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

True or False

35 35 7 2

22 83 14 34

8 6 100 15 F T

F T

T F

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

Example 6:2

3 13 6

y y

6 62

3 13 6

y y

6 1218 6

3 6

y y

2 18 2 6y y

2 2 42 22y yy y

12 18 2 68 18y y

2 2 24y y

0 24 0 24 No Solution

LCD = 6

9.3 – Solving Linear Equations