9.1 – Symbols and Sets of NumbersDefinitions:
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
Inequality Symbols
a b a is less than b
a b a is greater than b
a b a is greater than or
equal to b
a b a is less 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 2.5x
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.
0 1 12 43 67-11-25-92
1 43 67 12
11 12 11 92
9.1 – Symbols and Sets of Numbers
Translating Sentences into Mathematical Statements
Fourteen is greater than or equal to fourteen.
Zero is less than five.
Nine is not equal to ten.
The opposite of five is less than or equal to negative two.
0 5
5 2
9 10
14 14
9.1 – Symbols and Sets of Numbers
Identifying Common Sets of Numbers
Definitions:
9.1 – Symbols and Sets of Numbers
Integers: All positive numbers, negative numbers and zero without fractions and decimals.
{…, -3, -2, -1, 0, 1, 2, 3, 4, …}
Identifying Common Sets of Numbers
Definitions:
9.1 – Symbols and Sets of Numbers
Rational Numbers: Any number that can be expressed as a quotient of two integers.
and are integers and 0aa b b
b
Irrational Numbers: Any number that can not be expressed as a quotient of two integers.
, 5, 13, 3 22
Real Numbers
Irrational Rational
Non-integer rational #s
Integers
Negative numbers
Whole numbers
ZeroNatural numbers
9.1 – Symbols and Sets of Numbers
Given the following set of numbers, identify which elements belong in each classification:
2100, , 0, , 6, 913
5
Natural Numbers
Whole Numbers
Integers
Rational Numbers
Irrational Numbers
Real Numbers
6 913
6 9130
100 0 6 91325
100 0 6 913
All elements
9.1 – Symbols and Sets of Numbers
9.2 – Properties of Real NumbersCommutative Properties
Addition: a b b a Multiplication: a b b a
m r
12t
5 y
8 z
5y
8z
12 t
r m
Associative Properties
Addition: a b c a b c Multiplication: a b c a b c
92mr 17q r
5 3 6
2 7 3 5 3 6
2 7 3
17q r
92m r
9.2 – Properties of Real Numbers
Distributive Property of Multiplication
a b c ab ac
4 7k 4 6 2x y z
5 x y
3 2 7x 5 5x y6 21x
4 24 8x y z 4 7k
a b c ab ac
3k
9.2 – Properties of Real Numbers
Identity Properties:
0 0a a and a a Addition:
Multiplication: 1 1a a and a a
9.2 – Properties of Real Numbers
0 is the identity element for addition
1 is the identity element for multiplication
Additive Inverse Property: The numbers a and –a are additive inverses or opposites of each other if their sum is zero.
0a a
1and 0bb b
11b
b
Multiplicative Inverse Property: The numbers are reciprocals or multiplicative inverses of each other if their product is one.
9.2 – Properties of Real Numbers
Name the appropriate property for the given statements:
7 7 7a b a b
4 6 4 6x x
6 2 6 2z z
13 1
3
7 10 7 10y y
12 12y y
Distributive
Commutative prop. of addition
Associative property of multiplication
Commutative prop. of addition
Multiplicative inverse
Commutative and associative prop. of multiplication
9.2 – Properties of Real Numbers
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 3 1 20b
12 4 20b
41 4 20 42b
12 24b12 24
12 12
b
2b
Check:
3 2024 1
4 6 1 20
4 5 20
20 20
9.3 – Solving Linear Equations
Example 2: 4 8 2 9z z
4 16 72z z
4 1616 6 21 7z zz z
12 72z 12 72
12 12
z
6z
Check:
4 8 26 6 9
24 8 12 9
24 8 3
24 24
9.3 – Solving Linear Equations
Example 3:
4 16
y
4 6 16
6y
24 6y
24 24 42 6y
Check:
304 1
6
5 4 1
1 1
30y
624 6
6
y
LCD = 6
9.3 – Solving Linear Equations
Example 4: 0.4 7 0.1 3 6 0.8x x
0.4 2.8 0.3 0.6 0.8x x
0.1 2.2 0.8x
2.20 2.1 . 22.2 8 .0x
30x
0.1 3.0x
0.1 3.0
0.1 0.1
x
9.3 – Solving Linear Equations
Example 4:
0.4 7 0.1 330 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
Check:
0.8 0.8
9.2 8.4 0.8
0.4 7 0.1 3 6 0.8x x 9.3 – Solving Linear Equations
Example 5: 6 5 12 6 42x x
6 30 12 6 42x x
6 42 6 42x x
42 442 26 6 42x x
0 0
6 6x x66 66x x xx
Identity Equation – It has an infinite number of solutions.
9.3 – Solving Linear Equations