1. Analyzing Equations and Inequalities
Objectives:
- evaluate expressions and formulas using order of operations
- understand/use properties & classifications of real numbers
- solve equations and inequalities, including those containing absolute value

2. Expressions & Formulas
ORDER OF OPERATIONS
Parentheses
Exponents
Multiply/Divide from left to right
Add/Subtract from left to right

3. Order of Operations
Simplify: [9 ÷ (42 - 7)] - 8
Exponents [9 ÷ (16 - 7)] - 8
Parentheses [9 ÷ (9)] - 8
Divide [ 1 ] - 8
Subtract

4. Expressions and Formulas
How do you evaluate expressions and formulas?
Replace each variable with a value and then apply the order of operations.

5. Expressions Evaluate: a[b2(b + a)] if a = 12 and b= 1
Substitute: [12(1 + 12)]
Parentheses: [12(13)]
Exponents: [1(13)]
Parentheses: [13]
Multiply:

6. Properties of Real Numbers
The properties of real numbers
allow us to manipulate expressions
and equations and find the values
of a variable.

7. Number Classification
Natural numbers are the counting numbers.
Whole numbers are natural numbers and zero.
Integers are whole numbers and their opposites.
Rational numbers can be written as a fraction.
Irrational numbers cannot be written as a fraction.
All of these numbers are real numbers.

8. Number Classifications
Subsets of the Real Numbers
Q - Rational
Z - Integers
I - Irrational
W - Whole
N - Natural

9. Classify each number -1 6 real, rational, integer
real, rational, integer, whole, natural
real, irrational
real, rational
real, rational, integer, whole 6
-2.222

10. Properties of Real Numbers
Commutative Property
Think… commuting to work.
Deals with ORDER. It doesn’t matter what order you ADD or MULTIPLY.
a+b = b+a
4 • 6 = 6 • 4

11. Properties of Real Numbers
Associative Property
Think…the people you associate with, your group.
Deals with grouping when you Add or Multiply.
Order does not change.

12. Properties of Real Numbers
Associative Property
(a + b) + c = a + ( b + c)
(nm)p = n(mp)

13. Properties of Real Numbers
Additive Identity Property
s + 0 = s
Multiplicative Identity Property
1(b) = b

14. Properties of Real Numbers
Distributive Property
a(b + c) = ab + ac
(r + s)9 = 9r + 9s

15. Name the Property 5 = 5 + 0 5(2x + 7) = 10x + 35 8 • 7 = 7 • 8
24(2) = 2(24)
(7 + 8) + 2 = 2 + (7 + 8)
Additive Identity
Distributive
Commutative

16. Name the Property Associative 7 + (8 + 2) = (7 + 8) + 2 Multiplicative
Identity
Distributive
7 + (8 + 2) = (7 + 8) + 2
1 • v + -4 = v + -4
(6 - 3a)b = 6b - 3ab
4(a + b) = 4a + 4b

17. Properties of Real Numbers
Reflexive Property
a + b = a + b
The same expression is written on both sides of the equal sign.

18. Properties of Real Numbers
Symmetric Property
If a = b then b = a
If = 9 then 9 = 4 + 5

19. Properties of Real Numbers
Transitive Property
If a = b and b = c then a = c
If 3(3) = 9 and 9 = 4 +5, then 3(3) = 4 + 5

20. Properties of Real Numbers
Substitution Property
If a = b, then a can be replaced by b.
a(3 + 2) = a(5)

21. Name the property Distributive Substitution Reflexive Symmetric
5(4 + 6) =
5(4 + 6) = 5(10)
5(4 + 6) = 5(4 + 6)
If 5(4 + 6) = 5(10) then 5(10) = 5(4 + 6)
5(4 + 6) = 5(6 + 4)
If 5(10) = 5(4 + 6) and 5(4 + 6) = then 5(10) =
Distributive
Substitution
Reflexive
Symmetric
Commutative
Transitive

22. Solving Equations
To solve an equation, find replacements for the variables to make the equation true.
Each of these replacements is called a solution of the equation.
Equations may have {0, 1, 2 … solutions.

23. Solving Equations
3(2a + 25) - 2(a - 1) = 78
4(x - 7) = 2x x

24. Solving Equations
Solve: V = πr2h, for h
Solve: de - 4f = 5g, for e