Transcript Ch1-Sec 1.6

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1.6 – Slide 1
Chapter 1
The Real Number System
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1.6 – Slide 2
1.6
Multiplying and Dividing
Real Numbers
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1.6 – Slide 3
1.6 Multiplying and Dividing Real Numbers
Objectives
1.
2.
3.
4.
5.
6.
7.
Find the product of a positive number and a
negative number.
Find the product of two negative numbers.
Use the reciprocal of a number to apply the
definition of division.
Use the rules for order of operations when
multiplying and dividing signed numbers.
Evaluate expressions involving variables.
Translate words and phrases involving
multiplication and division.
Translate simple sentences into equations.
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1.6 – Slide 4
1.6 Multiplying and Dividing Real Numbers
Finding the Product of a Positive and Negative Number
Multiplication Property of 0
For any real number a,
a  0  0  a  0.
Since multiplication can also be considered repeated
addition, the product 3(–1) represents the sum
–1 + (–1) + (–1) = –3.
Add –1 three times.
The product of a positive number and a negative
number is negative. 6  3  18 and  6  3  18
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1.6 – Slide 5
1.6 Multiplying and Dividing Real Numbers
Finding the Product of a Positive and Negative Number
Example 1
Find each product using the multiplication rule.
(a) 9(–3) = –(9 · 3) = –27
(b) –6(8) = –(6 · 8) = –48
(c) –16(⅜) = –6
(d) 2.9(–3.2) = –9.28
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1.6 – Slide 6
1.6 Multiplying and Dividing Real Numbers
Finding the Product of Two Negative Numbers
The product of two negative numbers is positive.
Example 2
Find each product using the multiplication rule.
(a) –5(–7) = 35
(b) –6(–12) = 72
(c) –2(3)(–1) = –6(–1) = 6
(d) 3(–5)(–2) = –15(–2) = 30
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1.6 – Slide 7
1.6 Multiplying and Dividing Real Numbers
Using a Reciprocal to Apply the Definition of Division
Reciprocals
Pairs of numbers whose product is 1 are called
reciprocals of each other.
3
5
3 5
i.e. and are reciprocals because   1.
5
3
5 3
0 has no reciprocal.
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1.6 – Slide 8
1.6 Multiplying and Dividing Real Numbers
Using a Reciprocal to Apply the Definition of Division
Division
a
of real numbers a and b, with b ≠ 0, is
The quotient
b
a
1
 a .
b
b
Example :
Note
8
 1
 8     2
4
 4
0
If b  0, then  0.
b
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1.6 – Slide 9
1.6 Multiplying and Dividing Real Numbers
Using a Reciprocal to Apply the Definition of Division
Example 3
Find each quotient.
15
1
(a)
 15   5
3
3
8
(d)
 Undefined
0
3 2
3 3
9
(b)       
4 3
4 2
8
0
(e)
 0
5
1.8
 1 

1.8

(c)


6
0.3
 0.3 
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(f )
4  3
3
 4
1.6 – Slide 10
1.6 Multiplying and Dividing Real Numbers
Using a Reciprocal to Apply the Definition of Division
Example 4
Find each quotient.
(a)
10
 5
2
12
(b)
4
3
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1  3 2
(c)      
5  10  3
6.3
(d)
 90
0.07
1.6 – Slide 11
1.6 Multiplying and Dividing Real Numbers
Using a Reciprocal to Apply the Definition of Division
Dividing Signed Numbers
The quotient of two numbers having the same sign is
positive. The quotient of two numbers have different
signs is negative.
For any positive real numbers a and b,
a a
a

 .
b b
b
For any positive real numbers a and b,
a a
 .
b b
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1.6 – Slide 12
1.6 Multiplying and Dividing Real Numbers
Using the Order of Operations with Signed Numbers
Example 5
Simplify.
(a) –9(2) – (–3)(2) =
Find all products, working from left to right.
–9(2) – (–3)(2) = –18 – (–6)
= –18 + 6
= –12
(b) –6(–2) –3(–4) = 12 – (–12)
= 12 + 12
= 24
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1.6 – Slide 13
1.6 Multiplying and Dividing Real Numbers
Using the Order of Operations with Signed Numbers
Example 5 (concluded)
Simplify.
5(2)  3(4)
(c)
2(1  6)
Simplify the numerator and denominator separately.
5(2)  3(4) 10  12

2(1  6)
2(5)
22

10
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11

5
1.6 – Slide 14
1.6 Multiplying and Dividing Real Numbers
Evaluating Expressions Involving Variables
Example 6
Evaluate each expression, given that x = –1, y = –2, and m = –3.
(a) (3x + 4y)(–2m)
Substitute the given values for the variables. Then use the
order of operations to find the value of the expression.
(3x + 4y)(–2m) = [3(–1) + 4(–2)][–2(–3)]
= [–3 + (–8)][6]
= (–11)(6)
= –66
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1.6 – Slide 15
1.6 Multiplying and Dividing Real Numbers
Evaluating Expressions Involving Variables
Example 6 (continued)
Evaluate each expression, given that x = –1, y = –2, and m = –3.
(b) 2x2 – 3y2 = 2(–1)2 – 3(–2)2
= 2(1) – 3(4)
= 2 – 12
= –10
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1.6 – Slide 16
1.6 Multiplying and Dividing Real Numbers
Evaluating Expressions Involving Variables
Example 6 (concluded)
Evaluate each expression, given that x = –1, y = –2, and m = –3.
4 y 2  x 4(2) 2  (1)
(c)

m
(3)
4(4)  ( 1)
3
16  (1)

3
15

3
 5

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1.6 – Slide 17
1.6 Multiplying and Dividing Real Numbers
Translating Words and Phrases
Example
Numerical
Expression and
Simplification
Product of
The product of –5 and –2
–5(–2) =10
Times
13 times –4
Twice
(meaning “2
times”)
Twice 6
2(6) =12
Of (used with
½ of 10
fractions)
½(10) =5
Word or
Phrase
Percent of
12 % of –16
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13(–4) = –52
0.12(–16) =–1.92
1.6 – Slide 18
1.6 Multiplying and Dividing Real Numbers
Translating Words and Phrases
Word or
Phrase
Example
Quotient of
The quotient of –24 and 3
Divided by
–16 divided by –4
Ratio of
The ratio of 2 to 3
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Numerical
Expression and
Simplification
24
 8
3
16
4
4
2
3
1.6 – Slide 19
1.6 Multiplying and Dividing Real Numbers
Translating Words and Phrases
Example 7
Write a numerical expression for each phrase, and simplify the
expression.
(a) Three fourths of the difference between 8 and –2
3
3
[8  (2)]  (8  2)
4
4
3
 (10)
4
15
  7.5
2
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1.6 – Slide 20
1.6 Multiplying and Dividing Real Numbers
Translating Words and Phrases
Example 7 (concluded)
Write a numerical expression for each phrase, and simplify the
expression.
(b) 20% of the sum of 1200 and 400
0.20(1200 + 400) = 0.20(1600)
= 320
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1.6 – Slide 21
1.6 Multiplying and Dividing Real Numbers
Translating Words and Phrases
Example 8
Write a numerical expression for each phrase, and simplify the
expression.
(c) The quotient of 20 and the difference between –11 and –7
20
20
20


 5
[11  (7)] [11  7] 4
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1.6 – Slide 22
1.6 Multiplying and Dividing Real Numbers
Translating Simple Sentences into Equations
Example 9
Write each sentence in symbols, using x to represent the number.
(a) Five times a number is 40.
5x = 40
(b) The quotient of a number and –8 is 6.
x
6
8
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1.6 – Slide 23
1.6 Multiplying and Dividing Real Numbers
Translating Simple Sentences into Equations
CAUTION
It is important to recognize the distinction between the
types of problems found in Examples 7 and 8 and those
in Example 9. In Example 7 and 8, the phrases translate as expressions, while in Example 9, the sentences
translate as equations. Remember that an expression
is a phrase, while an equation is a sentence.
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1.6 – Slide 24