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