Transcript Symbols and Sets of Numbers

Symbols and Sets of Numbers
Equality Symbols
a  b  a is equal to b
a  b  a is not equal to b
Symbols and Sets of Numbers
Inequality Symbols
a  b  a is less than b
a  b  a is greater than b
a  b  a is less than or
equal to b
a  b  a is greater than or
equal to b
Symbols and Sets of Numbers
Equality and Inequality Symbols
are used to create mathematical
statements.
37
52
6  27
x  2.5
Symbols and Sets of Numbers
Order Property for Real Numbers
For any two real numbers, a and b, a is
less than b if a is to the left of b on the
number line.
-92
-25
1  43
11  12
-11
0
1
12
43
67  12
11  92
67
Symbols and Sets of Numbers
True or False
86
F
35  35  F 
100  15 T 
7  2
T 
22  83 T  14  34  F 
Symbols and Sets of Numbers
Translating Sentences into
Mathematical Statements
Fourteen is greater than or
equal to fourteen.
14  14
Zero is less than five.
05
Nine is not equal to ten.
9  10
The opposite of five is
less than or equal to
negative two.
5  2
Symbols and Sets of Numbers
Definitions:
–Natural Numbers: {1, 2, 3, 4, …}
Symbols and Sets of Numbers
Definitions:
–Natural Numbers: {1, 2, 3, 4, …}
–Whole Numbers: All natural
numbers plus zero, {0, 1, 2, 3, …}
Identifying Common Sets of
Numbers
Definitions:
Integers: All positive numbers,
negative numbers and zero
without fractions and decimals
{…, -3, -2, -1, 0, 1, 2, 3, 4, …}
Symbols and Sets of Numbers
Identifying Common Sets of
Numbers
Definitions:
Rational Numbers: Any number
that can be expressed as a
quotient of two integers.
a

 a and b are integers and b  0 
b

Symbols and Sets of Numbers
Identifying Common Sets of
Numbers
Definitions:
Irrational Numbers: Any
number that can not be
expressed as a quotient of two
integers.
Examples:  , 5, 13, 3 22
Symbols and Sets of Numbers
Real Numbers
Irrational
Rational
Non-integer
rational #s
Integers
Negative
numbers
Whole
numbers
Zero
Natural
numbers
Symbols and Sets of Numbers
Given the following set of numbers, identify which elements
belong in each classification:
2


100,  , 0,  , 6, 913
5


Natural Numbers
6 913
Whole Numbers
0 6 913
Integers
100 0 6 913
Rational Numbers
100  52 0 6 913
Irrational Numbers

Real Numbers
All elements
Properties of Real Numbers
Commutative Properties
a b  ba
Multiplication: a  b  b  a
5 y  y 5
Addition:
8 z 
z 8
t  12  12  t
m r  r  m
Properties of Real Numbers
Associative Properties
 a  b   c  a  b  c 
Multiplication:  a  b   c  a   b  c 
5   3  6   5  3  6
 2  7  3  2   7  3
 q  r   17  q   r  17 
 mr   92  m   r  92
Addition:
Properties of Real Numbers
Distributive Property of Multiplication
a  b  c   ab  ac
a  b  c   ab  ac
5   x  y   5x  5 y
3  2  7x   6  21x
4  x  6 y  2 z   4 x  24 y  8 z
4  k   7 
k 3
Properties of Real Numbers
Identity Properties:
Addition: a  0  a
and
0a  a
0 is the identity element for addition
Multiplication: a 1  a
and 1 a  a
1 is the identity element for multiplication
Properties of Real Numbers
Additive Inverse Property: The numbers
a and –a are additive inverses or opposites
of each other if their sum is zero.
a   a   0
Multiplicative Inverse Property: The
numbers b and b1  b  0 are reciprocals
or multiplicative inverses of each other
if their product is one.
1
b  1
b
Name the appropriate property for the given statements:
7  a  b   7a  7b
12  y  y  12
Distributive
Commutative
prop. of addition
4   6  x    4  6  x
Associative property
of multiplication
6   z  2  6   2  z 
Commutative prop. of
addition
1
3   1
3
Multiplicative inverse
 7  y  10  y   7 10
Commutative and
associative prop. of
Solving Linear Equations
Suggestions for Solving Linear Equations:
1. If fractions exist, multiply by the LCD to clear all
fractions.
2. If parentheses exist, used the distributive
property to remove them.
3. Simplify each side of the equation by combining
like-terms.
4. Get all variable terms to one side of the equation
and all numbers to the other side.
5. Use the appropriate properties to get the
variable’s coefficient to be 1.
6. Check the solution by substituting it into the
original equation.
Solving Linear Equations
Example 1:
4  3b 1  20
12b  4  20
12b  4  4  20  4
12b  24
12b 24

12 12
b2
Check:
4  3  2 1  20
4  6 1  20
4  5  20
20  20
Solving Linear Equations
Example 2:
4 z  8  2 z  9
4z  16z  72
4z 16z  16z 16z  72
12z  72
12 z 72

12 12
z  6
Check:
4  6  8  2  6  9
24  8  12  9
24  8  3
24  24
Solving Linear Equations
Example 3:
y
 4 1
6
y

6    4   6  1
6

6y
 24  6
6
y  24  6
y  24  24  6  24
y  30
Check:
30
 4 1
6
5 4 1
11
Solving Linear Equations
Example 4:
0.4  x  7   0.1 3x  6  0.8
0.4x  2.8  0.3x  0.6  0.8
0.1x  2.2  0.8
0.1x  2.2  2.2  0.8  2.2
0.1x  3.0
0.1x 3.0

0.1
0.1
x  30
Solving Linear Equations
Example 4: 0.4  x  7   0.1 3x  6   0.8
Check:
0.4  30  7   0.1 3  30  6  0.8
12.0  2.8  0.1 90  6  0.8
12.0  2.8  0.1 84   0.8
12.0  2.8  8.4  0.8
9.2  8.4  0.8
0.8  0.8
Solving Linear Equations
Example 5:
6  x  5  12  6 x  42
6x  30 12  6x  42
6x  42  6x  42
6x  42  42  6x  42  42
6x  6x
6x  6x  6x  6x
0  0 Identity Equation – It has an infinite
number of solutions.
Solving Linear Equations
Example 6:
y
2y
3 
1
3
6

6y
12 y
 18 
6
3
6
y

 2y 
6    3  6  
 1
3

 6


2 y  18  18  2 y  6  18
2 y  2 y  2 y  2 y  24
0  24
0  24
No Solution
2 y  18  2 y  6

2 y  2 y  24