Download Add two numbers with the same sign.

Chapter 1
Section 5
1.5 Adding and Subtracting Real Numbers
Objectives
1
Add two numbers with the same sign.
2
Add two numbers with different signs.
3
Use the definition of subtraction.
4
Use the rules for order of operations with real numbers.
5
Translate words and phrases involving addition and subtraction.
6
Use signed numbers to interpret data.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 1
Add two numbers with the same
sign.
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Slide 1.5-3
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.
Adding Numbers with the Same Sign
To 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:
4   3  7
To avoid confusion, two operation symbols should not be written
successively without a parenthesis between them.
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Slide 1.5-4
EXAMPLE 1 Adding Numbers on a Number Line
Use a number line to find each sum.
Solution:
1 4
2   5
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5
−7
Slide 1.5-5
EXAMPLE 2 Adding Two Negative Numbers
Find the sum.
Solution:
15   4
   15  4 
  15  4
 19
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 1.5-6
Objective 2
Add two numbers with different
signs.
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Slide 1.5-7
Add two numbers with different signs.
Adding Numbers with the Same Sign
To 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.
For instance, to add −12 and 6, find their absolute values:
12  12 and 6  6
Then find the difference between these absolute values:
12  6  6
The sum will be negative, since
so the final answer is
12  6 ,
12  6  6.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 1.5-8
EXAMPLE 3 Adding Numbers with Different Signs
Use a number line to find the sum.
Solution:
6   3
3
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Slide 1.5-9
EXAMPLE 4 Adding Numbers with Different Signs
Find the sum.
6   14
Solution:
Find their absolute values:
6 6
and
14  14
Then find the difference between these absolute values:
14  6  8
The sum will be negative, since
6  14 ,
so the final answer is 8.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 1.5-10
EXAMPLE 5 Adding Mentally
Check each answer.
Solution:
3  11 
5
   
4  8
8
Correct
3.8  9.5  5.7
Correct
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 1.5-11
Objective 3
Use the definition of subtraction.
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Slide 1.5-12
Use the definition of subtraction.
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.
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)
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Slide 1.5-13
Use the definition of subtraction. (cont’d)
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,
x  y  x  ( 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.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 1.5-14
EXAMPLE 6 Using the Definition of Subtraction
Subtract.
Solution:
8  5
 8  (5)
 13
8  (12)  8  (12)
4
5  3
 
4  7
35 12


28 28
5 3
 
4 7
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47
19

or 1
28
28
Slide 1.5-15
Use the definition of subtraction. (cont’d)
Uses of the Symbol −
We use the symbol − for three purposes:
1. to represent subtraction, as in 9  5  4;
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.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 1.5-16
Objective 4
Use the rules for order of operations
with real numbers.
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Slide 1.5-17
EXAMPLE 7 Adding and Subtracting with Grouping Symbols
Perform each indicated operation.
Solution:
6   1  4   2 
 6   1   4    2 
 6   5  2
 6   5   2  
 6   7 
 1
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 1.5-18
EXAMPLE 7 Adding and Subtracting with Grouping Symbols (cont’d)
Perform each indicated operation.
Solution:
1  1 1
   
6  3 4
1 1 1
  
6 3 4
2 4
3
 

12 12 12
2 3
 
12 12
1

12
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 1.5-19
Objective 5
Translate words and phrases
involving addition and subtraction.
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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.
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Slide 1.5-21
EXAMPLE 8 Translating Words and Phrases (Addition)
Write a numerical expression for the phrase and simplify the
expression.
7 increased by the sum of 8 and −3
Solution:
7  8   3 
 7   5
 12
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 1.5-22
Translate words and phrases involving addition and
subtraction. (cont’d)
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, a  b  b  a. For example, 5  3  3  5.
Think carefully before interpreting an expression involving subtraction.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 1.5-23
EXAMPLE 9 Translating Words and Phrases (Subtraction)
Write a numerical expression for each phrase, and simplify the
expression.
Solution:
The difference between −5 and −12
 5   12
 5  12
7
−2 subtracted from the sum of 4 and −4
  4  (4)  (2)
  4  4  2
2
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 1.5-24
EXAMPLE 10
Solving a Problem Involving Subtraction
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
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 1.5-25
Objective 6
Use signed numbers to interpret
data.
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Slide 1.5-26
EXAMPLE 11
Using a Signed Number to Interpret Data
Refer to Figure 17 and use a signed number to represent the change
in the CPI from 2002 to 2003.
Solution:
 121.4 119.3
 121.4  (119.3)
 2.1
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Slide 1.5-27