Elementary Algebra

Elementary Algebra Exam 1 Material

Familiar Sets of Numbers • Natural numbers • Numbers used in counting: 1, 2, 3, … (Does not include zero) • Whole numbers • Includes zero and all natural numbers: 0, 1, 2, 3, … (Does not include negative numbers) • Fractions • Ratios of whole numbers where bottom number can not be zero:

Prime Numbers • Natural Numbers, not including 1, whose only factors are themselves and 1 2, 3, 5, 7, 11, 13, 17, 19, 23, etc. • What is the next biggest prime number? 29

Composite Numbers • Natural Numbers, bigger than 1, that are not prime 4, 6, 8, 9, 10, 12, 14, 15, 16, etc. • Composite numbers can always be “factored” as a product (multiplication) of prime numbers

Factoring Numbers • To factor a number is to write it as aproduct of two or more other numbers, each of which is called a factor 12 = (3)(4) 3 & 4 are factors 12 = (6)(2) 6 & 2 are factors 12 = (12)(1) 12 and 1 are factors 12 = (2)(2)(3) 2, 2, and 3 are factors In the last case we say the 12 is “completely factored” because all the factors are prime numbers

Hints for Factoring Numbers • To factor a number we can get two factors by writing any multiplication problem that comes to mind that is equal to the given number • Any factor that is not prime can then be written as a product of two other factors • This process continues until all factors are prime • Completely factor 28 28 = (4)(7) 4 & 7 are factors, but 4 is not prime 28 = (2)(2)(7) 4 is written as (2)(2), both prime In the last case we say the 28 is “completely factored” because all the factors are prime numbers

Other Hints for Factoring • Some people prefer to begin factoring by thinking of the smallest prime number that evenly divides the given number • If the second factor is not prime, they again think of the smallest prime number that evenly divides it • This process continues until all factors are prime • Completely factor 120 120 = (2)(60) 60 is not prime, and is divisible by 2 120 = (2)(2)(30) 30 is not prime, and is divisible by 2 120 = (2)(2)(2)(15) 30 is not prime, and is divisible by 3 120 = (2)(2)(2)(3)(5) all factors are prime In the last case we say the 120 is “completely factored” because all the factors are prime numbers

Fundamental Principle of Fractions • If the numerator and denominator of a fraction contain a common factor, that factor may be divided out to reduce the fraction to lowest terms: • Reduce to lowest terms by factoring:

Summarizing the Process of Reducing Fractions • Completely factor both numerator and denominator • Apply the fundamental principle of fractions: divide out common factors that are found in both the numerator and the denominator

When to Reduce Fractions to Lowest Terms • Unless there is a specific reason not to reduce, fractions should always be reduced to lowest terms • A little later we will see that, when adding or subtracting fractions, it may be more important to have fractions with a common denominator than to have fractions in lowest terms

Multiplying Fractions • Factor each numerator and denominator • Divide out common factors • Write answer • Example:

Dividing Fractions • Invert the divisor and change problem tomultiplication • Example:

Adding Fractions Having a Common Denominator • Add the numerators and keep the common denominator • Example:

Adding Fractions Having a Different Denominators • Write equivalent fractions having a “least common denominator” • Add the numerators and keep the common denominator • Reduce the answer to lowest terms

Finding the Least Common Denominator, LCD, of Fractions • Completely factor each denominator • Construct the LCD by writing down each factor the maximum number of times it is found in any denominator

Example of Finding the LCD • Given two denominators, find the LCD: , • Factor each denominator: • Construct LCD by writing each factor the maximum number of times it’s found in any denominator:

Writing Equivalent Fractions • Given a fraction, an equivalent fraction is found by multiplying the numerator and denominator by a common factor • Given the following fraction, write an equivalent fraction having a denominator of 72: • Multiply numerator and denominator by 4:

Adding Fractions • Find a least common denominator, LCD, for the fractions • Write each fraction as an equivalent fraction having the LCD • Write the answer byadding numerators as indicated, and keeping the LCD • If possible, reduce the answer to lowest terms

Example • Find a least common denominator, LCD, for the rational expressions: • Write each fraction as an equivalent fraction having the LCD: • Write the answer by adding or subtracting numerators as indicated, and keeping the LCD: • If possible, reduce the answer to lowest terms

Subtracting Fractions • Find a least common denominator, LCD, for the fractions • Write each fraction as an equivalent fraction having the LCD • Write the answer bysubtracting numerators as indicated, and keeping the LCD • If possible, reduce the answer to lowest terms

Example • Find a least common denominator, LCD, for the rational expressions: • Write each fraction as an equivalent fraction having the LCD: • Write the answer by adding or subtracting numerators as indicated, and keeping the LCD: • If possible, reduce the answer to lowest terms

Improper Fractions& Mixed Numbers • A fraction is called “improper” if the numerator is bigger than the denominator • There is nothing wrong with leaving an improper fraction as an answer, but they can be changed to mixed numbers by doing the indicated division to get a whole number plus a fraction remainder • Likewise, mixed numbers can be changed to improper fractions by multiplying denominator times whole number, plus the numerator, all over the denominator

Doing Math Involving Improper Fractions & Mixed Numbers • Convert all numbers to improper fractions then proceed as previously discussed

Homework Problems • Section: 1.1 • Page: 11 • Problems: Odd: 7 – 29, 33 – 51, 55 – 69 • MyMathLab Homework 1.1 for practice • MyMathLab Homework Quiz 1.1 is due for a grade on the date of our next class meeting

Exponential Expressions “3” is called the base “4” is called the exponent • An exponent that is a natural number tells how many times to multiply the base by itself Example: What is the value of 34 ? (3)(3)(3)(3) = 81 • An exponent applies only to the base (what it touches) • Meanings of exponents that are not natural numbers will be discussed later

Order of Operations • Many math problems involve more than one math operation • Operations must be performed in the following order: • Parentheses (and other grouping symbols) • Exponents • Multiplication and Division (left to right) • Addition and Subtraction (left to right) • It might help to memorize: • Please Excuse My Dear Aunt Sally

Order of Operations • Example: • P • E • MD • AS

Example of Order of Operations • Evaluate the following expression:

Inequality Symbols • An inequality symbol is used to compare numbers: • Symbols include: greater than: greater than or equal to: less than: less than or equal to: not equal to: • Examples: .

Expressions InvolvingInequality Symbols • Expressions involving inequality symbols may be either true or false • Determine whether each of the following is true or false:

Translating to Expressions Involving Inequality Symbols • English expressions may sometimes be translated to math expressions involving inequality symbols: Seven plus three is less than or equal to twelve Nine is greater than eleven minus four Three is not equal to eight minus six

Equivalent Expressions Involving Inequality Symbols • A true expression involving a “greater than” symbol can be converted to an equivalent statement involving a “less then” symbol • Reverse the expressions and reverse the direction of the inequality symbol 5 > 2 is equivalent to: 2 < 5 • Likewise, a true expression involving a “less than symbol can be converted to an equivalent statement involving a “greater than” symbol by the same process • Reverse the expressions and reverse the direction of the inequality symbol 3 < 7 is equivalent to: 7 > 3

Homework Problems • Section: 1.2 • Page: 21 • Problems: Odd: 5 – 19, 23 – 49, 53 – 79, 83 – 85 • MyMathLab Homework 1.2 for practice • MyMathLab Homework Quiz 1.2 is due for a grade on the date of our next class meeting

Terminology of Algebra • Constant – A specific number Examples of constants: • Variable – A letter or other symbol used to represent a number whose value varies or is unknown Examples of variables:

Terminology of Algebra • Expression – constants and/or variables combined in a meaningful way with one or more math operation symbols for addition, subtraction, multiplication, division, exponents and roots Examples of expressions: • Only the first of these expressions can be simplified, because we don’t know the numbers represented by the variables

Terminology of Algebra • If we know the number value of each variable in an expression, we can “evaluate” the expression • Given the value of each variable in an expression, “evaluate the expression” means: • Replace each variable with empty parentheses • Put the given number inside the pair of parentheses that has replaced the variable • Do the math problem and simplify the answer

Example • Evaluate the expression for : • Consider the next similar, but slightly different, example

Example • Evaluate the expression for : • Notice the difference between this example and the previous one – it illustrates the importance of using a parenthesis in place of the variable

Example • Evaluate the expression for :

Translating English Phrases Into Algebraic Expressions • Many English phrases can be translated into algebraic expressions: • Use a variable to indicate an unspecified number • Identify key words that imply: • Add • Subtract • Multiply • Divide

English Phrase A number plus 5 The sum of 3 and a number 4 more than a number A number increased by 8 Algebra Expression Phrases that Translate to Addition

English Phrase 4 less than a number A number subtracted from 7 6 subtracted from a number a number decreased by 9 2 minus a number Algebra Expression Phrases that Translate to Subtraction

English Phrase 7 times a number the product of 4 and a number double a number the square of a number Algebra Expression Phrases that Translate to Multiplication

English Phrase the quotient of 2 and a number a number divided by 8 6 divided by a number Algebra Expression Phrases that Translate to Division

English Phrase 4 less than 3 times a number the quotient of 5 and twice a number 6 times the difference between a number and 5 Algebra Expression Phrases Translating to Expressions Involving Multiple Math Operations

English Phrase the difference between 4 and 7 times a number the quotient of a number and 5, subtracted from the number the product of 3, and a number increased by 4 Algebra Expression Phrases Translating to Expressions Involving Multiple Math Operations

Equations • Equation – a statement that two expressions are equal • Equations always contain an equal sign, but an expression does not have an equal sign • Like a statement in English, an equation may be true or false • Examples: .

Equations • Most equations contain one or more variables and the truthfulness of the equation depends on the numbers that replace the variables • Example: • What value of x makes this true? • A number that can replace a variable to make an equation true is called a solution

Distinguishing Between Expressions & Equations • Expressions contain constants, variables and math operations, but NO EQUAL SIGN • Equations always CONTAIN AN EQUAL SIGN that indicates that two expressions have the same value

Elementary Algebra