right)
11. 14 < 21 12. 17 < 35
13. 36\ge 19 14. 42\ge 27
15. 3n=24 16. 6n=36
17. y-1 > 6 18. y-4 > 8
19. 2\le 18 \div 6 20. 3\le 20 \div 4
21. a\ne 7\cdot 4 22. a\ne 1\cdot 12

Identify Expressions and Equations

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

23. 9\cdot 6=54 24. 7\cdot 9=63
25. 5\cdot 4+3 26. 6\cdot 3+5
27. x+7 28. x+9
29. y-5=25 30. y-8=32

Simplify Expressions with Exponents

In the following exercises, write in exponential form.

31. 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3 32. 4\cdot 4\cdot 4\cdot 4\cdot 4\cdot 4
33. x\cdot x\cdot x\cdot x\cdot x 34. y\cdot y\cdot y\cdot y\cdot y\cdot y

In the following exercises, write in expanded form.

35. {5}^{3} 36. {8}^{3}
37. {2}^{8} 38. {10}^{5}

Simplify Expressions Using the Order of Operations

In the following exercises, simplify.

39.

a.\phantom{\rule{0.2em}{0ex}}3+8\cdot 5\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.4em}{0ex}}

b.\phantom{\rule{0.2em}{0ex}}\text{(3+8)}\cdot \text{5}

40.

a.\phantom{\rule{0.2em}{0ex}}2+6\cdot 3\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.4em}{0ex}}

b.\phantom{\rule{0.2em}{0ex}}\text{(2+6)}\cdot \text{3}

41. {2}^{3}-12 \div \left(9-5\right) 42. {3}^{2}-18\div \left(11-5\right)
43. 3\cdot 8+5\cdot 2 44. 4\cdot 7+3\cdot 5
45. 2+8\left(6+1\right) 46. 4+6\left(3+6\right)
47. 4\cdot 12/8 48. 2\cdot 36/6
49. 6+10/2+2 50. 9+12/3+4
51. \left(6+10\right)\div \left(2+2\right) 52. \left(9+12\right)\div \left(3+4\right)
53. 20\div 4+6\cdot5 54. 33\div 3+8\cdot2
55. 20\div \left(4+6\right)\cdot 5 56. 33\div \left(3+8\right)\cdot 2
57. {4}^{2}+{5}^{2} 58. {3}^{2}+{7}^{2}
59. {\left(4+5\right)}^{2} 60. {\left(3+7\right)}^{2}
61. 3\left(1+9\cdot 6\right)-{4}^{2} 62. 5\left(2+8\cdot 4\right)-{7}^{2}
63. 2\left[1+3\left(10-2\right)\right] 64. 5\left[2+4\left(3-2\right)\right]

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 83'' Rashard Lewis 82''
Boris Diaw 80'' LeBron James 80''
Kawhi Leonard 79'' Chris Bosh 83''
Tony Parker 74'' Dwyane Wade 76''
Danny Green 78'' Ray Allen 77''
  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 50 mountains with an elevation of over 14,000\phantom{\rule{0.2em}{0ex}}\text{feet.} The table shows the ten tallest. Use this table to fill in the appropriate inequality symbol.

Mountain Elevation
Mt. Elbert 14,433'
Mt. Massive 14,421'
Mt. Harvard 14,420'
Blanca Peak 14,345'
La Plata Peak 14,336'
Uncompahgre Peak 14,309'
Crestone Peak 14,294'
Mt. Lincoln 14,286'
Grays Peak 14,270'
Mt. Antero 14,269'
  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?

Answers

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.

Attributions

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.

3

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 x+7 when

  1.  \phantom{\rule{0.2em}{0ex}}x=3
  2.  \phantom{\rule{0.2em}{0ex}}x=12
Solution

a. To evaluate, substitute 3 for x in the expression, and then simplify.

.
Substitute. .
Add. .

When x=3, the expression x+7 has a value of 10.

b. To evaluate, substitute 12 for x in the expression, and then simplify.

.
Substitute. .
Add. .

When x=12, the expression x+7 has a value of 19.

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 x were different. When we evaluate an expression, the value varies depending on the value used for the variable.

TRY IT 1.1

Evaluate:

y+4\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}

  1.  \phantom{\rule{0.2em}{0ex}}y=6\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}
  2.  \phantom{\rule{0.2em}{0ex}}y=15
Show Answer
  1.  10
  2.  19

TRY IT 1.2

Evaluate:

a-5\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}

  1.  \phantom{\rule{0.2em}{0ex}}a=9\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}
  2.  \phantom{\rule{0.2em}{0ex}}a=17
Show Answer
  1.  4
  2.  12

EXAMPLE 2

Evaluate 9x-2,\text{when}\phantom{\rule{0.2em}{0ex}}

  1.  \phantom{\rule{0.2em}{0ex}}x=5\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}
  2.  \phantom{\rule{0.2em}{0ex}}x=1
Solution

Remember ab means a times b, so 9x means 9 times x.

a. To evaluate the expression when x=5, we substitute 5 for x, and then simplify.

.
. .
Multiply. .
Subtract. .

b. To evaluate the expression when x=1, we substitute 1 for x, and then simplify.

.
. .
Multiply. .
Subtract. .

Notice that in part a) that we wrote 9\cdot 5 and in part b) we wrote 9\left(1\right). Both the dot and the parentheses tell us to multiply.

TRY IT 2.1

Evaluate:

8x-3,\text{when}\phantom{\rule{0.2em}{0ex}}

  1. \phantom{\rule{0.2em}{0ex}}x=2\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}
  2. \phantom{\rule{0.2em}{0ex}}x=1
Show Answer
  1.  13
  2.  5

TRY IT 2.2

Evaluate:

4y-4,\text{when}\phantom{\rule{0.2em}{0ex}}

  1. \phantom{\rule{0.2em}{0ex}}y=3\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}
  2. \phantom{\rule{0.2em}{0ex}}y=5
Show Answer
  1.  8
  2.  16

EXAMPLE 3

Evaluate {x}^{2} when x=10.

Solution

We substitute 10 for x, and then simplify the expression.

.
. .
Use the definition of exponent. .
Multiply. .

When x=10, the expression {x}^{2} has a value of 100.

TRY IT 3.1

Evaluate:

{x}^{2}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=8.

Show Answer

64

TRY IT 3.2

Evaluate:

{x}^{3}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=6.

Show Answer

216

EXAMPLE 4

\text{Evaluate}\phantom{\rule{0.2em}{0ex}}{2}^{x}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=5.

Solution

In this expression, the variable is an exponent.

.
. .
Use the definition of exponent. .
Multiply. .

When x=5, the expression {2}^{x} has a value of 32.

TRY IT 4.1

Evaluate:

{2}^{x}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=6.

Show Answer

64

TRY IT 4.2

Evaluate:

{3}^{x}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=4.

Show Answer

81

EXAMPLE 5

\text{Evaluate}\phantom{\rule{0.2em}{0ex}}3x+4y-6\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=10\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=2.

Solution

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

.
. .
Multiply. .
Add and subtract left to right. .

When x=10 and y=2, the expression 3x+4y-6 has a value of 32.

TRY IT 5.1

Evaluate:

2x+5y-4\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=11\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=3

Show Answer

33

TRY IT 5.2

Evaluate:

5x-2y-9\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=7\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=8

Show Answer

10

EXAMPLE 6

\text{Evaluate}\phantom{\rule{0.2em}{0ex}}2{x}^{2}+3x+8\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=4.

Solution

We need to be careful when an expression has a variable with an exponent. In this expression, 2{x}^{2} means 2\cdot x\cdot x and is different from the expression {\left(2x\right)}^{2}, which means 2x\cdot 2x.

.
. .
Simplify {4}^{2}. .
Multiply. .
Add. .

TRY IT 6.1

Evaluate:

3{x}^{2}+4x+1\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=3.

Show Answer

40

TRY IT 6.2

Evaluate:

6{x}^{2}-4x-7\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=2.

Show Answer

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 7,y,5{x}^{2},9a,\text{and}\phantom{\rule{0.2em}{0ex}}13xy.

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 3x is 3. When we write x, the coefficient is 1, since x=1\cdot x. The table below gives the coefficients for each of the terms in the left column.

Term Coefficient
7 7
9a 9
y 1
5{x}^{2} 5

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
7 7
y y
x+7 x,7
2x+7y+4 2x,7y,4
3{x}^{2}+4{x}^{2}+5y+3 3{x}^{2},4{x}^{2},5y,3

EXAMPLE 7

Identify each term in the expression 9b+15{x}^{2}+a+6. Then identify the coefficient of each term.

Solution

The expression has four terms. They are 9b,15{x}^{2},a, and 6.

The coefficient of 9b is 9.

The coefficient of 15{x}^{2} is 15.

Remember that if no number is written before a variable, the coefficient is 1. So the coefficient of a is 1.

The coefficient of a constant is the constant, so the coefficient of 6 is 6.

TRY IT 7.1

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

4x+3b+2

Show Answer

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:

9a+13{a}^{2}+{a}^{3}

Show Answer

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?

5x,7,{n}^{2},4,3x,9{n}^{2}

Which of these terms are like terms?

Terms are called like terms if they have the same variables and exponents. All constant terms are also like terms. So among the terms 5x,7,{n}^{2},4,3x,9{n}^{2},

7\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}4\phantom{\rule{0.2em}{0ex}}\text{are like terms.}
5x\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}3x\phantom{\rule{0.2em}{0ex}}\text{are like terms.}
{n}^{2}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}9{n}^{2}\phantom{\rule{0.2em}{0ex}}\text{are like 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:

  1. \phantom{\rule{0.2em}{0ex}}{y}^{3},7{x}^{2},14,23,4{y}^{3},9x,5{x}^{2}
  2. \phantom{\rule{0.2em}{0ex}}4{x}^{2}+2x+5{x}^{2}+6x+40x+8xy
Solution

a. \phantom{\rule{0.2em}{0ex}}{y}^{3},7{x}^{2},14,23,4{y}^{3},9x,5{x}^{2}

Look at the variables and exponents. The expression contains {y}^{3},{x}^{2},x, and constants.

The terms {y}^{3} and 4{y}^{3} are like terms because they both have {y}^{3}.

The terms 7{x}^{2} and 5{x}^{2} are like terms because they both have {x}^{2}.

The terms 14 and 23 are like terms because they are both constants.

The term 9x does not have any like terms in this list since no other terms have the variable x raised to the power of 1.

b. \phantom{\rule{0.2em}{0ex}}4{x}^{2}+2x+5{x}^{2}+6x+40x+8xy

Look at the variables and exponents. The expression contains the terms 4{x}^{2},2x,5{x}^{2},6x,40x,\text{and}\phantom{\rule{0.2em}{0ex}}8xy

The terms 4{x}^{2} and 5{x}^{2} are like terms because they both have {x}^{2}.

The terms 2x,6x,\text{and}\phantom{\rule{0.2em}{0ex}}40x are like terms because they all have x.

The term 8xy has no like terms in the given expression because no other terms contain the two variables xy.

TRY IT 8.1

Identify the like terms in the list or the expression:

9,2{x}^{3},{y}^{2},8{x}^{3},15,9y,11{y}^{2}

Show Answer

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

TRY IT 8.2

Identify the like terms in the list or the expression:

4{x}^{3}+8{x}^{2}+19+3{x}^{2}+24+6{x}^{3}

Show Answer

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 3x+6x would simplify to? If you thought 9x, you would be right!

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

The image shows the expression 3 x plus 6 x. The 3 x represents x plus x plus x. The 6 x represents x plus x plus x plus x plus x plus x. The expression 3 x plus 6 x becomes x plus x plus x plus x plus x plus x plus x plus x plus x. This simplifies to a total of 9 x's or the term 9 x.

Add the coefficients and keep the same variable. It doesn’t matter what x is. If you have 3 of something and add 6 more of the same thing, the result is 9 of them. For example, 3 oranges plus 6 oranges is 9 oranges. We will discuss the mathematical properties behind this later.

The expression 3x+6x 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.

The image shows the expression 3 x plus 4 y plus 2 x plus 6 y. The position of the middle terms, 4 y and 2 x, can be switched so that the expression becomes 3 x plus 2 x plus 4 y plus 6 y. Now the terms containing x are together and the terms containing y are together.

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: 3x+7+4x+5.

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:

7x+9+9x+8

Show Answer

16x + 17

TRY IT 9.2

Simplify:

5y+2+8y+4y+5

Show Answer

17y + 7

EXAMPLE 10

Simplify the expression: 7{x}^{2}+8x+{x}^{2}+4x.

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 8{x}^{2}+12x is in simplest form.

TRY IT 10.1

Simplify:

3{x}^{2}+9x+{x}^{2}+5x

Show Answer

4x2 + 14x

TRY IT 10.2

Simplify:

11{y}^{2}+8y+{y}^{2}+7y

Show Answer

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
Addition a plus b
the sum of a and b
a increased by b
b more than a
the total of a and b
b added to a
a+b
Subtraction a minus b
the difference of a and b
b subtracted from a
a decreased by b
b less than a
a-b
Multiplication a times b
the product of a and b
a\cdot b, ab, a\left(b\right), \left(a\right)\left(b\right)
Division a divided by b
the quotient of a and b
the ratio of a and b
b divided into a
a \div b, a/b, \frac{a}{b}, b\overline{)a}

Look closely at these phrases using the four operations:

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 20 and 4
  2.  the quotient of 10x and 3
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.

\begin{array}{cccc}\\ \text{the difference}\phantom{\rule{0.2em}{0ex}}\text{of}\phantom{\rule{0.2em}{0ex}}20\phantom{\rule{0.2em}{0ex}}and\phantom{\rule{0.2em}{0ex}}4\hfill \\ 20\phantom{\rule{0.2em}{0ex}}\text{minus}\phantom{\rule{0.2em}{0ex}}4\hfill \\ 20-4\hfill \end{array}

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

\begin{array}{ccccc}\\ \text{the quotient of}\phantom{\rule{0.2em}{0ex}}10x\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}3\hfill \\ \text{divide}\phantom{\rule{0.2em}{0ex}}10x\phantom{\rule{0.2em}{0ex}}\text{by}\phantom{\rule{0.2em}{0ex}}3\hfill \\ 10x\div 3\hfill \end{array}

This can also be written as \begin{array}{l}10x/3\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.4em}{0ex}}\frac{10x}{3}\hfill \end{array}

TRY IT 11.1

Translate the given word phrase into an algebraic expression:

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

TRY IT 11.2

Translate the given word phrase into an algebraic expression:

  1.  the sum of 17 and 19
  2.  the product of 7 and x
Show Answer
  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 8 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 7 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 y
  2.  Seven less than 9z
Solution

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

\begin{array}{l}\text{Eight more than}\phantom{\rule{0.2em}{0ex}}y\\ \text{Eight added to}\phantom{\rule{0.2em}{0ex}}y\\ y+8\end{array}

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

\begin{array}{l}\text{Seven less than}\phantom{\rule{0.2em}{0ex}}9z\\ \text{Seven subtracted from}\phantom{\rule{0.2em}{0ex}}9z\\ 9z-7\end{array}

TRY IT 12.1

Translate each word phrase into an algebraic expression:

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

TRY IT 12.2

Translate each word phrase into an algebraic expression:

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

EXAMPLE 13

Translate each word phrase into an algebraic expression:

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

a. There are two operation words: times tells us to multiply and sum tells us to add. Because we are multiplying 5 times the sum, we need parentheses around the sum of m and n.

five times the sum of m and n
\begin{array}{cccc}\\ \\ \phantom{\rule{4em}{0ex}}5\left(m+n\right)\hfill \end{array}

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 m and n.

the sum of five times m and n
\begin{array}{ccccc}\\ \\ \phantom{\rule{4em}{0ex}}5m+n\hfill \end{array}

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 p and q
  2. the sum of four times p and q
Show Answer
  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 x\phantom{\rule{0.2em}{0ex}}\text{and 8}\phantom{\rule{0.2em}{0ex}}
  2. two times the difference of x\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}8
Show Answer
  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 6 inches less than the width. Let w represent the width of the window. Write an expression for the height of the window.

Solution
Write a phrase about the height. 6 less than the width
Substitute w for the width. 6 less than w
Rewrite ‘less than’ as ‘subtracted from’. 6 subtracted from w
Translate the phrase into algebra. w-6

TRY IT 14.1

The length of a rectangle is 5 inches less than the width. Let w represent the width of the rectangle. Write an expression for the length of the rectangle.

Show Answer

w − 5

TRY IT 14.2

The width of a rectangle is 2 metres greater than the length. Let l represent the length of the rectangle. Write an expression for the width of the rectangle.

Show Answer

l + 2

EXAMPLE 15

Blanca has dimes and quarters in her purse. The number of dimes is 2 less than 5 times the number of quarters. Let q 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 q for the number of quarters. 2 less than five times q
Translate 5 times q. 2 less than 5q
Translate the phrase into algebra. 5q-2

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 q represent the number of quarters. Write an expression for the number of dimes.

Show Answer

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 n represent the number of nickels. Write an expression for the number of dimes.

Show Answer

4n + 8

ACCESS ADDITIONAL ONLINE RESOURCES

Key Concepts

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. 7x+8\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=2 2. 9x+7\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=3
3. 5x-4\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=6 4. 8x-6\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=7
5. {x}^{2}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=12 6. {x}^{3}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=5
7. {x}^{5}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=2 8. {x}^{4}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=3
9. {3}^{x}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=3 10. {4}^{x}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=2
11. {x}^{2}+3x-7\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=4 12. {x}^{2}+5x-8\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=6
13. 2x+4y-5\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=7,y=8 14. 6x+3y-9\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=6,y=9
15. {\left(x-y\right)}^{2}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=10,y=7 16. {\left(x+y\right)}^{2}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=6,y=9
17. {a}^{2}+{b}^{2}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}a=3,b=8 18. {r}^{2}-{s}^{2}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}r=12,s=5
19. 2l+2w\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}l=15,w=12 20. 2l+2w\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}l=18,w=14

Identify Terms, Coefficients, and Like Terms

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

21. 15{x}^{2}+6x+2 22. 11{x}^{2}+8x+5
23. 10{y}^{3}+y+2 24. 9{y}^{3}+y+5

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

25. 8a 26. 13m
27. 5{r}^{2} 28. 6{x}^{3}

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

29. {x}^{3},8x,14,8y,5,8{x}^{3} 30. 6z,3{w}^{2},1,6{z}^{2},4z,{w}^{2}
31. 9a,{a}^{2},16ab,16{b}^{2},4ab,9{b}^{2} 32. 3,25{r}^{2},10s,10r,4{r}^{2},3s

Simplify Expressions by Combining Like Terms

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

33. 10x+3x 34. 15x+4x
35. 17a+9a 36. 18z+9z
37. 4c+2c+c 38. 6y+4y+y
39. 9x+3x+8 40. 8a+5a+9
41. 7u+2+3u+1 42. 8d+6+2d+5
43. 7p+6+5p+4 44. 8x+7+4x-5
45. 10a+7+5a-2+7a-4 46. 7c+4+6c-3+9c-1
47. 3{x}^{2}+12x+11+14{x}^{2}+8x+5 48. 5{b}^{2}+9b+10+2{b}^{2}+3b-4

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 x and 4 58. 3 less than x
59. The product of 6 and y 60. The product of 9 and y
61. The sum of 8x and 3x 62. The sum of 13x and 3x
63. The quotient of y and 3 64. The quotient of y and 8
65. Eight times the difference of y and nine 66. Seven times the difference of y and one
67. Five times the sum of x and y 68.  times five less than twice x

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 b 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 3 more than the number of classical CDs. Let c 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 4 less than the number of boys. Let b represent the number of boys. Write an expression for the number of girls. 72. Marcella has 6 fewer male cousins than female cousins. Let f 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 n 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 t 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 x and y to help you explain. 78. Explain the difference between \text{``4} times the sum of x and y\text{''} and “the sum of 4 times x and y\text{.''}

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. \frac{y}{3} 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 0. The negative numbers are to the left of zero on the number line. See Figure 1.

A number line extends from negative 4 to 4. A bracket is under the values “negative 4” to “0” and is labeled “Negative numbers”. Another bracket is under the values 0 to 4 and labeled “positive numbers”. There is an arrow in between both brackets pointing upward to zero.
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.

A number line ranges from negative 4 to 4. An arrow above the number line extends from negative 1 towards 4 and is labeled “larger”. An arrow below the number line extends from 1 towards negative 4 and is labeled “smaller”.
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 \text{…}-3,-2,-1,0,1,2,3\text{…}

A number line extends from negative four to four. Points are plotted at negative four, negative three, negative two, negative one, zero, one, two, 3, and four.
Figure 3 All the marked numbers are called integers.

EXAMPLE 1

Order each of the following pairs of numbers, using < or >: a) 14___6 b) -1___9 c) -1___-4 d) 2___-20.

Solution

It may be helpful to refer to the number line shown.

A number line ranges from negative twenty to fifteen with ticks marks between numbers. Every fifth tick mark is labeled a number. Points are plotted at points negative twenty, negative 4, negative 1, 2, 6, 9 and 14.

a) 14 is to the right of 6 on the number line. 14___6  14 > 6
b) −1 is to the left of 9 on the number line. -1___9  -1<9
c) −1 is to the right of −4 on the number line. -1___-4
d) 2 is to the right of −20 on the number line. 2___-20 2 > -20

TRY IT 1.1

Order each of the following pairs of numbers, using < or > \text{:} a) 15___7 b) -2___5 c) -3___-7
d) 5___-17.

Show answer

a) > b) < c) > d) >

TRY IT 1.2

Order each of the following pairs of numbers, using < or > \text{:} a) 8___13 b) 3___-4 c) -5___-2
d) 9___-21.

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 -2 are the same distance from zero, they are called opposites. The opposite of 2 is -2, and the opposite of -2 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 -3.

A number line ranges from negative 4 to 4. There are two brackets above the number line. The bracket on the left spans from negative three to 0. The bracket on the right spans from zero to three. Points are plotted on both negative three and three.
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.

\begin{array}{cccc}10-4\hfill & & & \text{Between two numbers, it indicates the operation of}\phantom{\rule{0.2em}{0ex}}\mathit{\text{subtraction}}.\hfill \\ & & & \text{We read}\phantom{\rule{0.2em}{0ex}}10-4\phantom{\rule{0.2em}{0ex}}\text{as}\phantom{\rule{0.2em}{0ex}}\text{``}10\phantom{\rule{0.2em}{0ex}}\text{minus}\phantom{\rule{0.2em}{0ex}}4.\text{''}\hfill \\ -8\hfill & & & \text{In front of a number, it indicates a}\phantom{\rule{0.2em}{0ex}}\mathit{\text{negative}}\phantom{\rule{0.2em}{0ex}}\text{number.}\hfill \\ & & & \text{We read}\phantom{\rule{0.2em}{0ex}}-8\phantom{\rule{0.2em}{0ex}}\text{as ``negative eight.''}\hfill \\ -x\hfill & & & \text{In front of a variable, it indicates the}\phantom{\rule{0.2em}{0ex}}\mathit{\text{opposite}}.\phantom{\rule{0.2em}{0ex}}\text{We read}\phantom{\rule{0.2em}{0ex}}-x\phantom{\rule{0.2em}{0ex}}\text{as ``the opposite of}\phantom{\rule{0.2em}{0ex}}x.\text{''}\hfill \\ -\left(-2\right)\hfill & & & \text{Here there are two}\phantom{\rule{0.2em}{0ex}}\text{``}-\text{''}\phantom{\rule{0.2em}{0ex}}\text{signs. The one in the parentheses tells us the number is}\hfill \\ & & & \text{negative}\phantom{\rule{0.2em}{0ex}}2.\phantom{\rule{0.2em}{0ex}}\text{The one outside the parentheses tells us to take the}\phantom{\rule{0.2em}{0ex}}\mathit{\text{opposite}}\phantom{\rule{0.2em}{0ex}}\text{of}\phantom{\rule{0.2em}{0ex}}-2.\hfill \\ & & & \text{We read}\phantom{\rule{0.2em}{0ex}}-\left(-2\right)\phantom{\rule{0.2em}{0ex}}\text{as ``the opposite of negative two.''}\hfill \end{array}

10-4 Between two numbers, it indicates the operation of subtraction.
We read 10-4 as “10 minus 4.”
-8 In front of a number, it indicates a negative number.
We read −8 as “negative eight.”
-x In front of a variable, it indicates the opposite. We read -x as “the opposite of x.”
-\left(-2\right) 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 -\left(-2\right) as “the opposite of negative two.”

Opposite Notation

-a means the opposite of the number a.

The notation -a is read as “the opposite of a.”

EXAMPLE 2

Find: a) the opposite of 7 b) the opposite of -10 c) -\left(-6\right).

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 -3 c) -\left(-1\right).

Show answer

a)-4 b) 3 c) 1

 

TRY IT 2.2

Find: a) the opposite of 8 b) the opposite of -5 c) -\left(-5\right).

Show answer

a)-8 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 \text{…}-3,-2,-1,0,1,2,3\text{…}

Integers

The whole numbers and their opposites are called the integers.

The integers are the numbers

\text{…}-3,-2,-1,0,1,2,3\text{…}

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 -x is positive or negative. We can see this in Example 3.

EXAMPLE 3

Evaluate a) -x, when x=8 b) -x, when x=-8.

Solution
  1. .
    x
    . .
    Write the opposite of 8. .

     

  2. .
    x
    . .
    Write the opposite of −8. 8

TRY IT 3.1

Evaluate -n, when a) n=4 b) n=-4.

Show answer

a)-4 b) 4

TRY IT 3.2

Evaluate -m, when a) m=11 b) m=-11.

Show answer

a)-11 b) 11

Simplify: Expressions with Absolute Value

We saw that numbers such as 2\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}-2 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 |n|.

For example,

Figure 5 illustrates this idea.

The integers 5\phantom{\rule{0.2em}{0ex}}\text{and are}\phantom{\rule{0.2em}{0ex}}5 units away from 0.

A number line is shown ranging from negative 5 to 5. A bracket labeled “5 units” lies above the points negative 5 to 0. An arrow labeled “negative 5 is 5 units from 0, so absolute value of negative 5 equals 5.” is written above the labeled bracket. A bracket labeled “5 units” lies above the points “0” to “5”. An arrow labeled “5 is 5 units from 0, so absolute value of 5 equals 5.” and is written above the labeled bracket.
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 0\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}0 on the number line is zero units.

Property of Absolute Value

|n|\ge 0 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) |3| b) |-44| c) |0|.

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) |3|
\phantom{\rule{1.2em}{0ex}}3

b) |-44|
\phantom{\rule{1.5em}{0ex}}44

c) |0|
0\phantom{\rule{1.5em}{0ex}}

TRY IT 4.1

Simplify: a) |4| b) |-28| c) |0|.

Show answer

a) 4 b) 28 c) 0

TRY IT 4.2

Simplify: a) |-13| b) |47| c) |0|.

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 <, >, \phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}= for each of the following pairs of numbers:

a) |-5|___-|-5| b) 8___-|-8|c) -9___-|-9| d) –-16___-|-16|

Solution
|-5|___ –|-5|
a) Simplify.
Order.
5 ___ -5
5  > -5
|-5|  > –|-5|
b) Simplify.
Order.
8 ___ –|-8|
8 ___ -8
8 > -8
8 > –|-8|
c) Simplify.
Order.
9 ___ –|-9|

-9 ___ -9

-9 = -9

-9 = –|-9|

d) Simplify.
Order.
(-16) ___ –|-16|

16 ____ -16

16 > -16

(-16) > –|-16|

TRY IT 5.1

Fill in <, >, or = for each of the following pairs of numbers: a) |-9|___-|-9| b) 2___-|-2| c) -8___|-8|
d) \-(-9___-|-9|.

Show answer

a) > b) > c) < d) >

TRY IT 5.2

Fill in <, >, or = for each of the following pairs of numbers: a) 7___-|-7| b) \-(-10___-|-10|
c) |-4|___-|-4| d) -1___|-1|.

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.

Grouping Symbols
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: 24-|19-3\left(6-2\right)|.

Solution
24-|19-3\left(6-2\right)|
Work inside parentheses first: subtract 2 from 6. 24-|19-3\left(4\right)|
Multiply 3(4). 24-|19-12|
Subtract inside the absolute value bars. 24-|7|
Take the absolute value. 24-7
Subtract. 17

TRY IT 6.1

Simplify: 19-|11-4\left(3-1\right)|.

Show answer

16

TRY IT 6.2

Simplify: 9-|8-4\left(7-5\right)|.

Show answer

9

EXAMPLE 7

Evaluate: a) |x|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=-35 b) |-y|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}y=-20 c) -|u|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}u=12 d) -|p|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}p=-14.

Solution

a)|x|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=-35

|x|
. .
Take the absolute value. 35

b)|-y|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}y=-20

|-y|
. .
Simplify. |20|
Take the absolute value. 20

c)-|u|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}u=12

-|u|
. .
Take the absolute value. -12

d)-|p|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}p=-14

-|p|
. .
Take the absolute value. -14

TRY IT 7.1

Evaluate: a) |x|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=-17 b) |-y|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}y=-39 c) -|m|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}m=22 d) -|p|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}p=-11.

Show answer

a) 17 b) 39 c) -22 d) -11

TRY IT 7.2

Evaluate: a) |y|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}y=-23 b) |-y|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}y=-21 c) -|n|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}n=37 d) -|q|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}q=-49.

Show answer

a) 23 b) 21 c) -37 d) -49

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.

In this image we have a blue counter above a red counter with a circle around both. The equation to the right is 1 plus negative 1 equals 0.

We will use the counters to show how to add the four addition facts using the numbers 5,-5 and 3,-3.

\begin{array}{cccccccccc}\hfill 5+3\hfill & & & \hfill -5+\left(-3\right)\hfill & & & \hfill -5+3\hfill & & & \hfill 5+\left(-3\right)\hfill \end{array}

To add 5+3, we realize that 5+3 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 -5+\left(-3\right). Watch for similarities to the last example 5+3=8.

To add -5+\left(-3\right), we realize this means the sum of -5\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}-3.

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?

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.

This figure is divided into two columns. In the left column there are eight blue counters in a horizontal row. Under them is the text “8 positives.” Centred under this is the equation 5 plus 3 equals 8. In the right column are eight red counters in a horizontal row which are labled below with the phrase “8 negatives”. Centred under this is the equation negative 5 plus negative 3 equals negative 8, where negative 3 is in parentheses.

EXAMPLE 8

Add: a) 1+4 b) -1+\left(-4\right).

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) 2+4 b) -2+\left(-4\right).

Show answer

a) 6 b) -6

TRY IT 8.2

Add: a) 2+5 b) -2+\left(-5\right).

Show answer

a) 7 b) -7

So what happens when the signs are different? Let’s add -5+3. We realize this means the sum of -5 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+\left(-3\right).

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.

Two images are shown and labeled. The left image shows five red counters in a horizontal row drawn above three blue counters in a horizontal row, where the first three pairs of red and blue counters are circled. Above this diagram is written “negative 5 plus 3” and below is written “More negatives – the sum is negative.” The right image shows five blue counters in a horizontal row drawn above three red counters in a horizontal row, where the first three pairs of red and blue counters are circled. Above this diagram is written “5 plus negative 3” and below is written “More positives – the sum is positive.”

EXAMPLE 9

Add: a) -1+5 b) 1+\left(-5\right).

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) -2+4 b) 2+\left(-4\right).

Show answer

a) 2 b) -2

TRY IT 9.2

Add: a) -2+5 b) 2+\left(-5\right).

Show answer

a) 3 b) -3

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 37+\left(-53\right), 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 53-37=16, there are 16 more red counters.

Therefore, the sum of 37+\left(-53\right) is -16.

37+\left(-53\right)=-16

Let’s try another one. We’ll add -74+\left(-27\right). Again, imagine 74 red counters and 27 more red counters, so we’d have 101 red counters. This means the sum is -101.

-74+\left(-27\right)=-101

Let’s look again at the results of adding the different combinations of 5,-5 and 3,-3.

Addition of Positive and Negative Integers

\begin{array}{cccc}\hfill 5+3\hfill & & & \hfill -5+\left(-3\right)\hfill \\ \hfill 8\hfill & & & \hfill -8\hfill \\ \hfill \text{both positive, sum positive}\hfill & & & \hfill \text{both negative, sum negative}\hfill \end{array}

When the signs are the same, the counters would be all the same color, so add them.

\begin{array}{cccc}\hfill -5+3\hfill & & & \hfill 5+\left(-3\right)\hfill \\ \hfill -2\hfill & & & \hfill 2\hfill \\ \hfill \text{different signs, more negatives, sum negative}\hfill & & & \hfill \text{different signs, more positives, sum positive}\hfill \end{array}

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) 19+\left(-47\right) b) -14+\left(-36\right).

Solution
  1.  Since the signs are different, we subtract \text{19 from 47}\text{.} The answer will be negative because there are more negatives than positives.
    \begin{array}{cccc}& & & \hfill \phantom{\rule{0.3em}{0ex}}19+\left(-47\right)\hfill \\ \text{Add.}\hfill & & & \hfill \phantom{\rule{0.3em}{0ex}}-28\hfill \end{array}
  2.  Since the signs are the same, we add. The answer will be negative because there are only negatives.
    \begin{array}{cccc}& & & \hfill -14+\left(-36\right)\hfill \\ \text{Add.}\hfill & & & \hfill -50\hfill \end{array}

TRY IT 10.1

Simplify: a) -31+\left(-19\right) b) 15+\left(-32\right).

Show answer

a)-50 b)-17

TRY IT 10.2

Simplify: a) -42+\left(-28\right) b) 25+\left(-61\right).

Show answer

a)-70 b)-36

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: -5+3\left(-2+7\right).

Solution
-5+3\left(-2+7\right)
Simplify inside the parentheses. -5+3\left(5\right)
Multiply. -5+15
Add left to right. 10

TRY IT 11.1

Simplify: -2+5\left(-4+7\right).

Show answer

13

TRY IT 11.2

Simplify: -4+2\left(-3+5\right).

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 \text{``}5-3\text{''} as \text{``}5 take away 3.\text{''} When you use counters, you can think of subtraction the same way!

We will model the four subtraction facts using the numbers 5 and 3.

\begin{array}{cccccccccc}\hfill 5-3\hfill & & & \hfill -5-\left(-3\right)\hfill & & & \hfill -5-3\hfill & & & \hfill 5-\left(-3\right)\hfill \end{array}

To subtract 5-3, we restate the problem as \text{``}5 take away 3.\text{''}

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 -5-\left(-3\right). Watch for similarities to the last example 5-3=2.

To subtract -5-\left(-3\right), we restate this as \text{``}-5 take away -3\text{''}

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.

Two images are shown and labeled. The first image shows five blue counters, three of which are circled with an arrow. Above the counters is the equation “5 minus 3 equals 2.” The second image shows five red counters, three of which are circled with an arrow. Above the counters is the equation “negative 5, minus, negative 3, equals negative 2.”

EXAMPLE 12

Subtract: a) 7-5 b) -7-\left(-5\right).

Solution
a)
Take 5 positive from 7 positives and get 2 positives.
\begin{array}{c}7-5\\ 2\end{array}
b)
Take 5 negatives from 7 negatives and get 2 negatives.
\begin{array}{c}-7-\left(-5\right)\\ -2\end{array}

TRY IT 12.1

Subtract: a) 6-4 b) -6-\left(-4\right).

Show answer

a) 2 b) -2

TRY IT 12.2

Subtract: a) 7-4 b) -7-\left(-4\right).

 

Show answer

a) 3 b) -3

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.

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, 5-\left(-3\right). 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) -3-1 b) 3-\left(-1\right).

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) -6-4 b) 6-\left(-4\right).

Show answer

a)-10 b) 10

TRY IT 13.2

Subtract: a) -7-4 b) 7-\left(-4\right).

Show answer

a)-11 b) 11

Have you noticed that subtraction of signed numbers can be done by adding the opposite? In Example 13, -3-1 is the same as -3+\left(-1\right) and 3-\left(-1\right) is the same as 3+1. You will often see this idea, the subtraction property, written as follows:

Subtraction Property

a-b=a+\left(-b\right)
Subtracting a number is the same as adding its opposite.

Look at these two examples.

Two images are shown and labeled. The first image shows four gray spheres drawn next to two gray spheres, where the four are circled in red, with a red arrow leading away to the lower left. This drawing is labeled above as “6 minus 4” and below as “2.” The second image shows four gray spheres and four red spheres, drawn one above the other and circled in red, with a red arrow leading away to the lower left, and two gray spheres drawn to the side of the four gray spheres. This drawing is labeled above as “6 plus, open parenthesis, negative 4, close parenthesis” and below as “2.”

6-4\phantom{\rule{0.2em}{0ex}}\text{gives the same answer as}\phantom{\rule{0.2em}{0ex}}6+\left(-4\right).

Of course, when you have a subtraction problem that has only positive numbers, like 6-4, you just do the subtraction. You already knew how to subtract 6-4 long ago. But knowing that 6-4 gives the same answer as 6+\left(-4\right) helps when you are subtracting negative numbers. Make sure that you understand how 6-4 and 6+\left(-4\right) give the same results!

EXAMPLE 14

Simplify: a) 13-8 and 13+\left(-8\right) b) -17-9 and -17+\left(-9\right).

Solution
a)
Subtract.
\begin{array}{c}13-8\\ 5\end{array} \begin{array}{c}13+\left(-8\right)\\ 5\end{array}
b)
Subtract.
\begin{array}{c}-17-9\\ -29\end{array} \begin{array}{c}-17+\left(-9\right)\\ -26\end{array}

TRY IT 14.1

Simplify: a) 21-13 and 21+\left(-13\right) b) -11-7 and -11+\left(-7\right).

Show answer

a) 8 b) -18

TRY IT 14.2

Simplify: a) 15-7 and 15+\left(-7\right) b) -14-8 and -14+\left(-8\right).

Show answer

a) 8 b)-22

Look at what happens when we subtract a negative.

This figure is divided vertically into two halves. The left part of the figure contains the expression 8 minus negative 5, where negative 5 is in parentheses. The expression sits above a group of 8 blue counters next to a group of five blue counters in a row, with a space between the two groups. Underneath the group of five blue counters is a group of five red counters, which are circled. The circle has an arrow pointing away toward bottom left of the image, symbolizing subtraction. Below the counters is the number 13. The right part of the figure contains the expression 8 plus 5. The expression sits above a group of 8 blue counters next to a group of five blue counters in a row, with a space between the two groups. Underneath the counters is the number 13.

8-\left(-5\right)\phantom{\rule{0.2em}{0ex}}\text{gives the same answer as}\phantom{\rule{0.2em}{0ex}}8+5

Subtracting a negative number is like adding a positive!

You will often see this written as a-\left(-b\right)=a+b.

Does that work for other numbers, too? Let’s do the following example and see.

EXAMPLE 15

Simplify: a) 9-\left(-15\right) and 9+15 b) -7-\left(-4\right) and -7+4.

Solution

a)
\begin{array}{ccccccccc}& & & & & \hfill 9-\left(-15\right)\hfill & & & \hfill 9+15\hfill \\ \text{Subtract.}\hfill & & & & & \hfill 24\hfill & & & \hfill 24\hfill \end{array}

b)
\begin{array}{ccccccccc}& & & & & \hfill -7-\left(-4\right)\hfill & & & \hfill -7+4\hfill \\ \text{Subtract.}\hfill & & & & & \hfill -3\hfill & & & \hfill -3\hfill \end{array}

a)
Subtract.
\begin{array}{c}9-\left(-15\right)\\ 24\end{array} \begin{array}{c}9+15\\ 24\end{array}
b)
Subtract.
\begin{array}{c}-7-\left(-4\right)\\ -3\end{array} \begin{array}{c}-7+4\\ -3\end{array}

TRY IT 15.1

Simplify: a) 6-\left(-13\right) and 6+13 b) -5-\left(-1\right) and -5+1.

Show answer

a)19 b)-4

TRY IT 15.2

Simplify: a) 4-\left(-19\right) and 4+19 b) -4-\left(-7\right) and -4+7.

Show answer

a) 23 b) 3

Let’s look again at the results of subtracting the different combinations of 5,-5 and 3,-3.

Subtraction of Integers

\begin{array}{cccc}\hfill 5-3\hfill & & & \hfill -5-\left(-3\right)\hfill \\ \hfill 2\hfill & & & \hfill -2\hfill \\ \hfill 5\phantom{\rule{0.2em}{0ex}}\text{positives take away}\phantom{\rule{0.2em}{0ex}}3\phantom{\rule{0.2em}{0ex}}\text{positives}\hfill & & & \hfill 5\phantom{\rule{0.2em}{0ex}}\text{negatives take away}\phantom{\rule{0.2em}{0ex}}3\phantom{\rule{0.2em}{0ex}}\text{negatives}\hfill \\ \hfill 2\phantom{\rule{0.2em}{0ex}}\text{positives}\hfill & & & \hfill 2\phantom{\rule{0.2em}{0ex}}\text{negatives}\hfill \end{array}

When there would be enough counters of the colour to take away, subtract.\begin{array}{cccc}\hfill -5-3\hfill & & & \hfill 5-\left(-3\right)\hfill \\ \hfill -8\hfill & & & \hfill 8\hfill \\ \hfill 5\phantom{\rule{0.2em}{0ex}}\text{negatives, want to take away}\phantom{\rule{0.2em}{0ex}}3\phantom{\rule{0.2em}{0ex}}\text{positives}\hfill & & & \hfill 5\phantom{\rule{0.2em}{0ex}}\text{positives, want to take away}\phantom{\rule{0.2em}{0ex}}3\phantom{\rule{0.2em}{0ex}}\text{negatives}\hfill \\ \hfill \text{need neutral pairs}\hfill & & & \hfill \text{need neutral pairs}\hfill \end{array}

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: 7-\left(-4-3\right)-9.

Solution
7-\left(-4-3\right)-9
Simplify inside the parentheses first. 7-\left(-7\right)-9
Subtract left to right. 14-9
Subtract. 5

TRY IT 16.1

Simplify: 8-\left(-3-1\right)-9.

Show answer

3

TRY IT 16.2

Simplify: 12-\left(-9-6\right)-14.

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

Glossary

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 |n|.
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: -a means the opposite of the number. The notation -a is read “the opposite of a.”

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) 9___4
b) -3___6
c) -8___-2
d) 1___-10
2.
a) -7___3
b) -10___-5
c) 2___-6
d) 8___9

In the following exercises, find the opposite of each number.

3.
a) 2
b) -6
4.
a) 9
b) -4

In the following exercises, simplify.

5. -\left(-4\right) 6. -\left(-8\right)
7. -\left(-15\right) 8. -\left(-11\right)

In the following exercises, evaluate.

9. - c when
a) c=12
b) c=-12
10. - d when
a) d=21
b) d=-21

Simplify Expressions with Absolute Value

In the following exercises, simplify.

11.
a) |-32|
b) |0|
c) |16|
12.
a) |0|
b) |-40|
c) |22|

In the following exercises, fill in <, >, or = for each of the following pairs of numbers.

13.
a) -6___|-6|
b) -|-3|___-3
14.
a) |-5|___-|-5|
b) 9___-|-9|

In the following exercises, simplify.

15. -\left(-5\right)\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}-|-5| 16. -|-9|\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}-\left(-9\right)
17. 8|-7| 18. 5|-5|
19. |15-7|-|14-6| 20. |17-8|-|13-4|
21. 18-|2\left(8-3\right)| 22. 18-|3\left(8-5\right)|

In the following exercises, evaluate.

23.
a) -|p|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}p=19
b) -|q|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}q=-33
24.
a) -|a|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}a=60
b) -|b|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}b=-12

Add Integers

In the following exercises, simplify each expression.

25. -21+\left(-59\right) 26. -35+\left(-47\right)
27. 48+\left(-16\right) 28. 34+\left(-19\right)
29. -14+\left(-12\right)+4 30. -17+\left(-18\right)+6
31. 135+\left(-110\right)+83 32. 6 -38+27+\left(-8\right)+126
33. 19+2\left(-3+8\right) 34. 24+3\left(-5+9\right)

Subtract Integers

In the following exercises, simplify.

35. 8-2 36. -6-\left(-4\right)
37. -5-4 38. -7-2
39. 8-\left(-4\right) 40. 7-\left(-3\right)
41.
a) 44-28
b) 44+\left(-28\right)
42.
a) 35-16
b) 35+\left(-16\right)
43.
a) 27-\left(-18\right)
b) 27+18
44.
a) 46-\left(-37\right)
b) 46+37

In the following exercises, simplify each expression.

45. 15-\left(-12\right) 46. 14-\left(-11\right)
47. 48-87 48. 45-69
49. -17-42 50. -19-46
51. -103-\left(-52\right) 52. -105-\left(-68\right)
53. -45-\left(54\right) 54. -58-\left(-67\right)
55. 8-3-7 56. 9-6-5
57. -5-4+7 58. -3-8+4
59. -14-\left(-27\right)+9 60. 64+\left(-17\right)-9
61. \left(2-7\right)-\left(3-8\right)\left(2\right) 62. \left(1-8\right)-\left(2-9\right)
63. -\left(6-8\right)-\left(2-4\right) 64. -\left(4-5\right)-\left(7-8\right)
65. 25-\left[10-\left(3-12\right)\right] 66. 32-\left[5-\left(15-20\right)\right]
67. 6.3-4.3-7.2 68. 5.7-8.2-4.9
69. {5}^{2}-{6}^{2} 70. {6}^{2}-{7}^{2}

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 58° Celsius, recorded in the Sahara Desert in 1922. The lowest recorded temperature was 90° below 0° 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 -504
Tuesday +142
Wednesday -449
Thursday +410
Friday +369

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 -201
Tuesday -16
Wednesday -23
Thursday +172
Friday -34

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 \text{``}-\text{''} sign in algebra? Explain how they differ.
79. Explain why the sum of -8 and 2 is negative, but the sum of 8 and -2 is positive. 80. Give an example from your life experience of adding two negative numbers.

Answers

1. a) > b) < c) < d) > 3. a)-2b) 6 5. 4
7. 15 9. a)-12b) 12 11. a) 32 b) 0 c) 16
13. a) < b) = 15. 5,-5 17. 56
19. 0 21. 8 23. a) -19 b) -33
25. -80 27. 32 29. -22
31. 108 33. 29 35. 6
37. -9 39. 12 41. a) 16 b) 16
43. a) 45 b) 45 45. 27 47. -39
49. -59 51. -51 53. -99
55. -2 57. -2 59. 22
61. -15 63. 0 65. 6
67. -5.2 69. -11 71. a) 20,320 b) -282
73. a) $5.6 million b) -\$7.5 million 75. -32 77. Answers may vary
79. Answers may vary

Attributions

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.

5

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 a \cdot b means add a, b times. Here, we are using the model just to help us discover the pattern.

Two images are shown side-by-side. The image on the left has the equation five times three at the top. Below this it reads “add 5, 3 times.” Below this depicts three rows of blue counters, with five counters in each row. Under this, it says “15 positives.” Under thisis the equation“5 times 3 equals 15.” The image on the right reads “negative 5 times three. The three is in parentheses. Below this it reads, “add negative five, three times.” Under this are fifteen red counters in three rows of five. Below this it reads” “15 negatives”. Below this is the equation negative five times 3 equals negative 15.”

The next two examples are more interesting.

What does it mean to multiply 5 by -3? 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.

This figure has two columns. In the top row, the left column contains the expression 5 times negative 3. This means take away 5, three times. Below this, there are three groups of five red negative counters, and below each group of red counters is an identical group of five blue positive counters. What are left are fifteen negatives, represented by 15 red counters. Underneath the counters is the equation 5 times negative 3 equals negative 15. In the top row, the right column contains the expression negative 5 times negative 3. This means take away negative 5, three times. Below this, there are three groups of five blue positive counters, and below each group of blue counters is an identical group of five red negative counters. What are left are fifteen positives, represented by 15 blue counters. Underneath the blue counters is the equation negative 5 times negative 3 equals 15.

In summary:

\begin{array}{cccccccc}\hfill 5 \cdot 3& =\hfill & 15\hfill & & & \hfill -5\left(3\right)& =\hfill & -15\hfill \\ \hfill 5\left(-3\right)& =\hfill & -15\hfill & & & \hfill \left(-5\right)\left(-3\right)& =\hfill & 15\hfill \end{array}

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

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
\begin{array}{ccc}\hfill 7\cdot4& =\hfill & 28\hfill \\ \hfill -8\left(-6\right)& =\hfill & 48\hfill \end{array}
Different signs Product Example
Positive \cdot negative
Negative \cdot positive
Negative
Negative
\begin{array}{ccc}\hfill 7\left(-9\right)& =\hfill & -63\hfill \\ \hfill -5\cdot10& =\hfill & -50\hfill \end{array}

EXAMPLE 1

Multiply: a) -9\cdot3 b) -2\left(-5\right) c) 4\left(-8\right) d) 7\cdot6.

Solution
a)
Multiply, noting that the signs are different so the product is negative.
\begin{array}{c}-9\cdot3\\ -27\end{array}
b)
Multiply, noting that the signs are the same so the product is positive.
\begin{array}{c}-2\left(-5\right)\\ 10\end{array}
c)
Multiply, with different signs.
\begin{array}{c}4\left(-8\right)\\ -32\end{array}
d)
Multiply, with same signs.
\begin{array}{c}7\cdot6\\ 42\end{array}

TRY IT 1.1

Multiply: a) -6\cdot8 b) -4\left(-7\right) c) 9\left(-7\right) d) 5\cdot12.

Show answer

a)-48 b) 28 c) -63 d) 60

TRY IT 1.2

Multiply: a) -8\cdot7 b) -6\left(-9\right) c) 7\left(-4\right) d) 3\cdot13.

Show answer

a)-56 b) 54 c) -28 d) 39

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

\begin{array}{ccccccc}& & & \hfill -1\cdot 4\hfill & & & \hfill -1\left(-3\right)\hfill \\ \text{Multiply.}\hfill & & & \hfill -4\hfill & & & \hfill 3\hfill \\ & & & \hfill -4\phantom{\rule{0.2em}{0ex}}\text{is the opposite of}\phantom{\rule{0.2em}{0ex}}4.\hfill & & & \hfill 3\phantom{\rule{0.2em}{0ex}}\text{is the opposite of}\phantom{\rule{0.2em}{0ex}}-3.\hfill \end{array}

Each time we multiply a number by -1, we get its opposite!

Multiplication by -1

-1a=\text{-}a

Multiplying a number by -1 gives its opposite.

EXAMPLE 2

Multiply: a) -1\cdot7 b) -1\left(-11\right).

Solution
a)
Multiply, noting that the signs are different so the product is negative.
\begin{array}{c}-1\cdot7\\ -7\\ -7\phantom{\rule{0.2em}{0ex}}\text{is the opposite of}\phantom{\rule{0.2em}{0ex}}7.\end{array}
b)
Multiply, noting that the signs are the same so the product is positive.
\begin{array}{c}-1\left(-11\right)\\ 11\\ 11\phantom{\rule{0.2em}{0ex}}\text{is the opposite of}\phantom{\rule{0.2em}{0ex}}-11.\end{array}

TRY IT 2.1

Multiply: a) -1\cdot9 b) -1 \cdot \left(-17\right).

Show answer

a)-9 b) 17

TRY IT 2.2

Multiply: a) -1\cdot8 b) -1 \cdot \left(-16\right).

Show answer

a)-8 b) 16

Divide Integers

What about division? Division is the inverse operation of multiplication. So, 15\div 3=5 because 5\cdot3=15. 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.

\begin{array}{cccccccccccccc}\hfill 5\cdot3& =\hfill & 15\phantom{\rule{0.2em}{0ex}}\text{so}\phantom{\rule{0.2em}{0ex}}15\div 3\hfill & =\hfill & 5\hfill & & & & & \hfill -5\left(3\right)& =\hfill & -15\phantom{\rule{0.2em}{0ex}}\text{so}\phantom{\rule{0.2em}{0ex}}-15\div3\hfill & =\hfill & -5\hfill \\ \hfill \left(-5\right)\left(-3\right)& =\hfill & 15\phantom{\rule{0.2em}{0ex}}\text{so}\phantom{\rule{0.2em}{0ex}}15\div \left(-3\right)\hfill & =\hfill & -5\hfill & & & & & \hfill 5\left(-3\right)& =\hfill & -15\phantom{\rule{0.2em}{0ex}}\text{so}\phantom{\rule{0.2em}{0ex}}-15\div \left(-3\right)\hfill & =\hfill & 5\hfill \end{array}

Division follows the same rules as multiplication!

For division of two signed numbers, when the:

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) -27\div 3 b) -100\div \left(-4\right).

Solution
a)
Divide. With different signs, the quotient is negative.
\begin{array}{c}-27\div 3\\ -9\end{array}
b)
Divide. With signs that are the same, the quotient is positive.
\begin{array}{c}-100\div\left(-4\right)\\ 25\end{array}

TRY IT 3.1

Divide: a) -42\div 6 b) -117\div\left(-3\right).

Show answer

a)-7 b) 39

TRY IT 3.2

Divide: a) -63\div 7 b) -115\div\left(-5\right).

Show answer

a)-9 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: 7\left(-2\right)+4\left(-7\right)-6.

Solution
7\left(-2\right)+4\left(-7\right)-6
Multiply first. -14+\left(-28\right)-6
Add. -42-6
Subtract. -48

TRY IT 4.1

Simplify: 8\left(-3\right)+5\left(-7\right)-4.

Show answer

-63

TRY IT 4.2

Simplify: 9\left(-3\right)+7\left(-8\right)-1.

Show answer

-84

EXAMPLE 5

Simplify: a) {\left(-2\right)}^{4} b) \text{-}{2}^{4}.

Solution
a)
Write in expanded form.
Multiply.
Multiply.
Multiply.
\begin{array}{c}{\left(-2\right)}^{4}\\ \left(-2\right)\left(-2\right)\left(-2\right)\left(-2\right)\\ 4\left(-2\right)\left(-2\right)\\ -8\left(-2\right)\\ 16\end{array}
b)
Write in expanded form. We are asked to find the opposite of\phantom{\rule{0.2em}{0ex}}{2}^{4}.
Multiply.
Multiply.
Multiply.
\begin{array}{c}\text{-}{2}^{4}\\ \text{-}\left(2\cdot2\cdot2\cdot2\right)\\ \text{-}\left(4\cdot2\cdot2\right)\\ \text{-}\left(8\cdot2\right)\\ 16\end{array}

Notice the difference in parts a) and b). In part a), the exponent means to raise what is in the parentheses, the \left(-2\right) to the {4}^{\text{th}} power. In part b), the exponent means to raise just the 2 to the {4}^{\text{th}} power and then take the opposite.

TRY IT 5.1

Simplify: a) {\left(-3\right)}^{4} b) \text{-}{3}^{4}.

Show answer

a) 81 b) -81

TRY IT 5.2

Simplify: a) {\left(-7\right)}^{2} b) \text{-}{7}^{2}.

Show answer

a) 49 b) -49

The next example reminds us to simplify inside parentheses first.

EXAMPLE 6

Simplify: 12-3\left(9-12\right).

Solution
12-3\left(9-12\right)
Subtract in parentheses first. 12-3\left(-3\right)
Multiply. 12-\left(-9\right)
Subtract. 21

TRY IT 6.1

Simplify: 17-4\left(8-11\right).

Show answer

29

TRY IT 6.2

Simplify: 16-6\left(7-13\right).

Show answer

52

EXAMPLE 7

Simplify: 8\left(-9\right)\div{\left(-2\right)}^{3}.

Solution
8\left(-9\right)\div{\left(-2\right)}^{3}
Exponents first. 8\left(-9\right)\div\left(-8\right)
Multiply. -72\div \left(-8\right)
Divide. 9

TRY IT 7.1

Simplify: 12\left(-9\right)\div{\left(-3\right)}^{3}.

Show answer

4

TRY IT 7.2

Simplify: 18\left(-4\right)\div{\left(-2\right)}^{3}.

Show answer

9

EXAMPLE 8

Simplify: -30\div 2+\left(-3\right)\left(-7\right).

Solution
-30\div 2+\left(-3\right)\left(-7\right)
Multiply and divide left to right, so divide first. -15+\left(-3\right)\left(-7\right)
Multiply. -15+21
Add. 6

TRY IT 8.1

Simplify: -27\div 3+\left(-5\right)\left(-6\right).

Show answer

21

TRY IT 8.2

Simplify: -32\div 4+\left(-2\right)\left(-7\right).

Show answer

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 n=-5, evaluate: a) n+1 b) \text{-}n+1.

Solution

a)

.
. .
Simplify. −4

b)

.
. .
Simplify. .
Add. 6

TRY IT 9.1

When n=-8, evaluate a) n+2 b) \text{-}n+2.

Show answer

a)-6 b) 10

TRY IT 9.2

When y=-9, evaluate a) y+8 b) \text{-}y+8.

Show answer

a)-1 b) 17

EXAMPLE 10

Evaluate {\left(x+y\right)}^{2} when x=-18 and y=24.

Solution
.
. {\left(-18+24\right)}^{2}
Add inside parenthesis. (6)2
Simplify. 36

TRY IT 10.1

Evaluate {\left(x+y\right)}^{2} when x=-15 and y=29.

Show answer

196

TRY IT 10.2

Evaluate {\left(x+y\right)}^{3} when x=-8 and y=10.

Show answer

8

EXAMPLE 11

Evaluate 20-z when a) z=12 and b) z=-12.

Solution

a)

.
. .
Subtract. 8

b)

.
. .
Subtract. 32

TRY IT 11.1

Evaluate: 17-k when a) k=19 and b) k=-19.

Show answer

a)-2 b) 36

TRY IT 11.2

Evaluate: -5-b when a) b=14 and b) b=-14.

Show answer

a)-19 b) 9

EXAMPLE 12

Evaluate: 2{x}^{2}+3x+8 when x=4.

Solution

Substitute 4\phantom{\rule{0.2em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}x. Use parentheses to show multiplication.

.
Substitute. .
Evaluate exponents. .
Multiply. .
Add. 52

TRY IT 12.1

Evaluate: 3{x}^{2}-2x+6 when x=-3.

Show answer

39

TRY IT 12.2

Evaluate: 4{x}^{2}-x-5 when x=-2.

Show answer

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 -12, increased by 3

Solution
the sum of 8 and -12, increased by 3.
Translate. \left[8+\left(-12\right)\right]+3
Simplify. Be careful not to confuse the brackets with an absolute value sign. \left(-4\right)+3
Add. -1

TRY IT 13.1

Translate and simplify the sum of 9 and -16, increased by 4

Show answer

\left(9+\left(-16\right)\right)+4-3

TRY IT 13.2

Translate and simplify the sum of -8 and -12, increased by 7

Show answer

\left(-8+\left(-12\right)\right)+7-13

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.

a-b
a minus b
the difference of a and b
b subtracted from a
b less than a

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

EXAMPLE 14

Translate and then simplify a) the difference of 13 and -21 b) subtract 24 from -19.

Solution
a)
Translate.
Simplify.
\begin{array}{c}\text{the}\phantom{\rule{0.2em}{0ex}}\mathbf{\text{difference}}\phantom{\rule{0.2em}{0ex}}\mathit{\text{of}}\phantom{\rule{0.2em}{0ex}}13\phantom{\rule{0.2em}{0ex}}\mathit{\text{and}}\phantom{\rule{0.2em}{0ex}}-21\\ 13-\left(-21\right)\\ 34\end{array}
b)
Translate. Remember, “subtract b from a means a-b.
Simplify.
\begin{array}{c}\mathbf{\text{subtract}}\phantom{\rule{0.2em}{0ex}}24\phantom{\rule{0.2em}{0ex}}\mathbf{\text{from}}\phantom{\rule{0.2em}{0ex}}-19\\ -19-24\\ -43\end{array}

TRY IT 14.1

Translate and simplify a) the difference of 14 and -23 b) subtract 21 from -17.

Show answer

a)14-\left(-23\right);37 b)-17-21;-38

TRY IT 14.2

Translate and simplify a) the difference of 11 and -19 b) subtract 18 from -11.

Show answer

a)11-\left(-19\right);30 b)-11-18;-29

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 -2 and 14

Solution
\text{the product}\phantom{\rule{0.2em}{0ex}}\mathit{\text{of}}\phantom{\rule{0.2em}{0ex}}-2\phantom{\rule{0.2em}{0ex}}\mathit{\text{and}}\phantom{\rule{0.2em}{0ex}}14
Translate. \left(-2\right)\left(14\right)
Simplify. -28

TRY IT 15.1

Translate to an algebraic expression and simplify if possible: the product of -5 and 12

Show answer

-5\left(12\right);-60

TRY IT 15.2

Translate to an algebraic expression and simplify if possible: the product of 8 and -13.

Show answer

-8\left(13\right);-104

EXAMPLE 16

Translate to an algebraic expression and simplify if possible: the quotient of -56 and -7.

Solution
\text{the quotient}\phantom{\rule{0.2em}{0ex}}\mathit{\text{of}}\phantom{\rule{0.2em}{0ex}}-56\phantom{\rule{0.2em}{0ex}}\mathit{\text{and}}\phantom{\rule{0.2em}{0ex}}-7
Translate. -56\div \left(-7\right)
Simplify. 8

TRY IT 16.1

Translate to an algebraic expression and simplify if possible: the quotient of -63 and -9.

Show answer

-63\div \left(-9\right);7

TRY IT 16.2

Translate to an algebraic expression and simplify if possible: the quotient of -72 and -9.

Show answer

-72\div \left(-9\right);8

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 -9 degrees. What was the difference of the morning and afternoon temperatures?

Solution

This is a table with two columns. The left column includes steps to solve the problem. The right column includes the math to solve the problem. In the first row, the left column says “Step 1. Read the problem. Make sure all the words and ideas are understood.” The right column is blank.In the second row, the left column says “Step 2. Identify what we are asked to find”. The right column says, “the difference of the morning and afternoon temperatures.”In the third row, the left column says, “Step 3. Write a phrase that gives the information to find it.” Next to this in the right column, it says “the difference of 11 and negative 9.”In the fourth row, the left column says, “Step 4. Translate the phrase to an expression.” The right column contains 11 minus negative 9.In the fifth row, the left column says, “Step 5. Simplify the expression.” The right column contains 20.The final row says, “Step five. Write a complete sentence that answers the question.” Next to this in the right column, it says “the difference in temperatures was 20 degrees.”

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?

Show answer

The difference in temperatures was 45 degrees.

TRY IT 17.2

The temperature in Quesnel, BC, was -6 degrees at lunchtime. By sunset the temperature had dropped to -15 degrees. What was the difference in the lunchtime and sunset temperatures?

Show answer

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. 3\left(-15\right)
Step 5. Simplify the expression. -45
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?

Show answer

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

Practice Makes Perfect

Multiply Integers

In the following exercises, multiply.

1. -4\cdot8 2. -3\cdot9
3. 9\left(-7\right) 4. 13\left(-5\right)
5. -1\cdot6 6. -1\cdot3
7. -1\left(-14\right) 8. -1\left(-19\right)

Divide Integers

In the following exercises, divide.

9. -24\div6 10. 35\div\left(-7\right)
11. -52\div\left(-4\right) 12. -84\div\left(-6\right)
13. -180\div15 14. -192\div12

Simplify Expressions with Integers

In the following exercises, simplify each expression.

15. 5\left(-6\right)+7\left(-2\right)-3 16. 8\left(-4\right)+5\left(-4\right)-6
17. {\left(-2\right)}^{6} 18. {\left(-3\right)}^{5}
19. \text{-}{4}^{2} 20. \text{-}{6}^{2}
21. -3\left(-5\right)\left(6\right) 22. -4\left(-6\right)\left(3\right)
23. \left(8-11\right)\left(9-12\right) 24. \left(6-11\right)\left(8-13\right)
25. 26-3\left(2-7\right) 26. 23-2\left(4-6\right)
27. 65\div\left(-5\right)+\left(-28\right)\div\left(-7\right) 28. 52\div\left(-4\right)+\left(-32\right)\div\left(-8\right)
29. 9-2\left[3-8\left(-2\right)\right] 30. 11-3\left[7-4\left(-2\right)\right]
31. {\left(-3\right)}^{2}-24\div\left(8-2\right) 32. {\left(-4\right)}^{2}-32\div\left(12-4\right)

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

33. y+\left(-14\right) when
a) y=-33
b) y=30
34. x+\left(-21\right) when
a) x=-27
b) x=44
35.
a)a+3 when a=-7
b) \text{-}a+3 when a=-7
36.
a)d+\left(-9\right) when d=-8
b) \text{-}d+\left(-9\right) when d=-8
37. m+n when
m=-15,n=7
38. p+q when
p=-9,q=17
39. r+s when r=-9,s=-7 40. t+u when t=-6,u=-5
41. {\left(x+y\right)}^{2} when
x=-3,y=14
42. {\left(y+z\right)}^{2} when
y=-3,z=15
43. -2x+17 when
a) x=8
b) x=-8
44. -5y+14 when
a) y=9
b) y=-9
45. 10-3m when
a) m=5
b) m=-5
46. 18-4n when
a) n=3
b) n=-3
47. 2{w}^{2}-3w+7 when
w=-2
48. 3{u}^{2}-4u+5 when u=-3
49. 9a-2b-8 when
a=-6\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b=-3
50. 7m-4n-2 when
m=-4\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}n=-9

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 -15, increased by 7 52. the sum of -8 and -9, increased by 23
53. the difference of 10 and -18 54. subtract 11 from -25
55. the difference of -5 and -30 56. subtract -6 from -13
57. the product of -3 and 15 58. the product of -4 and 16
59. the quotient of -60 and -20 60. the quotient of -40 and -20
61. the quotient of -6 and the sum of a and b 62. the quotient of -7 and the sum of m and n
63. the product of -10 and the difference of p\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}q 64. the product of -13 and the difference of c\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}d

Use Integers in Applications

In the following exercises, solve.

65. Temperature On January 15, the high temperature in Lytton, British Columbia, was 84° . That same day, the high temperature in Fort Nelson, British Columbia was -12°. What was the difference between the temperature in Lytton and the temperature in Embarrass? 66. Temperature On January 21, the high temperature in Palm Springs, California, was 89°, and the high temperature in Whitefield, New Hampshire was -31°. 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 -\$38 in his checking account. He deposits $225 to the account. What is the new balance? 72. Checking Account Reymonte has a balance of -\$49 in his checking account. He deposits $281 to the account. What is the new balance?

Everyday Math

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 -{2}^{4}\ne {\left(-2\right)}^{4}? 78. Why is -{4}^{3}={\left(-4\right)}^{3}?

Answers

1. -32 3. -63 5. -6
7. 14 9. -4 11. 13
13. -12 15. -47 17. 64
19. -16 21. 90 23. 9
25. 41 27. -9 29. -29
31. 5 33. a)-47 b) 16 35. a)-4 b) 10
37. -8 39. -16 41. 121
43. a) 1 b) 33 45. a)-5b) 25 47. 21
49. -56 51. \left(3+\left(-15\right)\right)+7;-5 53. 10-\left(-18\right);28
55. -5-\left(-30\right);25 57. -3\cdot 15;-45 59. -60\div\left(-20\right);3
61. \frac{-6}{a+b} 63. -10\left(p-q\right) 65. 96°
67. 21 69. \text{-}\$28 71. $187
73. \text{-}\$3600 75. Answers may vary 77. Answers may vary

Attributions

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.

6

1.6 Chapter Review

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. x+ 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.

73. the sum of 8 and 12 74. the sum of 9 and 1
75. the difference of x and 4 76. the difference of x and 3
77. the product of 6 and y 78. the product of 9 and y
79. Derek bought a skirt and a blouse. The skirt cost $15 more than the blouse. Let b represent the cost of the blouse. Write an expression for the cost of the skirt. 80. Marcella has 6 fewer boy cousins than girl cousins. Let g represent the number of girl cousins. Write an expression for the number of boy cousins.

Use Negatives and Opposites of Integers

In the following exercises, order each of the following pairs of numbers, using < or >.

81.
a) 6___2
b) -7___4
c) -9___-1
d) 9___-3
82.
a) -5___1
b) -4___-9
c) 6___10
d) 3___-8

In the following exercises,, find the opposite of each number.

83. a) -8 b) 1 84. a) -2 b) 6

In the following exercises, simplify.

85. (–19) 86. (–53)

In the following exercises, simplify.

87. −m when
a) m = 3
b) m=-3
88. −p when
a) p = 6
b) p = -6

Simplify Expressions with Absolute Value

In the following exercises,, simplify.

89. a) |7| b) |-25| c) |0| 90. a) |5| b) |0| c) |-19|

In the following exercises, fill in <, >, or = for each of the following pairs of numbers.

91.
a) – 8 ___ |–8|
b) – |–2|___ –2
92.
a) –3|___ – | –3|
b) 4 ___ – | –4|

In the following exercises, simplify.

93. |8 – 4| 94. |9 – 6|
95. 8 (14 – 2 |- 2|) 96. 6(13 – 4 |-2|)

In the following exercises, evaluate.

97. a) |x| when x = -28 b) |-x| when x =-15 98. a) |y| when y = -37 b) |-z| when z=-24

Add Integers

In the following exercises, simplify each expression.

99. -200 + 65 100. -150 + 45
101. 2 + (-8) + 6 102. 4 + (-9) + 7
103. 140 + (-75) + 67 104. -32 + 24 + (-6) + 10

Subtract Integers

In the following exercises, simplify.

105. 9 – 3 106. -5 – (-1)
107. a) 15 – 6 b) 15 + (-6) 108. a) 12 – 9 b) 12 + (-9)
109. a) 8 – (-9) b) 8 + 9 110. a) 4 – (-4) b) 4 + 4

In the following exercises, simplify each expression.

111. 10 – (-19) 112. 11 – ( -18)
113. 31 – 79 114. 39 – 81
115. -31 – 11 116. -32 – 18
117. -15 – (-28) + 5 118. 71 + (-10) – 8
119. -16 – (-4 + 1) – 7 120. -15 – (-6 + 4) – 3

Multiply Integers

In the following exercises, multiply.

121. -5 (7) 122. -8 (6)
123. -18(-2) 124. -10 (-6)

Divide Integers

In the following exercises, divide.

125. -28 ÷ 7 126. 56 ÷ ( -7)
127. -120 ÷ -20) 128. -200 ÷ 25

Simplify Expressions with Integers

In the following exercises, simplify each expression.

129. -8 (-2) -3 (-9) 130. -7 (-4) – 5(-3)
131. (-5)3 132. (-4)3
133. -4 · 2 · 11 134. -5 · 3 · 10
135. -10(-4) ÷ (-8) 136. -8(-6) ÷ (-4)
137. 31 – 4(3-9) 138. 24 – 3(2 – 10)

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

139. x + 8 when
a) x = -26
b) x = -95
140. y + 9 when
a) y = -29
b) y = -84
141. When b = -11, evaluate:
a) b + 6
b) −b + 6
142. When c = -9, evaluate:
a) c + (-4)
b) −c + (-4)
143. p2 – 5p + 2 when p = -1 144. q2 – 2q + 9 when q = -2
145. 6x – 5y + 15 when x = 3 and y = -1 146. 3p – 2q + 9 when p = 8 and q = -2

Translate English Phrases to Algebraic Expressions

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

147. the sum of -4 and -17, increased by 32 148. a) the difference of 15 and -7 b) subtract 15 from -7
149. the quotient of -45 and -9 150. the product of -12 and the difference of c and d.

Use Integers in Applications

In the following exercises, solve.

151. Temperature The high temperature one day in Miami Beach, Florida, was 76° F. That same day, the high temperature in Buffalo, New York was −8° F. What was the difference between the temperature in Miami Beach and the temperature in Buffalo? 152. CheckingAccount Adrianne has a balance of -$22 in her checking account. She deposits $301 to the account. What is the new balance?

Review Exercise Answers

1. a) tens b) ten thousands c) hundreds d) ones e) thousands 3. a) ten millions b) tens c) hundred thousands d) millions e) ten thousands 5. six thousand, one hundred four
7. three million, nine hundred seventy-five thousand, two hundred eighty-four 9. 315 11. 90,425,016
13. a)410b)8,560 15. a)865,000 b)865,000c)860,000 17. by 2,3,6
19. by 3,5 21. by 2,5,10 23. 2 · 2 · 3 · 5 · 7
25. 3 · 3 · 5 · 5 27. 2 · 2 · 2 · 3 · 5 · 13 29. 2 · 2 · 2 · 3 · 3
31. 2 · 2 · 3 · 3 · 7 33. 30 35. 120
37. 25 minus 7, the difference of twenty-five and seven 39. 45 divided by 5, the quotient of forty-five and five 41. forty-two is greater than or equal to twenty-seven
43. 3 is less than or equal to 20 divided by 4, three is less than or equal to the quotient of twenty and four 45. expression 47. 243
49. 13 51. 10 53. 41
55. 34 57. 81 59. 58
61. 12 63. 12 and 3, n2 and 3n2 65. 11×2, 3x, 6
67. 26a 69. 12x + 8 71. 12p + 2
73. 8 + 12 75. x – 4 77. 6y
79. b + 15 81. a) > b) < c) < d) > 83. a) 8 b) -1
85. 19 87. a) -3 b) 3 89. a) 7 b) 25 c) 0
91. a) < b) = 93. 4 95. 80
97. a) 28 b) 15 99. -135 101. 0
103. 132 105. 6 107. a) 9 b) 9
109. a) 17 b) 17 111. 29 113. -48
115. -42 117. 18 119. -20
121. -35 123. 36 125. -4
127. 6 129. 43 131. -125
133. -88 135. -5 137. 55
139. a) -18 b) -87 141. a) -5 b) 17 143. 8
145. 38 147. (-4 + (-17)) + 32; 11 149. \dfrac{-45}{-9} ;5
151. 84 degrees F

Practice Test

1. Write as a whole number using digits: two hundred five thousand, six hundred seventeen. 2. Find the prime factorization of 504.
3. Find the Least Common Multiple of 18 and 24. 4. Combine like terms: 5n + 8 + 2n – 1.

In the following exercises, evaluate.

5. −|x| when x = -2 6. 11 – a when a = -3
7. Translate to an algebraic expression and simplify: twenty less than negative 7. 8. Monique has a balance of  −$18 in her checking account. She deposits $152 to the account. What is the new balance?
9. Round 677.1348 to the nearest hundredth. 10. Simplify  expression  -6 (-2) –  3 · 4 ÷ (-6)
11.  Simplify  expression   4(-2) + 4 ·2 –  {(-3)}^{3} 12. Simplify  expression  -8(-3) ÷ (-6)
13.Simplify  expression  21 – 5(2 – 7) 14. Simplify  expression  2 + 2(3 – 10) – {(2)}^{3}

Practice Test Answers

1. 205,617 2.  2 · 2 · 2 · 3 · 3 · 7 3. 72
4. 7n + 7 5. -2 6. 14
7. -7 – 20; -27 8. $ 134 9. 677.13
10. 10 11. 27 12. -4
13. 46 14. -20

II

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.

This figure consists of a Venn diagram. To start there is a large rectangle marked Real Numbers. The right half of the rectangle consists of Irrational Numbers. The left half consists of Rational Numbers. Within the Rational Numbers rectangle, there are Integers …, negative 2, negative 1, 0, 1, 2, …. Within the Integers rectangle, there are Whole Numbers 0, 1, 2, 3, … Within the Whole Numbers rectangle, there are Counting Numbers 1, 2, 3, …

7

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 \frac{1}{3} 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 \frac{2}{3} represents two of three equal parts. In the fraction \frac{2}{3}, the 2 is called the numerator and the 3 is called the denominator.

Two circles are shown, each divided into three equal pieces by lines. The left hand circle is labeled “one third” in each section. Each section is shaded. The circle on the right is shaded in two of its three sections.
Figure 1.

The circle on the left has been divided into 3 equal parts. Each part is \frac{1}{3} of the 3 equal parts. In the circle on the right, \frac{2}{3} of the circle is shaded (2 of the 3 equal parts).

Fraction

A fraction is written \frac{a}{b}, where b\ne 0 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 \frac{6}{6} pieces, or, in other words, one whole pie.

A circle is shown and is divided into six section. All sections are shaded.

So \frac{6}{6}=1. This leads us to the property of one that tells us that any number, except zero, divided by itself is 1

Property of One

\begin{array}{cccc}\frac{a}{a}=1\hfill & & & \left(a\ne 0\right)\hfill \end{array}

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

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

The fractions \frac{6}{6} and \frac{8}{8} 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 \frac{1}{2} is equivalent to \frac{4}{8}. In other words, they are equivalent fractions.

A circle is shown that is divided into eight equal wedges by lines. The left side of the circle is a pizza with four sections making up the pizza slices. The right side has four shaded sections. Below the diagram is the fraction four eighths.
Figure 2.

Since the same amount is of each pizza is shaded, we see that \frac{1}{2} is equivalent to \frac{4}{8}. They are equivalent fractions.

Equivalent Fractions

Equivalent fractions are fractions that have the same value.

How can we use mathematics to change \frac{1}{2} into \frac{4}{8}? 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 8 pieces instead of just 2. Mathematically, what we’ve described could be written like this as \frac{1\cdot 4}{2\cdot 4}=\frac{4}{8}. See (Figure 3).

A circle is shown and is divided in half by a vertical black line. It is further divided into eighths by the addition of dotted red lines.
Figure 3.

Cutting each half of the pizza into 4 pieces, gives us pizza cut into 8 pieces: \frac{1\cdot4}{2\cdot4}=\frac{4}{8}.

This model leads to the following property:

Equivalent Fractions Property

If a,b,c are numbers where b\ne 0,c\ne 0, then

\frac{a}{b}=\frac{a\cdot c}{b\cdot c}

If we had cut the pizza differently, we could get

An image shows three rows of fractions. In the first row are the fractions “1, times 2, divided by 2, times 2, equals two fourths”. Next to this is the word “so” and the fraction “one half, equals two fourths. The second row reads “1, times 3, divided by 2 times 3, equals three sixths”. Next to this is the word “so” and the fraction “one half equals, three sixths”. The third row reads “1 times 10, divided by 2 times 10, ten twentieths”. Next to this is the word “so” and the fraction “one half equals, ten twentieths”.

So, we say \frac{1}{2},\frac{2}{4},\frac{3}{6},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{10}{20} are equivalent fractions.

EXAMPLE 1

Find three fractions equivalent to \frac{2}{5}.

Solution

To find a fraction equivalent to \frac{2}{5}, we multiply the numerator and denominator by the same number. We can choose any number, except for zero. Let’s multiply them by 2, 3, and then 5.

A row of fractions reads “2 times 2, divided by 5 times 2, equals four tenths”. Next to this is “2, times 3, divided by 5 times 3, equals six fifteenths”. Next to this is “2 times 5, divided by 5 times 5, equals ten twenty-fifths”.

So, <