Chapter 1 Section 5. Objectives 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Adding and Subtracting Real Numbers Add two numbers with the same

Chapter 1 Section 5

Objectives

1

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Adding and Subtracting Real Numbers

Add two numbers with the same sign.

Add two numbers with different signs.

Use the definition of subtraction.

Use the rules for order of operations with real numbers.

Translate words and phrases involving addition and subtraction.

Use signed numbers to interpret data.

1.5

2

3

4

5

6


Add two numbers with the same sign.

Objective 1

Slide 1.5-3


Adding Numbers with the Same SignTo add two numbers with the same sign, add the absolute values of the numbers. The sum has the same sign as the numbers being added.

Example:

Add two numbers with the same sign.The sum of two negative numbers is a negative number whose distance from 0 is the sum of the distance of each number from 0.

That is, the sum of two negative numbers is the negative of the sum of their absolute values.

4 3 7

To avoid confusion, two operation symbols should not be written successively without a parenthesis between them.

Slide 1.5-4


Use a number line to find each sum.

Solution:

1 4

2 5

5

−7

Slide 1.5-5

EXAMPLE 1 Adding Numbers on a Number Line


Find the sum.

Solution:

15 4 15 4

15 4

19

Slide 1.5-6

EXAMPLE 2 Adding Two Negative Numbers



Objective 2

Slide 1.5-7


For instance, to add −12 and 6, find their absolute values:

and

Then find the difference between these absolute values:

The sum will be negative, since ,

so the final answer is .

Adding Numbers with the Same SignTo add two numbers with different signs, find the absolute values of the numbers and subtract the lesser absolute value from the greater. Give the answer the same sign as the number having the greater absolute value.

12 12 6 6

12 6 6

12 6

12 6 6

Slide 1.5-8



Solution:

Use a number line to find the sum.

6 3

3

Slide 1.5-9

EXAMPLE 3 Adding Numbers with Different Signs


Find their absolute values:

and

Then find the difference between these absolute values:

The sum will be negative, since ,

so the final answer is .

Solution:

Find the sum.

8

6 14

Slide 1.5-10

EXAMPLE 4 Adding Numbers with Different Signs

6 14

6 6

14 6 8

14 14


Solution:

Correct3 11 5

4 8 8

3.8 9.5 5.7

Check each answer.

Correct

Slide 1.5-11

EXAMPLE 5 Adding Mentally



Objective 3

Slide 1.5-12


We can illustrate the subtraction of 4 from 7, written 7 − 4, with a number line.


The procedure to find the difference 7 − 4 is exactly the same procedure that would be used to find the sum.

7 4 7 ( 4)

Slide 1.5-13

The answer to a subtraction problem is called a difference. In the subtraction x −y, x is called the minuend and y is called the subtrahend.


The previous equation suggests that subtracting a positive number from a greater positive number is the same as adding the additive inverse of the lesser number to the greater.

Definition of Subtraction

For any real numbers x and y,

To subtract y from x, add the additive inverse (or opposite) of y to x. That is, change the subtrahend to its opposite and add.

( .)x y x y

Slide 1.5-14

Use the definition of subtraction. (cont’d)


Solution:

Subtract.

8 ( 12)

8 ( 5) 13

8 (12) 4

8 5

Slide 1.5-15

EXAMPLE 6 Using the Definition of Subtraction

5 3

4 7

5 3

4 7

35 12

28 28

47 19 or 1

28 28


Uses of the Symbol −

We use the symbol − for three purposes:

1. to represent subtraction, as in

2. to represent negative numbers, such as −10, −2, and −3;

3. to represent the opposite (or negative) of a number, as in “the opposite (or negative) of 8 is −8.”

We may see more than one use of − in the same expression, such as −6 − (−9), where −9 is subtracted from −6. The meaning of the symbol depends on its position in the algebraic expression.

9 5 4;

Slide 1.5-16

Use the definition of subtraction. (cont’d)


Use the rules for order of operations with real numbers.

Objective 4

Slide 1.5-17


Perform each indicated operation.

Solution:

6 1 4 2 6 1 4 2 6 5 2

6 5 2 6 7

1

Slide 1.5-18

EXAMPLE 7 Adding and Subtracting with Grouping Symbols


Perform each indicated operation.

Solution:

1 1 1

6 3 4

Slide 1.5-19

EXAMPLE 7 Adding and Subtracting with Grouping Symbols (cont’d)

1 1 1

6 3 4

2 4 3

12 12 12

2 3

12 12

1

12


Translate words and phrases involving addition and subtraction.

Objective 5

Slide 1.5-20


Translate words and phrases involving addition and subtraction.The word sum indicates addition. The table lists other words and phrases that indicate addition in problem solving.

Slide 1.5-21


Solution:

Write a numerical expression for the phrase and simplify the expression.

7 increased by the sum of 8 and −3

7 8 3 7 5

12

Slide 1.5-22

EXAMPLE 8 Translating Words and Phrases (Addition)


The word difference indicates subtraction. Other words and phrases that indicate subtraction in problem solving are given in the table.

In subtracting two numbers, be careful to write them in the correct order, because in general, . For example, .

Think carefully before interpreting an expression involving subtraction.

a b b a 5 3 3 5

Slide 1.5-23

Translate words and phrases involving addition and subtraction. (cont’d)


Write a numerical expression for each phrase, and simplify the expression.

The difference between −5 and −12

−2 subtracted from the sum of 4 and −4

Solution:

4 ( 4) ( 2)

4 4 2

2

Slide 1.5-24

EXAMPLE 9 Translating Words and Phrases (Subtraction)

5 12 5 12

7


The highest Fahrenheit temperature ever recorded in Barrow, Alaska, was 79°F, while the lowest was −56°F. What is difference between these highest and lowest temperatures? (Source: World Almanac and Book of Facts.)

Solution:

79 ( 56)

79 56

135 F

Slide 1.5-25

EXAMPLE 10 Solving a Problem Involving Subtraction


Use signed numbers to interpret data.

Objective 6

Slide 1.5-26


Solution:

Refer to Figure 17 and use a signed number to represent the change in the CPI from 2002 to 2003.

121.4 119.3

121.4 ( 119.3) 2.1

Slide 1.5-27

EXAMPLE 11 Using a Signed Number to Interpret Data

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Chapter 1 Section 5. Objectives 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Adding and Subtracting Real Numbers Add two numbers with the same