Analyzing Equations and Inequalities Objectives: - evaluate expressions and formulas using order of...

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

Expressions & Formulas

ORDER OF OPERATIONS• Parentheses • Exponents • Multiply/Divide from left to right• Add/Subtract from left to right

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

Expressions and Formulas

How do you evaluate expressions and formulas?

Replace each variable with a value and then apply the order of operations.

ExpressionsEvaluate: a[b2(b + a)]

if a = 12 and b= 1• Substitute: 12[12(1 + 12)]• Parentheses: 12[12(13)]• Exponents: 12[1(13)]• Parentheses: 12[13]• Multiply: 156

Example 1 8 – 2{20 [1 + (3)2]}

The value is 4

Find the value of

Example 2 [384 – 3(7 – 2)3]

3

The value is 3

Simplify:

Variables - symbols, usually letters, used to represent unknown quantities.Algebraic Expressions – expressions that contain at least one variable.

You Evaluate an algebraic expression by replacing each variable with a number and using the Order of Operations

Example 3

a3 + b(c – 1)2 – c2 if a = -2, b = 2.5, and c = 3

The value is - 7

Evaluate the expression:

Example 4

s – t(s2 – t) if s = 2 and t = 3.4

The value is – 0.04

Evaluate:

The fraction bar is both an operation symbol and a grouping symbol.

Example 5

5

82

3

y

zxyIf x = 5, y = - 2, and z = - 1

The value is - 9

Evaluate:

Example 6

Evaluate:

2

2

6

5

p

apn

if a = 5, n = - 2, and p = - 1

The value is 55

4

Formula: a mathematical sentence that expresses the relationship between certain quantities.

Example 7

Find the area of a trapezoid with base lengths of 13 meters and 25 meters and a height of 8 meters.

Area = 152 sq meters

Example 8The formula for the orbital period T of

a satellite is where r is the radius of the satellite and v is the velocity of the satellite. Find the period a a satellite in orbit above the earth is the radius of the orbit is 4268 miles and the velocity is 4.4 miles per second. Express the orbital period in hours.

v

rT

2

The orbital period is about 1.7 hours.

Properties of Real Numbers

The properties of real numbers

allow us to manipulate expressions

and equations and find the values

of a variable.

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.

Number Classifications

Subsets of the Real Numbers

I - IrrationalZ - Integers

W - Whole

N - Natural

Q - Rational

Classify each number -1 real, rational, integer

real, rational, integer, whole, natural

real, irrational

real, rational

real, rational, integer, whole

real, rational

6

1

2

-2.2220

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

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.

Properties of Real Numbers

Associative Property

• (a + b) + c = a + ( b + c)

• (nm)p = n(mp)

Properties of Real Numbers

Additive Identity Property• s + 0 = s

Multiplicative Identity Property • 1(b) = b

Distributive Property

• a(b + c) = ab + ac

• (r + s)9 = 9r + 9s

Properties of Real Numbers

• 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

Name the Property

Name the Property

• 7 + (8 + 2) = (7 + 8) + 2• 1 • v + -4 = v + -4

• (6 - 3a)b = 6b - 3ab• 4(a + b) = 4a + 4b

• Associative• Multiplicative

Identity• Distributive • Distributive

Properties of Real Numbers

Reflexive Property

• a + b = a + b

The same expression is written on both sides of the equal sign.

Properties of Real Numbers

• If a = b then b = a

• If 4 + 5 = 9 then 9 = 4 + 5

Symmetric Property

Properties of Real NumbersTransitive Property

• If a = b and b = c then a = c

• If 3(3) = 9 and 9 = 4 +5, then 3(3) = 4 + 5

Properties of Real Numbers

Substitution Property

• If a = b, then a can be replaced by b.

• a(3 + 2) = a(5)

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

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.

}

}

Solving Equations

• 3(2a + 25) - 2(a - 1) = 78

• 4(x - 7) = 2x + 12 + 2x

1 3 5 1 3772 4 6 4 6

x x x

Solving Equations

• Solve: V = πr2h, for h

• Solve: de - 4f = 5g, for e