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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

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Elementary Algebra

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  1. Elementary Algebra Exam 1 Material

  2. 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:

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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:

  9. 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

  10. 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

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

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

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

  14. 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

  15. 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

  16. 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:

  17. 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:

  18. 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

  19. 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

  20. 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

  21. 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

  22. 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

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

  24. 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

  25. 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

  26. 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

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

  28. Example of Order of Operations • Evaluate the following expression:

  29. 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: .

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

  31. 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

  32. 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

  33. 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

  34. 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:

  35. 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

  36. 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

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

  38. 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

  39. Example • Evaluate the expression for :

  40. Example • Evaluate the expression for :

  41. 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

  42. 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

  43. 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

  44. 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

  45. 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

  46. 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

  47. 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

  48. 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: .

  49. 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

  50. 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

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