Chapter P Pre-calculus notes Prerequisites: Fundamental Concepts of Algebra Date: P.1: Algebraic Expressions, Mathematical Models, and Real Numbers Algebraic expression: a combination of variables and numbers using the operations of addition, subtraction, multiplication, or division, as well as powers or roots Examples: Exponential notation: If n is a counting number (1,2,3, etc.), = b is the _______________ and n is the _____________________ Evaluating an algebraic expression: find the value of the expression for a given value of the variable Order of Operations 1. Start at the innermost set of parentheses and word outward. 2. Evaluate all exponential expressions. 3. Perform multiplication and division left to right. 4. Perform addition and subtraction left to right. Ex. 1: Evaluate 8 + 6( − 3) 2 = 6. When an equal sign is placed between two algebraic expressions, an _____________________ is formed. A formula is an ______________________ that uses variables to express a relationship between two or more quantities. Mathematical modeling: the process of finding formulas to describe real-world phenomena Ex. 2: If the average cost of tuition and fees, T, for public four-year colleges, adjusted for inflation is modeled by the formula = 17 2 + 261 + 3257 where x is the number of years since the end of the school year in 2000. Use the formula to project the average cost of tuition and fees at public U.S. colleges for the school year ending 2010.
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Chapter P Pre-calculus notes
Prerequisites: Fundamental Concepts of Algebra Date:
P.1: Algebraic Expressions, Mathematical Models, and Real Numbers
Algebraic expression: a combination of variables and numbers using the operations of addition,
subtraction, multiplication, or division, as well as powers or roots
Examples:
Exponential notation: If n is a counting number (1,2,3, etc.), 𝑏𝑛 =
b is the _______________ and n is the _____________________
Evaluating an algebraic expression: find the value of the expression for a given value of the variable
Order of Operations
1. Start at the innermost set of parentheses and word outward.
2. Evaluate all exponential expressions.
3. Perform multiplication and division left to right.
4. Perform addition and subtraction left to right.
Ex. 1: Evaluate 8 + 6(𝑥 − 3)2 𝑓𝑜𝑟 𝑥 = 6.
When an equal sign is placed between two algebraic expressions, an _____________________ is
formed.
A formula is an ______________________ that uses variables to express a relationship between two or
more quantities.
Mathematical modeling: the process of finding formulas to describe real-world phenomena
Ex. 2: If the average cost of tuition and fees, T, for public four-year colleges, adjusted for inflation is
modeled by the formula 𝑇 = 17𝑥2 + 261𝑥 + 3257 where x is the number of years since the end of
the school year in 2000. Use the formula to project the average cost of tuition and fees at public U.S.
colleges for the school year ending 2010.
Set: a collection of objects whose contents can be clearly determined
The objects in a set are called the ________________________ of the set.
Ways to represent a set:
Roster method (listing all the elements, separated by commas.
Set-builder notation:
Intersection of sets A and B: the set of elements common to both set A and set B
Notation:
Venn diagram:
Ex. 3: Find the intersection: {3,4,5,6,7} ∩ {6,8,10,12}
Union of sets A and B: the set of elements that are members of set A or of set B or of both sets.
Notation:
Venn diagram:
Ex. 4: Find the union: {3,4,5,6,7} ∪ {3,7,8,9}
A set with no elements is called _______________________ or ________________________.
Hmk: Math XL: Log in and that complete “Day 1 assignment” to practice using the features of the program. Also use the code to view the e-book and complete the following assignment:
Pgs. 15-16: #3-48 (mult of 3)—check odd answers—make sure to write out the problems and show all
of your work.
Chapter P Pre-calculus notes
Prerequisites: Fundamental Concepts of Algebra Date:
P.1: Algebraic Expressions, Mathematical Models, and Real Numbers (cont.)
Subsets of the Real Numbers
Name Symbol Description Examples
Natural Numbers
Whole Numbers
Integers
Rational Numbers
Irrational Numbers
Real numbers: the set of numbers that are either rational or irrational.
Ex. 5: Consider the following set of numbers: {−9, −1.3,0,0.3,𝜋
2, √9, √10}
List the members in the set that are:
Natural numbers
Whole numbers
Integers
Rational numbers
Irrational members
Real numbers
Ordering the real numbers
Symbols : ≤ and ≥
Absolute Value: |a|
Informal def:
Formal def: |𝑥| = {
Ex. 6: Rewrite without absolute value bars:
a) |1 − √2| b) |𝜋 − 3| c) |𝑥|
𝑥 if x>0.
Distance between Points on a Real Number Line: a and b are any two pts on a real number line, then
the distance between a and b is given by:
Ex. 7: Find the distance between -4 and 5. _________________ = _______________
Properties of Real Numbers and Algebraic Expressions
Name Meaning Example
Commutative Property of Addition/Multiplication
Associative Property of Addition/Multiplication
Distributive Property
Identity Property of Addition
Identity Property of Multiplication
Inverse Property of Addition
Inverse Property of Multiplication
Definition of Subtraction: a – b =
-b is called the _____________________________ of b.
Definition of Division: 𝑎 ÷ 𝑏 =
1
𝑏 is called the _______________________________ of b.
Simplifying Algebraic Expressions: Combine like terms (add their coefficients)
Ex. 8: Simplify: 6(2𝑥2 + 4𝑥) + 10(4𝑥2 + 3𝑥)
Properties of Negatives: Let a and b represent real numbers, variables, or algebraic expressions.
Property Example
(-1)a =
-(-a) =
(-a)b =
a(-b) =
-(a+b) =
-(a-b) =
If a negative sign or a subtraction symbol appears outside parentheses, drop the parentheses and
change the sign of every term within the parentheses. (Distribute the negative.)
Ex. 9: Simplify: 6 + 4[7 − (𝑥 − 2)]
Ex. 10: Name the property illustrated by each statement.