Download Chapter 1 Section 3 - Lamar County School District

Chapter 1
Real Numbers and
Introduction to
Algebra
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Bellwork:
Translate the following into mathematical sentences.
Then, determine if the sentence is true or false.
1.
The absolute value of
negative two is less than
negative four.
1.
The sum of five and
negative two is a whole
number.
3.
The quotient of three
and four is greater than
the sum of four and
negative six.
3.
Eight is not equal to the
the absolute value of
negative eight.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
2
Bellwork:
Translate the following into mathematical sentences.
Then, determine if the sentence is true or false.
1.
1.
The absolute value of
negative two is less than
negative four.
|-2| < -4
The sum of five and
negative two is a whole
number.
3.
The quotient of three
and four is greater than
the sum of four and
negative six.
3.
Eight is not equal to the
the absolute value of
negative eight.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
3
Bellwork:
Translate the following into mathematical sentences.
Then, determine if the sentence is true or false.
1.
1.
The absolute value of
negative two is less than
negative four.
|-2| < -4 FALSE
The sum of five and
negative two is a whole
number.
3.
The quotient of three
and four is greater than
the sum of four and
negative six.
3.
Eight is not equal to the
the absolute value of
negative eight.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
4
Bellwork:
Translate the following into mathematical sentences.
Then, determine if the sentence is true or false.
1.
1.
The absolute value of
negative two is less than
negative four.
|-2| < -4 FALSE
The sum of five and
negative two is a whole
number.
5 + -2
3.
The quotient of three
and four is greater than
the sum of four and
negative six.
3.
Eight is not equal to the
the absolute value of
negative eight.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
5
Bellwork:
Translate the following into mathematical sentences.
Then, determine if the sentence is true or false.
1.
1.
The absolute value of
negative two is less than
negative four.
|-2| < -4 FALSE
The sum of five and
negative two is a whole
number.
5 + -2 = 3
3.
The quotient of three
and four is greater than
the sum of four and
negative six.
3.
Eight is not equal to the
the absolute value of
negative eight.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
6
Bellwork:
Translate the following into mathematical sentences.
Then, determine if the sentence is true or false.
1.
1.
The absolute value of
negative two is less than
negative four.
|-2| < -4 FALSE
The sum of five and
negative two is a whole
number.
5 + -2 = 3 TRUE
3.
The quotient of three
and four is greater than
the sum of four and
negative six.
3.
Eight is not equal to the
the absolute value of
negative eight.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
7
Bellwork:
Translate the following into mathematical sentences.
Then, determine if the sentence is true or false.
1.
1.
The absolute value of
negative two is less than
negative four.
|-2| < -4 FALSE
The sum of five and
negative two is a whole
number.
5 + -2 = 3 TRUE
3.
The quotient of three
and four is greater than
the sum of four and
negative six.
¾ > 4+-6
3.
Eight is not equal to the
the absolute value of
negative eight.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
8
Bellwork:
Translate the following into mathematical sentences.
Then, determine if the sentence is true or false.
1.
1.
The absolute value of
negative two is less than
negative four.
|-2| < -4 FALSE
The sum of five and
negative two is a whole
number.
5 + -2 = 3 TRUE
3.
3.
The quotient of three
and four is greater than
the sum of four and
negative six.
¾ > 4+-6
¾ > -2
Eight is not equal to the
the absolute value of
negative eight.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
9
Bellwork:
Translate the following into mathematical sentences.
Then, determine if the sentence is true or false.
1.
1.
The absolute value of
negative two is less than
negative four.
|-2| < -4 FALSE
The sum of five and
negative two is a whole
number.
5 + -2 = 3 TRUE
3.
3.
The quotient of three
and four is greater than
the sum of four and
negative six.
¾ > 4+-6
TRUE
¾ > -2
Eight is not equal to the
the absolute value of
negative eight.
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
10
Bellwork:
Translate the following into mathematical sentences.
Then, determine if the sentence is true or false.
1.
1.
The absolute value of
negative two is less than
negative four.
|-2| < -4 FALSE
The sum of five and
negative two is a whole
number.
5 + -2 = 3 TRUE
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
3.
3.
The quotient of three
and four is greater than
the sum of four and
negative six.
¾ > 4+-6
TRUE
¾ > -2
Eight is not equal to the
the absolute value of
negative eight.
8 ≠ |-8|
11
Bellwork:
Translate the following into mathematical sentences.
Then, determine if the sentence is true or false.
1.
1.
The absolute value of
negative two is less than
negative four.
|-2| < -4 FALSE
The sum of five and
negative two is a whole
number.
5 + -2 = 3 TRUE
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
3.
3.
The quotient of three
and four is greater than
the sum of four and
negative six.
¾ > 4+-6
TRUE
¾ > -2
Eight is not equal to the
the absolute value of
negative eight.
FALSE 8 ≠ |-8|
12
1.3
Exponents, Order of
Operations, and Variable
Expression
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Objectives:
Define
and use exponents and order of
operations
Evaluate algebraic expressions
Determine whether a number is a solution to
an equation
Translate phrases into expressions and
sentences into equations
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14
Using Exponential Notation
We may use exponential notation to write products in
a more compact form.
2  2  2  2  2 can be written as 2 5
2 5 can be written as 2  2  2  2  2
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15
Using Exponential Notation
We may use exponential notation to write products in
a more compact form.
2  2  2  2  2 can be written as 2 5
2 5 can be written as 2  2  2  2  2
5
2
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16
Using Exponential Notation
We may use exponential notation to write products in
a more compact form.
2  2  2  2  2 can be written as 2 5
2 5 can be written as 2  2  2  2  2
base –
repeated factor
(number being
multiplied)
5
2
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17
Using Exponential Notation
We may use exponential notation to write products in
a more compact form.
2  2  2  2  2 can be written as 2 5
2 5 can be written as 2  2  2  2  2
base –
repeated factor
(number being
multiplied)
5
2
exponent number of times
the base is a factor
(repeatedly multiplied)
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18
Example 1
Evaluate.
a. 26 = 2 · 2 · 2 · 2 · 2 · 2 = 64
b. 54 = 5 · 5 · 5 · 5 = 625
c. 91 = 9
 2   2  2  2  2 4

d.       
 5   5  5  5  5 25
2

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19
Example 1
Evaluate.
a. 26 = 2 · 2 · 2 · 2 · 2 · 2 = 64
b. 54 = 5 · 5 · 5 · 5 = 625
c. 91 = 9
 2   2  2  2  2 4

d.       
 5   5  5  5  5 25
2

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20
Example 1
Evaluate.
a. 26 = 2 · 2 · 2 · 2 · 2 · 2 = 64
b. 54 = 5 · 5 · 5 · 5 = 625
c. 91 = 9
 2   2  2  2  2 4

d.       
 5   5  5  5  5 25
2

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21
Example 1
Evaluate.
a. 26 = 2 · 2 · 2 · 2 · 2 · 2 = 64
b. 54 = 5 · 5 · 5 · 5 = 625
c. 91 = 9
 2   2  2  2  2 4

d.       
 5   5  5  5  5 25
2

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22
Example 1
Evaluate.
a. 26 = 2 · 2 · 2 · 2 · 2 · 2 = 64
b. 54 = 5 · 5 · 5 · 5 = 625
c. 91 = 9
 2   2  2  2  2 4

d.       
 5   5  5  5  5 25
2

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23
Example 1
Evaluate.
a. 26 = 2 · 2 · 2 · 2 · 2 · 2 = 64
b. 54 = 5 · 5 · 5 · 5 = 625
c. 91 = 9
 2   2  2  2  2 4

d.       
 5   5  5  5  5 25
2

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24
Example 1
Evaluate.
a. 26 = 2 · 2 · 2 · 2 · 2 · 2 = 64
b. 54 = 5 · 5 · 5 · 5 = 625
c. 91 = 9
 2   2  2  2  2 4

d.       
 5   5  5  5  5 25
2

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25
Example 1
Evaluate.
a. 26 = 2 · 2 · 2 · 2 · 2 · 2 = 64
b. 54 = 5 · 5 · 5 · 5 = 625
c. 91 = 9
 2   2  2  2  2 4

d.       
 5   5  5  5  5 25
2

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26
Example 1
Evaluate.
a. 26 = 2 · 2 · 2 · 2 · 2 · 2 = 64
b. 54 = 5 · 5 · 5 · 5 = 625
c. 91 = 9
 2   2  2  2  2 4

d.       
 5   5  5  5  5 25
2

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27
Using the Order of Operations
Order of Operations
1. Perform all operations within grouping symbols
first, starting with the innermost set.
2. Evaluate exponential expressions.
3. Multiply or divide in order from left to right.
4. Add or subtract in order from left to right.
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28
Example 2
Simplify. 6  3  52
6  3  52  6  3  52
 2  25
 27
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29
Example 2
Simplify. 6  3  52
6  3  52  6  3  52
 2  25
 27
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30
Example 2
Simplify. 6  3  52
6  3  52  6  3  52
 2  25
 27
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31
Example 2
Simplify. 6  3  52
6  3  52  6  3  52
 2  25
 27
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32
Example 3
Simplify. 1  2[5(2  3  1)  10]
1  2[5(2  3  1)  10]  1  2[5( 2  3  1)  10]
 1  2[5(7)  10]
 1  2[35  10]
 1  50
 51
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33
Example 3
Simplify. 1  2[5(2  3  1)  10]
1  2[5(2  3  1)  10]  1  2[5( 2  3  1)  10]
 1  2[5(7)  10]
 1  2[35  10]
 1  50
 51
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34
Example 3
Simplify. 1  2[5(2  3  1)  10]
1  2[5(2  3  1)  10]  1  2[5( 2  3  1)  10]
 1  2[5(7)  10]
 1  2[35  10]
 1  50
 51
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35
Example 3
Simplify. 1  2[5(2  3  1)  10]
1  2[5(2  3  1)  10]  1  2[5( 2  3  1)  10]
 1  2[5(7)  10]
 1  2[35  10]
 1  50
 51
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36
Example 3
Simplify. 1  2[5(2  3  1)  10]
1  2[5(2  3  1)  10]  1  2[5( 2  3  1)  10]
 1  2[5(7)  10]
 1  2[35  10]
 1  50
 51
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37
Example 3
Simplify. 1  2[5(2  3  1)  10]
1  2[5(2  3  1)  10]  1  2[5( 2  3  1)  10]
 1  2[5(7)  10]
 1  2[35  10]
 1  50
 51
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38
Example 4
Simplify. 6  92  3
3
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39
Example 4
Simplify. 6  92  3
3
Write 32 as 9.
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40
Example 4
Simplify. 6  92  3
3
693
6  9 3

2
3
(9)
Write 32 as 9.
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41
Example 4
Simplify. 6  92  3
3
693
6  9 3

2
3
(9)
Write 32 as 9.
Divide 9 by 3.
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42
Example 4
Simplify. 6  92  3
3
693
6  9 3

2
3
(9)

6  (3)
9
Write 32 as 9.
Divide 9 by 3.
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43
Example 4
Simplify. 6  92  3
3
693
6  9 3

2
3
(9)

6  (3)
9
Write 32 as 9.
Divide 9 by 3.
Add 3 to 6.
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44
Example 4
Simplify. 6  92  3
3
693
6  9 3

2
3
(9)


6  (3)
9
9
9
Write 32 as 9.
Divide 9 by 3.
Add 3 to 6.
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45
Example 4
Simplify. 6  92  3
3
693
6  9 3

2
3
(9)


6  (3)
9
9
9
Write 32 as 9.
Divide 9 by 3.
Add 3 to 6.
Divide 9 by 9.
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46
Example 4
Simplify. 6  92  3
3
693
6  9 3

2
3
(9)

6  (3)
9
Write 32 as 9.
Divide 9 by 3.

9
9
Add 3 to 6.

1
Divide 9 by 9.
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47
Evaluating Algebraic Expressions
Variable: A letter to represent all the numbers fitting a
pattern.
Algebraic Expression: A collection of numbers,
variables, operation symbols, and grouping symbols.
Evaluating the Expression: Replacing a variable in an
expression by a number and then finding the value of the
expression
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48
Evaluating Algebraic Expressions
Variable: A letter to represent all the numbers fitting a
pattern.
Algebraic Expression: A collection of numbers,
variables, operation symbols, and grouping symbols.
Evaluating the Expression: Replacing a variable in an
expression by a number and then finding the value of the
expression
Plug it in!
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49
Example 5
Evaluate when z = ‒3.
7  3z
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50
Example 5
Evaluate when z = ‒3.
Plug it in!
7  3z
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51
Example 5
Evaluate when z = ‒3.
Plug it in!
7  3z
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52
Example 5
Evaluate when z = ‒3.
Plug it in!
7  3z  7  3(3)
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53
Example 5
Evaluate when z = ‒3.
Plug it in!
7  3z  7  3(3)
Simplify.
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54
Example 5
Evaluate when z = ‒3.
Plug it in!
7  3z  7  3(3)
 7  (9)
Simplify.
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55
Example 5
Evaluate when z = ‒3.
Plug it in!
7  3z  7  3(3)
 7  (9)
Simplify.
 7 9
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56
Example 5
Evaluate when z = ‒3.
Plug it in!
7  3z  7  3(3)
 7  (9)
Simplify.
 7 9
 2
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57
Solutions of Equations
**Recall:
An equation is when two algebraic expressions or
mathematical statements are connected by an equal sign.
Solving: In an equation containing a variable, finding
which values of the variable make the equation a true
statement.
Solution: In an equation, a value for the variable that
makes the equation a true statement.
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58
Example 6
Determine whether ‒7 is a solution of x + 23 = ‒16.
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59
Example 6
Determine whether ‒7 is a solution of x + 23 = ‒16.
Plug it in!
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60
Example 6
Determine whether ‒7 is a solution of x + 23 = ‒16.
Plug it in!
Simplify.
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61
Example 6
Determine whether ‒7 is a solution of x + 23 = ‒16.
(7)  23  16
Plug it in!
Simplify.
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62
Example 6
Determine whether ‒7 is a solution of x + 23 = ‒16.
(7)  23  16
Plug it in!
Simplify.
– 7 is not a solution.
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63
Translating Phrases
Addition
(+)
sum
plus
added to
more than
increased
by
total
Subtraction
(-)
difference
minus
Multiplication
(∙)
Division
(÷)
Equal Sign
product
quotient
equals
times
divide
subtracted
from
multiply
into
gives
is/was/
should be
less than
twice
ratio
decreased by of
less
yields
divided by amounts to
represents
is the same
as
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64
Example 7
Write as an algebraic expression. Use x to represent “a
number.”
a. 5 decreased by a number
5–x
b. The quotient of a number and 12
x
12  x or
12
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65
Example 7
Write as an algebraic expression. Use x to represent “a
number.”
a. 5 decreased by a number
5–x
b. The quotient of a number and 12
x
12  x or
12
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66
Example 7
Write as an algebraic expression. Use x to represent “a
number.”
a. 5 decreased by a number
5–x
b. The quotient of a number and 12
x
12  x or
12
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67