Course 2

2-1 Integers2-1 Integers

Course 2

Warm UpWarm Up

Problem of the DayProblem of the Day

Lesson PresentationLesson Presentation

Course 2

2-1 Integers

Warm UpCompare. Use <, >, or =

1. 7 5

2. 32 65

3. 82 28

4. 64 48

>

<

>

>

Course 2

2-1 Integers

Problem of the Day

Place 4, 5, 6, 7, 8, and 9 in the empty circles so that each side has the same sum.

1

3 284

5

96

7

Course 2

2-1 Integers

Learn to compare and order integers and to determine absolute value.

Course 2

2-1 Integers

Vocabulary

oppositeintegerabsolute value

Course 2

2-1 Integers

The opposite of a number is the samedistance from 0 on a number line as theoriginal number, but on the other sideof 0. Zero is its own opposite.

1 2 3 4 5 • •

–5–4–3–2–1 0

–4 and 4 are opposites

–4 4

Negative integers Positive integers

0 is neither positivenor negative

Course 2

2-1 Integers

The integers are the set of whole numbers and their opposites. By using integers, you can express elevations above, below, and at sea level. Sea level has an elevation of 0 feet.

The whole numbers are the counting numbers and zero: 0, 1, 2, 3, . . . .

Remember!

Course 2

2-1 Integers

Graph the integer 7 and its opposite on a number line.

Additional Example 1: Graphing Integers and Their Opposites on a Number Line

The opposite of –7 is 7.

1 2 3 4 5 6 7 –7–6–5–4–3–2–1 0

7 units 7 units

Course 2

2-1 Integers

Graph the integer 5 and its opposite on a number line.

Check It Out: Example 1

The opposite of –5 is 5.

1 2 3 4 5 6 7 –7–6–5–4–3–2–1 0

5 units 5 units

Course 2

2-1 Integers

You can compare and order integers by graphing them on a number line. Integers increase in value as you move to the right along a number line. They decrease in value as you move to the left.

Course 2

2-1 Integers

Compare the integers. Use < or >.

Additional Example 2A: Comparing Integers Using a Number Line

4 is farther to the right than -4, so 4 > -4.

1 2 3 4 5 6 7 –7–6–5–4–3–2–1 0

4 -4>

The symbol < means “is less than,” and the symbol > means “is greater than.”

Remember!

Course 2

2-1 Integers

Compare the integers. Use < or >.

Additional Example 2B: Comparing Integers Using a Number Line

-9 is farther to the right than -15, so -15 < -9.

-15 -9

>-15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

Course 2

2-1 Integers

Compare the integers. Use < or >.

Check It Out: Example 2A

6 is farther to the right than -6, so 6 > -6.

1 2 3 4 5 6 7 –7–6–5–4–3–2–1 0

6 -6>

Course 2

2-1 Integers

Compare the integers. Use < or >.

Check It Out: Example 2B

-4 is farther to the right than -11, so -4 > -11.

-4 -11>

-15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

Course 2

2-1 Integers

Use a number line to order the integers from least to greatest.

Additional Example 3: Ordering Integers Using a Number Line.

–3, 6, –5, 2, 0, –8

The numbers in order from least to greatest are –8, –5, –3, 0, 2, and 6.

1 2 3 4 5 6 7 8 –8 –7–6 –5–4 –3 –2 –1 0

Course 2

2-1 Integers

Check It Out: Example 3

Use a number line to order the integers from least to greatest.

–5, 4, –3, 2, –1, –2

The numbers in order from least to greatest are –5, –3, –2, –1, 2, and 4.

1 2 3 4 5 6 7 8 –8 –7–6–5–4 –3–2 –1 0

Course 2

2-1 Integers

A number’s absolute value is its distance from 0 on a number line. Since distance can never be negative, absolute values are never negative. They are always positive or zero.

Course 2

2-1 Integers

Use a number line to find each absolute value.

Additional Example 4A: Finding Absolute Value

|8|

1 2 3 4 5 6 7 8 –8 –7–6–5–4 –3–2 –1 0

8 units

8 is 8 units from 0, so |8| = 8.

Course 2

2-1 Integers

The symbol is read as “the absolute value of.” For example -3 is the absolute value of -3.

Reading Math

Course 2

2-1 Integers

Additional Example 4B: Finding Absolute Value

Use a number line to find each absolute value.

|–12|

–12 –11 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2

–12 is 12 units from 0, so |–12| = 12.

12 units

Course 2

2-1 Integers

Use a number line to find each absolute value.

|3|

1 2 3 4 5 6 7 8 –8 –7–6–5–4 –3–2 –1 0

3 units

3 is 3 units from 0, so |3| = 3.

Check It Out: Example 4A

Course 2

2-1 Integers

Use a number line to find the absolute value.

|–9|

–9 is 9 units from 0, so |–9| = 9.

9 units

Check It Out: Example 4B

–12 –11 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2

Course 2

2-1 Integers

Lesson Quiz: Part I

Compare. Use <, >, or =.

1. –32 32

2. 26 |–26|

3. –8 –12

4. Use a number line to order the integers –2, 3, –4, 5, and –1 from least to greatest.

=

<

>

1 2 3 4 5 •

–5–4 –3 –2–1 0•• •• –4, –2, –1, 3, 5

Course 2

2-1 Integers

Lesson Quiz: Part II

Use a number line to find the absolute value.

5. -3

3

1 2 3 4 5 –5 –4 –3 –2 –1 0•

3 units

Course 2

2-1 Integers2-2 Adding Integers

Course 2

Warm UpWarm Up

Problem of the DayProblem of the Day

Lesson PresentationLesson Presentation

Course 2

2-1 Integers

Warm UpFind each absolute value.

1. |8|

2. |–6|

3. |–9|

4. |–7|

5. |–12|

6. |53|

8

69

Course 2

2-2 Adding Integers

7

12

53

Course 2

2-1 Integers

Problem of the Day

Jan’s yearly salary is $30,000, and it will be increased by $3,000 each year. Phil’s salary is $20,000, and it will be increased by $5,000 each year. In how many years will Jan and Phil both have the same yearly salary?5 years

Course 2

2-2 Adding Integers

Course 2

2-1 Integers

Learn to add integers.

Course 2

2-2 Adding Integers

Course 2

2-1 Integers

Use a number line to find each sum.

Additional Example 1A: Modeling Integer Addition

Course 2

2-2 Adding Integers

2 –14 –12 –10 –8 –6 –4 –2 0

–7 + (–4)–7+(–4)

–7 + (–4) = –11

Start at 0. Move left 7 spaces. Then move left4 more spaces.

Course 2

2-1 Integers

Use a number line to find each sum.

Additional Example 1B: Modeling Integer Addition

Course 2

2-2 Adding Integers

–12 + 19

–12

19

–12 + 19 = 7

Start at 0. Move left 12 spaces. Then move right 19 spaces.

2 4 6 8

–12 –10 –8 –6 –4 –2 0

Course 2

2-1 Integers

Check It Out: Example 1A

Insert Lesson Title Here

Course 2

2-2 Adding Integers

Use a number line to find each sum.

2 –14 –12 –10 –8 –6 –4 –2 0

–4 + (–5)–4+(–5)

–4 + (–5) = –9

Start at 0. Move left 4 spaces. Then move left5 more spaces.

Course 2

2-1 Integers

Check It Out: Example 1B

Insert Lesson Title Here

Course 2

2-2 Adding Integers

Use a number line to find each sum.

–7 + 12

–7

12

–7 + 12 = 5

Start at 0. Move left 7 spaces. Then move right 12 spaces.

2 4 6 8

–12 –10 –8 –6 –4 –2 0

Course 2

2-1 Integers

Course 2

2-2 Adding Integers

Adding IntegersAdding Integers

To add two integers with the same sign, find the sum of their absolute values. Use the sign of the two integers.

To add two integers with different signs, find the difference of their absolute values. Use the sign of the integer with the greater absolute value.

Course 2

2-1 Integers

Find each sum.

Additional Example 2A: Adding Integers Using Absolute Values

Course 2

2-2 Adding Integers

–4 + 8

–4 + 8

The signs are different. Find the difference of the absolute values.

Think: 8 – 4 = 4.

Use the sign of the integer with the greater absolute value.

4

Course 2

2-1 Integers

Additional Example 2B: Adding Integers Using Absolute Values

Course 2

2-2 Adding Integers

Find each sum.

23 + (–35)

23 + (–35)

The signs are different. Find the difference of the absolute values.

Think: 35 – 23 = 12.

Use the sign of the integer with the greater absolute value.

–12

Course 2

2-1 Integers

Check It Out: Example 2A

Insert Lesson Title Here

Course 2

2-2 Adding Integers

Find each sum.

–5 + 3

–5 + 3 Think: 5 – 3 = 2.

Use the sign of the integer with the greater absolute value.

–2

The signs are different. Find the difference of the absolute values.

Course 2

2-1 Integers

Check It Out: Example 2B

Insert Lesson Title Here

Course 2

2-2 Adding Integers

Find each sum.

–13 + (–24)

–13 + (–24)

The signs are the same. Find the sum of the absolute values.

Think: 13 + 24 = 37.

Use the sign of the two integers.–37

Course 2

2-1 Integers

Evaluate x + y for x = –42, y = 71.

Additional Example 3: Evaluating Expressions with Integers

Course 2

2-2 Adding Integers

x + y

–42 + 71

Substitute for x and y.

The signs are different. Think 71- 42 = 29

Use the sign of the integer with the greater absolute value (positive).29

When adding integers, think: if the signs are the same, find the sum. If the signs are different, find the difference.

Helpful Hint

Course 2

2-1 Integers

Check It Out: Example 3

Insert Lesson Title Here

Course 2

2-2 Adding Integers

Evaluate x + y for x = –24, y = 17.

x + y

–24 + 17

Substitute for x and y.

The signs are different. Think: 24 – 17 = 7.

Use the sign of the integer with the greater absolute value (negative).

–7

Course 2

2-1 Integers

The jazz band’s income from a bake sale was $286. Expenses were $21. Use integer addition to find the band’s total profit or loss.

Additional Example 4: Application

Course 2

2-2 Adding Integers

286 + (–21)

286 – 21

265

The band earned $265.

Use negative for the expenses.

Find the differences of the absolute values.

The answer is positive.

Course 2

2-1 Integers

Check it Out: Example 4

The French Club was raising money for a trip to Washington D.C. Their car wash raised $730. They had expenses of $52. Use integer addition to find the club’s total profit or loss.

Insert Lesson Title Here

Course 2

2-2 Adding Integers

730 + (–52)

730 – 52

678

The club earned $678.

Use negative for the expenses.

Find the differences of the absolute values.

The answer is positive.

Course 2

2-1 Integers

Lesson Quiz: Part I

Use a number line to find the sum.

1. –7 + (–6)

Find each sum.

2. –15 + 24 + (–9)

3. –24 + 7 + (–3)

4. Evaluate x + y for x = –2 and y = –15.

0

–13

Insert Lesson Title Here

–20

–17

Course 2

2-2 Adding Integers

2 –14 –12 –10 –8 –6 –4 –2 0

Course 2

2-1 Integers

Lesson Quiz: Part II

Insert Lesson Title Here

Course 2

2-2 Adding Integers

5. The math club’s income from a bake sale was $217. Advertising expenses were $32. Use integer addition to find the club’s total profit or loss?

$185 profit

Course 2

2-1 Integers2-3 Subtracting Integers

Course 2

Warm UpWarm Up

Problem of the DayProblem of the Day

Lesson PresentationLesson Presentation

Course 2

2-1 Integers

Warm UpAdd.

1. –2 + 6 2. –3 + (–4)

3. 7 + (–5) 4. 3 + (–4)

5. –6 + (–1) 6. –6 + 1

4 –7

2

Course 2

2-3 Subtracting Integers

–1

–7 –5

Course 2

2-1 Integers

Problem of the Day

Ray earned $172 shoveling walks and $188 babysitting. He spent $21 for a shovel and rock salt and $26 for toys for children. Which job was more profitable?

babysitting

Course 2

2-3 Subtracting Integers

Course 2

2-1 Integers

Learn to subtract integers.

Course 2

2-3 Subtracting Integers

Course 2

2-1 Integers

Course 2

2-3 Subtracting Integers

During its flight to and from Earth, the space shuttle may be exposed to temperatures as cold as –250°F and as hot as 3,000°F.

To find the differences in these temperatures, you need to know how to subtract integers with different signs.

You can model the difference between two integers by using a number line. When you subtract a positive number, the difference is less than the original number, so you move to the left. To subtract a negative number, move to the right.

Course 2

2-1 Integers

Course 2

2-3 Subtracting Integers

If the number being subtracted is less than the number it is being subtracted from, the answer will be positive. If the number being subtracted is greater, the answer will be negative.

Helpful Hint

Course 2

2-1 Integers

Use a number line to find each difference.

Additional Example 1A: Modeling Integer Subtraction

Course 2

2-3 Subtracting Integers

4 – 1

–2–1 0 1 2 3 4 5 6

4

–1

4 – 1 = 3

Start at 0. Move right 4 spaces. To subtract 1,move to the left.

Course 2

2-1 Integers

Additional Example 1B: Modeling Integer Subtraction

1 2 –6–5–4 –3 –2–1 0

Use a number line to find each difference.

–3 – 1

Course 2

2-3 Subtracting Integers

–3–1

–3 – 1 = –4

Start at 0. Move 3 spaces left. To subtract 1,move to the left.

Course 2

2-1 Integers

Additional Example 1C: Modeling Integer Subtraction

Course 2

2-3 Subtracting Integers

Use a number line to find each difference.

–2 – (–4)

1 2 3 4 –4 –3 –2–1 0

–2

–(–4)

–2 – (–4) = 2

Start at 0. Move left 2 spaces. To subtract –4,move to the right.

Course 2

2-1 Integers

Check It Out: Example 1A

Insert Lesson Title Here

Course 2

2-3 Subtracting Integers

Use a number line to find each difference.

3 – 2

1 2 3 4 5 6 –2–1 0

3

–2

3 – 2 = 1

Start at 0. Move right 3 spaces. To subtract 2,move to the left.

Course 2

2-1 Integers

Check It Out: Example 1B

Insert Lesson Title Here

Course 2

2-3 Subtracting Integers

1 2 –6–5–4 –3 –2–1 0

Use a number line to find each difference.

–2 – 4–2–4

–2 – 4 = –6

Start at 0. Move 2 spaces left. To subtract 4,move to the left.

Course 2

2-1 Integers

Check It Out: Example 1C

Insert Lesson Title Here

Course 2

2-3 Subtracting Integers

Use a number line to find each difference.

–4 – (–2)

1 2 3 4 –4 –3 –2–1 0

–4

–(–2)

–4 – (–2) = –2

Start at 0. Move left 4 spaces. To subtract –2,move to the right.

Course 2

2-1 Integers

Course 2

2-3 Subtracting Integers

Addition and subtraction are inverseoperations—they “undo” each other.Instead of subtracting a number youcan add its opposite.

Course 2

2-1 Integers

Find each difference.

Additional Example 2: Subtracting Integers by Adding the Opposite

Course 2

2-3 Subtracting Integers

A. 5 – (–2)

5 + 2

7

B. –3 – 7

–3 + (–7)

–10C. – 1 – (–8)

–1 + 8

7

Add the opposite of –2.

Add the opposite of 7.

Add the opposite of –8.

Course 2

2-1 Integers

Check It Out: Example 2

Insert Lesson Title Here

Find each difference.

A. 4 – 2

4 + (–2)

2

B. –2 – (–6)

–2 + 6

4C. –1 – 4

–1 + (–4)

–5

Add the opposite of 2.

Add the opposite of –6.

Add the opposite of 4.

Course 2

2-3 Subtracting Integers

Course 2

2-1 Integers

Evaluate x – y for each set of values.

Additional Example 3: Evaluating Expressions with Integers

Course 2

2-3 Subtracting Integers

A. x = –3 and y = 2

x – y

–3 – 2 = –3 + (–2)

= –5

B. x = 4 and y = –6

x – y

4 – (–6) = 4 + 6

= 10

Substitute for x and y.

Add the opposite of 2.

Substitute for x and y.

Add the opposite –6.

Course 2

2-1 Integers

Check It Out: Example 3A & B

Insert Lesson Title Here

Course 2

2-3 Subtracting Integers

Evaluate x – y for each set of values.

A. x = –4 and y = –3

x – y

–4 – (–3) = –4 + 3

= –1

B. x = –4 and y = 5

x – y

–4 – 5 = –4 + (–5)

= –9

Substitute for x and y.

Add the opposite of –3.

Substitute for x and y.

Add the opposite of 5.

Course 2

2-1 Integers

Find the difference between 32°F and –10°F.

Additional Example 4: Temperature Application

Course 2

2-3 Subtracting Integers

32 – (–10)

32 + 10 = 42

The difference in temperature is 42˚F.

Add the opposite of –10.

Course 2

2-1 Integers

Check It Out: Example 4

Insert Lesson Title Here

Course 2

2-3 Subtracting Integers

Find the difference between 8°F and –5°F.

8 – (–5)

8 + 5 = 13

The difference in temperature is 13˚F.

Add the opposite of 5.

Course 2

2-1 Integers

Lesson Quiz: Part I

Use a number line to find the difference.

1. 3 – 9

Find each difference.

2. –7 – 4

3. –3 – (–5)

4. Evaluate x – y + z for x = –4, y = 5, and z = –10.

–11

–6

Insert Lesson Title Here

2

–19

Course 2

2-3 Subtracting Integers

Course 2

2-1 Integers

Lesson Quiz: Part II

Insert Lesson Title Here

Course 2

2-3 Subtracting Integers

5. On January 1, 2002, the high temperature was 81˚F in Kona, Hawaii. The low temperature was –29˚F in Barrow, Alaska. What was the difference between the two temperatures?

110˚F