Adding Integers
LESSON
1.5
Adding Integers Now
BEFORE
You added decimals.
WHY?
You’ll add integers.
So you can find a hockey player’s plus-minus rating, as in Ex. 40.
Vocabulary additive inverse, p. 30
Scuba Diver A scuba diver studying marine life is 4 feet below sea level. From that depth, the diver descends 72 feet to the ocean floor and then rises 61 feet. The diver rests there to avoid decompression illness. Where is the diver relative to sea level? In Example 3, you will see how to answer this question by adding integers. One way to add integers is to use a number line. To add a positive integer, move to the right. To add a negative integer, move to the left. 6 5
4
Example 1
3
2
1
0
1
2
3
4
5
6
Adding Integers Using a Number Line
Use a number line to find the sum.
a. 3 (9)
Study Strategy In the activity on page 28, you used two arrows to add two integers on a number line. The first arrow started at 0 and ended at the first number in the sum. In Example 1, you see that you can draw only one arrow if you start at the first number in the sum.
Move 9 units to the left.
End at 6. 8
7
6
5
4
3
2
1
Start at 3.
0
1
2
3
4
5
Answer The final position is 6. So, 3 (9) 6.
b. 5 3 Move 3 units to the right.
Start at 5. 6
5
4
3
2
End at 2. 1
0
1
Answer The final position is 2. So, 5 3 2. Checkpoint Use a number line to find the sum. 1. 11 6
Video Tutor Go to thinkcentral.com
2. 1 (8) Lesson 1.5
3. 10 (5) Adding Integers
29
Absolute Values You can use absolute values to find the sum of two or more integers.
Adding Integers Words
Numbers
1. Same Sign Add the absolute values
8 12 20 6 (4) 10
and use the common sign. 2. Different Signs Subtract the lesser
absolute value from the greater absolute value and use the sign of the number with greater absolute value. 3. Opposites The sum of a number and
5 (8) 3 11 13 2 7 (7) 0
its opposite is 0. Additive Inverse Property The opposite of a number is also called its additive inverse . Item 3 in the notebook can be written algebraically as a (a) 0 and is called the additive inverse property.
Example 2
Adding Two Integers
a. Find the sum 54 (28). Same sign: Add ⏐ⴚ54⏐ and ⏐ⴚ28⏐.
54 (28) 82
Both integers are negative, so the sum is negative.
b. Find the sum 38 (17). Different signs: Subtract ⏐ⴚ17⏐ from ⏐38⏐.
38 (17) 21
⏐38⏐ ⏐ⴚ17⏐, so the sum has the same sign as 38.
Example 3
Adding More Than Two Integers
To answer the question about the position of the scuba diver at the top of page 29, you can find the sum 4 (72) 61. 4 (72) 61 76 61 15
Add ⴚ4 and ⴚ72. Add ⴚ76 and 61.
Answer The sum is 15, so the diver is 15 feet below sea level. Checkpoint Find the sum. 4. 41 26 30
Chapter 1
Variables, Expressions, and Integers
5. 19 (11)
6. 52 (30) (46)
Example 4
Watch Out When you substitute a negative number for a variable, you may have to enclose the number in parentheses, as in part (b) of Example 4, to avoid confusion with an operation sign.
Evaluating Variable Expressions
Evaluate the expression when x ⴝ ⴚ22 and y ⴝ ⴚ12.
a. x (9)
b. x 17 y
Solution
a. x (9) ⴚ22 (9) 31
Substitute ⴚ22 for x. Add.
b. x 17 y ⴚ22 17 (ⴚ12) 5 (12) 17
Substitute for x and for y. Add ⴚ22 and 17. Add ⴚ5 and ⴚ12.
Checkpoint Evaluate the expression when a ⴝ ⴚ18 and b ⴝ ⴚ3. 7. a (8)
1.5
8. 32 a
9. a b 30
Exercises More Practice, p. 839
Go to thinkcentral.com Practice Exercises
Guided Practice Vocabulary Check
1. Copy and complete: To add two integers without using a number line,
you need to use the _?_ of each number. 2. How can you tell whether the sum of 71 and 43 is positive or negative
without actually finding the sum?
Skill Check
Use a number line to find the sum. 3. 9 11
4. 2 (13)
5. 15 (7)
7. 15 3
8. 11 (2)
Find the sum. 6. 24 (16)
Evaluate the expression when x ⴝ ⴚ9. 9. x 3
10. 6 x
11. x (3)
12. Food Science Food scientists tested the effects that freezing and
thawing have on the texture of a cheese filling for ravioli. The filling was frozen to a temperature of 18C. The temperature was then raised 108C. What was the final temperature of the filling? 13. Error Analysis
Describe and correct the error in using a number line to find the sum of 2 and 5.
–5 –8 –7 – 6 – 5 – 4 – 3 –2 –1 0 Lesson 1.5
Adding Integers
31
Practice and Problem Solving Homework Help Example 1 2 3 4
Exercises 14–23, 39 24–29, 40 30–32, 41 33–38
14. Match the correct sum with the addition shown on the number line.
7
6
5
A. 6 8
4
3
2
1
0
B. 6 (8)
1
2
3
C. 6 (8)
Use a number line to find the sum.
Lesson Resources Go to thinkcentral.com • More Examples • @HomeTutor
15. 1 (17)
16. 4 13
17. 7 (3)
18. 13 (3)
19. 9 (5)
20. 6 (7)
21. 8 (2)
22. 3 6
23. 5 (4)
24. 54 40
25. 20 (32)
26. 66 (16)
27. 19 (45)
28. 32 17
29. 72 (30)
30. 7 (9) 15
31. 40 33 12
32. 55 (28) (6)
Find the sum.
Evaluate the expression when x ⴝ ⴚ8, y ⴝ 4, and z ⴝ ⴚ5. 33. x 15
34. y (75)
35. 19 z
36. x y
37. x z
38. y z
39. Critical Thinking Use a single arrow on a number line to represent the
sum 4 (10). What does the length of the arrow indicate? What does the direction of the arrow indicate?
40. Hockey In the National Hockey League, a player is assigned a positive
point each time his team scores while he is on the ice. He is assigned a negative point each time the opposing team scores while he is on the ice. (No points are assigned if the scoring team has more players on the ice than the other team.) The sum of the positive and negative points is called the player’s plus-minus rating. The table shows the points awarded to a player in two games.
a. Find the player’s plus-minus rating for game 1; for game 2. b. Find the total plus-minus rating for the two games.
Game
Positive points
Negative points
1 2
3 2
1 5
c. Interpret and Apply Did the player have a better plus-minus rating in game 1 or in game 2? Explain. 41. Overdraft Your checking account shows an overdraft, or a negative
Tech Help To enter a negative number on a calculator, use , not .
32
Chapter 1
balance. Your present balance is $25. You deposit $108, then withdraw $20 at an ATM. Explain why the expression 25 108 (20) represents your new balance. Do you expect the new balance to be positive or negative? Explain.
Use a calculator to find the sum. Estimate the sum by rounding to check that your answer is reasonable. 42. 345 (978)
Variables, Expressions, and Integers
43. 2172 (4087)
44. 1117 539
45. Lake Vostok Lake Vostok, an Vostok Station
unfrozen lake buried under Antarctic ice, is about 1200 meters deep. Scientists drilled down 3623 meters into the ice to test for signs of life, but stopped 120 meters above the top of the lake to avoid contaminating it.
Lake Vostok
Ice Sheet
a. Find the position of the top of the lake relative to the ice surface. b. Find the position of the bottom of the lake relative to the ice surface. In the
Real World
Lake
46.
Writing Write three different pairs of integers, each of which has a sum of 24. Explain how you chose the integers.
47.
Writing
Lake Vostok The lowest
temperature ever recorded on Earth, about 89C, was recorded in the region of Lake Vostok. The lowest temperature ever recorded in the United States was about 27C higher. What was the record low temperature in the United States?
Bedrock
Justify the rules for adding integers.
Find the sum. 48. 35 16 (12) 7
49. 2 10(3) 5
50. 90 (24) (6) 5
51. 9 16 (12) 3
Evaluate the expression when a ⴝ ⴚ14, b ⴝ 5, and c ⴝ ⴚ8. 52. a b c
53. 15 b c
54. 8 a (4) c
Critical Thinking Using the given information and the fact that x and y are integers, tell whether the sum x ⴙ y is even or odd. Explain your reasoning. 55. x and y are even.
56. x and y are odd.
57. x is even; y is odd.
58. For what values of x is ⏐x⏐ x 0? Explain. 59. Absolute value bars are grouping symbols. Use the order of operations
to evaluate the expression 3 ⏐x 2⏐when x 12.
Challenge Find values of a and b for which the statement is true. 60. ⏐a b⏐ ⏐a⏐ ⏐b⏐
Mixed Review
61. ⏐a b⏐< ⏐a⏐ ⏐b⏐
Write a variable expression to represent the phrase. (Lesson 1.1) 62. The sum of a number and 14.5 63. The difference of a number and 2.75 64. Area You are building a house on a square-shaped lot. The side length
of the lot is 70 yards. Find the area of the lot. (Lesson 1.2) Copy and complete the statement using , or ⴝ. (Lesson 1.4) 66. 12 _?_ ⏐12⏐
65. ⏐15⏐_?_ 15
Standardized Test Practice
67. ⏐2⏐_?_ 2
68. Multiple Choice What is the value of the expression x ⏐y⏐when
x 3 and y 12?
A. 15
B. 9
C. 9
D. 15
69. Short Response For what integer value(s) of x is the value of the
expression 10 ⏐x⏐greater than 0? Explain your reasoning. Lesson 1.5
Adding Integers
33
1.5
Adding Integers Now
BEFORE
You added decimals.
WHY?
You’ll add integers.
So you can find a hockey player’s plus-minus rating, as in Ex. 40.
Vocabulary additive inverse, p. 30
Scuba Diver A scuba diver studying marine life is 4 feet below sea level. From that depth, the diver descends 72 feet to the ocean floor and then rises 61 feet. The diver rests there to avoid decompression illness. Where is the diver relative to sea level? In Example 3, you will see how to answer this question by adding integers. One way to add integers is to use a number line. To add a positive integer, move to the right. To add a negative integer, move to the left. 6 5
4
Example 1
3
2
1
0
1
2
3
4
5
6
Adding Integers Using a Number Line
Use a number line to find the sum.
a. 3 (9)
Study Strategy In the activity on page 28, you used two arrows to add two integers on a number line. The first arrow started at 0 and ended at the first number in the sum. In Example 1, you see that you can draw only one arrow if you start at the first number in the sum.
Move 9 units to the left.
End at 6. 8
7
6
5
4
3
2
1
Start at 3.
0
1
2
3
4
5
Answer The final position is 6. So, 3 (9) 6.
b. 5 3 Move 3 units to the right.
Start at 5. 6
5
4
3
2
End at 2. 1
0
1
Answer The final position is 2. So, 5 3 2. Checkpoint Use a number line to find the sum. 1. 11 6
Video Tutor Go to thinkcentral.com
2. 1 (8) Lesson 1.5
3. 10 (5) Adding Integers
29
Absolute Values You can use absolute values to find the sum of two or more integers.
Adding Integers Words
Numbers
1. Same Sign Add the absolute values
8 12 20 6 (4) 10
and use the common sign. 2. Different Signs Subtract the lesser
absolute value from the greater absolute value and use the sign of the number with greater absolute value. 3. Opposites The sum of a number and
5 (8) 3 11 13 2 7 (7) 0
its opposite is 0. Additive Inverse Property The opposite of a number is also called its additive inverse . Item 3 in the notebook can be written algebraically as a (a) 0 and is called the additive inverse property.
Example 2
Adding Two Integers
a. Find the sum 54 (28). Same sign: Add ⏐ⴚ54⏐ and ⏐ⴚ28⏐.
54 (28) 82
Both integers are negative, so the sum is negative.
b. Find the sum 38 (17). Different signs: Subtract ⏐ⴚ17⏐ from ⏐38⏐.
38 (17) 21
⏐38⏐ ⏐ⴚ17⏐, so the sum has the same sign as 38.
Example 3
Adding More Than Two Integers
To answer the question about the position of the scuba diver at the top of page 29, you can find the sum 4 (72) 61. 4 (72) 61 76 61 15
Add ⴚ4 and ⴚ72. Add ⴚ76 and 61.
Answer The sum is 15, so the diver is 15 feet below sea level. Checkpoint Find the sum. 4. 41 26 30
Chapter 1
Variables, Expressions, and Integers
5. 19 (11)
6. 52 (30) (46)
Example 4
Watch Out When you substitute a negative number for a variable, you may have to enclose the number in parentheses, as in part (b) of Example 4, to avoid confusion with an operation sign.
Evaluating Variable Expressions
Evaluate the expression when x ⴝ ⴚ22 and y ⴝ ⴚ12.
a. x (9)
b. x 17 y
Solution
a. x (9) ⴚ22 (9) 31
Substitute ⴚ22 for x. Add.
b. x 17 y ⴚ22 17 (ⴚ12) 5 (12) 17
Substitute for x and for y. Add ⴚ22 and 17. Add ⴚ5 and ⴚ12.
Checkpoint Evaluate the expression when a ⴝ ⴚ18 and b ⴝ ⴚ3. 7. a (8)
1.5
8. 32 a
9. a b 30
Exercises More Practice, p. 839
Go to thinkcentral.com Practice Exercises
Guided Practice Vocabulary Check
1. Copy and complete: To add two integers without using a number line,
you need to use the _?_ of each number. 2. How can you tell whether the sum of 71 and 43 is positive or negative
without actually finding the sum?
Skill Check
Use a number line to find the sum. 3. 9 11
4. 2 (13)
5. 15 (7)
7. 15 3
8. 11 (2)
Find the sum. 6. 24 (16)
Evaluate the expression when x ⴝ ⴚ9. 9. x 3
10. 6 x
11. x (3)
12. Food Science Food scientists tested the effects that freezing and
thawing have on the texture of a cheese filling for ravioli. The filling was frozen to a temperature of 18C. The temperature was then raised 108C. What was the final temperature of the filling? 13. Error Analysis
Describe and correct the error in using a number line to find the sum of 2 and 5.
–5 –8 –7 – 6 – 5 – 4 – 3 –2 –1 0 Lesson 1.5
Adding Integers
31
Practice and Problem Solving Homework Help Example 1 2 3 4
Exercises 14–23, 39 24–29, 40 30–32, 41 33–38
14. Match the correct sum with the addition shown on the number line.
7
6
5
A. 6 8
4
3
2
1
0
B. 6 (8)
1
2
3
C. 6 (8)
Use a number line to find the sum.
Lesson Resources Go to thinkcentral.com • More Examples • @HomeTutor
15. 1 (17)
16. 4 13
17. 7 (3)
18. 13 (3)
19. 9 (5)
20. 6 (7)
21. 8 (2)
22. 3 6
23. 5 (4)
24. 54 40
25. 20 (32)
26. 66 (16)
27. 19 (45)
28. 32 17
29. 72 (30)
30. 7 (9) 15
31. 40 33 12
32. 55 (28) (6)
Find the sum.
Evaluate the expression when x ⴝ ⴚ8, y ⴝ 4, and z ⴝ ⴚ5. 33. x 15
34. y (75)
35. 19 z
36. x y
37. x z
38. y z
39. Critical Thinking Use a single arrow on a number line to represent the
sum 4 (10). What does the length of the arrow indicate? What does the direction of the arrow indicate?
40. Hockey In the National Hockey League, a player is assigned a positive
point each time his team scores while he is on the ice. He is assigned a negative point each time the opposing team scores while he is on the ice. (No points are assigned if the scoring team has more players on the ice than the other team.) The sum of the positive and negative points is called the player’s plus-minus rating. The table shows the points awarded to a player in two games.
a. Find the player’s plus-minus rating for game 1; for game 2. b. Find the total plus-minus rating for the two games.
Game
Positive points
Negative points
1 2
3 2
1 5
c. Interpret and Apply Did the player have a better plus-minus rating in game 1 or in game 2? Explain. 41. Overdraft Your checking account shows an overdraft, or a negative
Tech Help To enter a negative number on a calculator, use , not .
32
Chapter 1
balance. Your present balance is $25. You deposit $108, then withdraw $20 at an ATM. Explain why the expression 25 108 (20) represents your new balance. Do you expect the new balance to be positive or negative? Explain.
Use a calculator to find the sum. Estimate the sum by rounding to check that your answer is reasonable. 42. 345 (978)
Variables, Expressions, and Integers
43. 2172 (4087)
44. 1117 539
45. Lake Vostok Lake Vostok, an Vostok Station
unfrozen lake buried under Antarctic ice, is about 1200 meters deep. Scientists drilled down 3623 meters into the ice to test for signs of life, but stopped 120 meters above the top of the lake to avoid contaminating it.
Lake Vostok
Ice Sheet
a. Find the position of the top of the lake relative to the ice surface. b. Find the position of the bottom of the lake relative to the ice surface. In the
Real World
Lake
46.
Writing Write three different pairs of integers, each of which has a sum of 24. Explain how you chose the integers.
47.
Writing
Lake Vostok The lowest
temperature ever recorded on Earth, about 89C, was recorded in the region of Lake Vostok. The lowest temperature ever recorded in the United States was about 27C higher. What was the record low temperature in the United States?
Bedrock
Justify the rules for adding integers.
Find the sum. 48. 35 16 (12) 7
49. 2 10(3) 5
50. 90 (24) (6) 5
51. 9 16 (12) 3
Evaluate the expression when a ⴝ ⴚ14, b ⴝ 5, and c ⴝ ⴚ8. 52. a b c
53. 15 b c
54. 8 a (4) c
Critical Thinking Using the given information and the fact that x and y are integers, tell whether the sum x ⴙ y is even or odd. Explain your reasoning. 55. x and y are even.
56. x and y are odd.
57. x is even; y is odd.
58. For what values of x is ⏐x⏐ x 0? Explain. 59. Absolute value bars are grouping symbols. Use the order of operations
to evaluate the expression 3 ⏐x 2⏐when x 12.
Challenge Find values of a and b for which the statement is true. 60. ⏐a b⏐ ⏐a⏐ ⏐b⏐
Mixed Review
61. ⏐a b⏐< ⏐a⏐ ⏐b⏐
Write a variable expression to represent the phrase. (Lesson 1.1) 62. The sum of a number and 14.5 63. The difference of a number and 2.75 64. Area You are building a house on a square-shaped lot. The side length
of the lot is 70 yards. Find the area of the lot. (Lesson 1.2) Copy and complete the statement using , or ⴝ. (Lesson 1.4) 66. 12 _?_ ⏐12⏐
65. ⏐15⏐_?_ 15
Standardized Test Practice
67. ⏐2⏐_?_ 2
68. Multiple Choice What is the value of the expression x ⏐y⏐when
x 3 and y 12?
A. 15
B. 9
C. 9
D. 15
69. Short Response For what integer value(s) of x is the value of the
expression 10 ⏐x⏐greater than 0? Explain your reasoning. Lesson 1.5
Adding Integers
33
