Transcript Symbols and Sets of Numbers

9.1 – Symbols and Sets of Numbers
Definitions:
Natural Numbers: {1, 2, 3, 4, …}
Whole Numbers: All natural numbers plus zero,
{0, 1, 2, 3, …}
Equality Symbols
a  b  a is equal to b
a  b  a is not equal to b
9.1 – 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
9.1 – Symbols and Sets of Numbers
Equality and Inequality Symbols are used to create
mathematical statements.
37
52
6  27
x  2.5
9.1 – 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
-11
0
1
12
1  43
67  12
11  12
11  92
43
67
9.1 – Symbols and Sets of Numbers
True or False
86
F
35  35  F 
22  83 T 
100  15
7  2
T 
T 
14  34  F 
9.1 – 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
9.1 – Symbols and Sets of Numbers
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, …}
9.1 – 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

Irrational Numbers: Any number that can not
be expressed as a quotient of two integers.
,
5,
13,
3
22
9.1 – Symbols and Sets of Numbers
Real Numbers
Irrational
Rational
Non-integer
rational #s
Integers
Negative
numbers
Whole
numbers
Zero
Natural
numbers
9.1 – 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
9.2 – Properties of Real Numbers
Commutative Properties
Addition: a  b  b  a
Multiplication: a  b  b  a
5 y  y 5
8 z  z 8
t  12  12  t
m r  r  m
9.2 – Properties of Real Numbers
Associative Properties
Addition:  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
9.2 – 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  4  k  7  k  3
9.2 – Properties of Real Numbers
Identity Properties:
Addition: a  0  a and 0  a
0 is the identity element for addition
a
Multiplication: a 1  a and 1 a  a
1 is the identity element for multiplication
9.2 – 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
9.2 – Properties of Real Numbers
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
 7  y  10  y   7 10
Multiplicative inverse
Commutative and associative
prop. of multiplication
9.3 – 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 the variable of interest to one side of the equation and all
terms 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.
9.3 – 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
9.3 – 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
9.3 – Solving Linear Equations
Example 3:
y
 4  1 LCD = 6
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
9.3 – 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
9.3 – 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
9.3 – 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.
9.3 – Solving Linear Equations
Example 6:
y
2y
3 
1
3
6
LCD = 6
6y
12 y
y

 2y 
 18 
6
6    3  6  
 1 
3
6
3

 6

 2 y  18  18  2 y  6  18
 2 y  2 y  2 y  2 y  24
2 y  18  2 y  6
2 y  2 y  24
0  24
0  24
No Solution