Analyzing Equations and Inequalities

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 -7

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 ab2(b a)
  if a 12 and b 1
• Substitute 1212(1 12)
• Parentheses 1212(13)
• Exponents 121(13)
• Parentheses 1213
• Multiply 156

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

• real, rational, integer
• real, rational, integer, whole, natural
• real, irrational
• real, rational
• real, rational, integer, whole
• real, rational

6
0
-2.222
10
Properties of Real Numbers

• Commutative Property
• Think commuting to work.
• Deals with ORDER. It doesnt matter what order
  you ADD or MULTIPLY.

• ab ba
• 4 6 6 4

11
Properties of Real Numbers

• Associative Property
• Thinkthe 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
• Commutative
• Commutative

16
Name the Property

• Associative
• Multiplicative
• Identity
• Distributive
• 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 4 5 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

• 5(4 6) 20 30
• 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) 20 30 then
  5(10) 20 30

• 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 12 2x

24
Solving Equations

• Solve V pr2h, for h
• Solve de - 4f 5g, for e