Transcript 1.1 Exponents, Order of Operations, and Inequality

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1.1 – Slide 1
Chapter 1
The Real Number System
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1.1 – Slide 2
1.1
Exponents, Order of Operations,
and Inequality
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1.1 – Slide 3
1.1 Exponents, Order of Operations, and Inequality
Objectives
1.
2.
3.
4.
5.
6.
Use exponents.
Use the rules for order of operations.
Use more than one grouping symbol.
Know the meanings of ≠, <, >, ≤, and ≥.
Translate word statements to symbols.
Write statements that change the direction
of inequality symbols.
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1.1 – Slide 4
1.1 Exponents, Order of Operations, and Inequality
Exponents allow us a way to abbreviate repeated factors.
Example 1
Exponent
3·3·3·3 = 3
4
4 factors of 3
CAUTION:
3
4
means
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3 · 3 · 3 · 3,
Base
not
4 · 3.
1.1 – Slide 5
1.1 Exponents, Order of Operations, and Inequality
Exponents
Example 2
5
3
= 5 · 5 · 5 = 125
Example 3
3
7
4
=
3
7
·
3
7
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·
3
7
·
3
7
=
81
2401
1.1 – Slide 6
1.1 Exponents, Order of Operations, and Inequality
Order of Operations
Order of Operations
If grouping symbols are present, simplify within them, innermost
first (and above and below fraction bars separately), in the
following order.
Step 1
Apply all exponents.
Step 2
Do any multiplications or divisions in the order in
which they occur, working from left to right.
Step 3
Do any additions or subtractions in the order in which
they occur, working from left to right.
If no grouping symbols are present, start with Step 1.
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1.1 – Slide 7
1.1 Exponents, Order of Operations, and Inequality
Order of Operations
Example 4
Find the value of the expression.
8·2–5
= 16 – 5
Multiply.
= 11
Subtract.
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1.1 – Slide 8
1.1 Exponents, Order of Operations, and Inequality
Order of Operations
Example 5
Find the value of the expression.
4(3 + 8)
= 4 ( 11 )
Add inside parentheses.
= 44
Multiply.
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1.1 – Slide 9
1.1 Exponents, Order of Operations, and Inequality
Order of Operations
Example 6
Find the value of the expression.
5·3–2·4
= 15 – 8
=7
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Multiply, working from
left to right.
Subtract.
1.1 – Slide 10
1.1 Exponents, Order of Operations, and Inequality
Order of Operations
Example 7 Find the value of the expression.
2
3 ( 10 – 3 · 2 ) + 30 – 5
= 3 ( 10 – 6 ) + 30 – 5
= 3 ( 4 ) + 30 – 5
2
2
Multiply inside parentheses.
Subtract inside parentheses.
= 3 ( 4 ) + 30 – 25
Apply the exponent.
= 12 + 30 – 25
Multiply.
= 42 – 25
Add.
= 17
Subtract.
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1.1 – Slide 11
1.1 Exponents, Order of Operations, and Inequality
Order of Operations
Example 8 Find the value of the expression.
3[9+2(4–1)]
=3[9+2(3)]
Subtract inside parentheses.
=3[9+6]
Multiply inside brackets.
= 3 [ 15 ]
Add inside brackets.
= 45
Multiply.
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1.1 – Slide 12
1.1 Exponents, Order of Operations, and Inequality
Order of Operations
Example 9 Find the value of the expression.
3(2+6)–9
2(9)–3(5)
=
3(8)–9
2(9)–3(5)
Add inside parentheses.
=
24 – 9
18 – 15
Multiply.
=
15
3
Subtract.
= 5
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Divide.
1.1 – Slide 13
1.1 Exponents, Order of Operations, and Inequality
Inequalities
Inequality
Example
Meaning
≠
2≠5
2 does not equal 5
<
3<4
3 is less than 4
≤
5≤7
5 is less than or equal to 7
>
7>3
7 is greater than 3
≥
8≥8
8 is greater than or equal to 8
To keep the meanings of the symbols < and > clear, remember
that the symbol always points to the lesser number.
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1.1 – Slide 14
1.1 Exponents, Order of Operations, and Inequality
Inequalities
#
Example
True or False?
(a)
6≠1
True
(b)
9≥5
True
(c)
8<4
False
(d)
1>2
False
(e)
6≤6
True
If either the < part or the = part is true, then the inequality ≤ is
true.
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1.1 – Slide 15
1.1 Exponents, Order of Operations, and Inequality
Inequalities
IMPORTANT: To compare fractions, write them with a common
denominator.
Is
5
7
15
21
True
15
21
False
15
21
≥
2
3
≥
14
21
First, rewrite fractions with a
common denominator.
>
14
21
=
14
21
Now compare the two parts.
Since one of the parts is true,
5
2
≥
is true.
7
3
?
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1.1 – Slide 16
1.1 Exponents, Order of Operations, and Inequality
Translating Word Statements to Symbols
#
Statement
(a)
Twelve is equal to ten plus two.
(b)
Nine is less
. than ten.
9 < 10
(c)
Fifteen is not equal to eighteen.
15 ≠ 18
(d)
Seven is greater than four.
7>4
(e)
Thirteen is less than or equal to forty.
13 ≤ 40
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Symbols
12 = 10 + 2
1.1 – Slide 17
1.1 Exponents, Order of Operations, and Inequality
Converting Inequalities
To convert between < and >, reverse both the order of the
numbers and the direction of the symbol.
Example 10
Interchange numbers.
6 < 10
becomes
10 > 6
Reverse symbol.
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1.1 – Slide 18
1.1 Exponents, Order of Operations, and Inequality
Converting Inequalities
To convert between < and >, reverse both the order of the
numbers and the direction of the symbol.
Example 11
Interchange numbers.
15 > 2
becomes
2 < 15
Reverse symbol.
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1.1 – Slide 19
1.1 Exponents, Order of Operations, and Inequality
Sentences vs. Expressions
CAUTION
The symbols of equality and inequality are used to write
mathematical sentences. They differ from the symbols for
operations (+, −, ·, and ÷), discussed earlier, which are used to
write mathematical expressions that represent a number. For
example, compare the sentence 4 < 10, which gives the
relationship between 4 and 10, with the expression 4 + 10,
which tells how to operate on 4 and 10 to get the number 14.
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1.1 – Slide 20