1. Intermediate Algebra Prerequisite Topics Review Quick review of basic algebra skills that you should have developed before taking this class 18 problems that are typical of things you should already know how to do

2. 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

3. Example of Order of Operations Evaluate the following expression:

4. Problem 1 Perform the indicated operation: Answer:

5. 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

6. 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

7. Example Evaluate the expression for :

8. Problem 2 Evaluate for x = -2, y = -4 and z = 3 Answer:

9. Like Terms Recall that a term is a _________, a ________, or a _______ of a ________ and _________ Like Terms: terms are called “like terms” if they have exactly the same variables with exactly the same exponents, but may have different coefficients Example of Like Terms:

10. Simplifying Expressions by Combining Like Terms Any expression containing more than one term may contain like terms, if it does, all like terms can be combined into a single like term by adding or subtracting as indicated by the sign in front of each term Example: Simplify:

11. Simplifying an Expression Get rid of parentheses by multiplying or distributing Combine like terms Example:

12. Problem 3 Simplify: Answer:

13. Linear Equations Linear equation – an equation where, after parentheses are gone, every term is either a constant, or of the form: cx where c is a constant and x is a variable with exponent1 Linear equations never have a variable in a denominator or under a radical (square root sign) Examples of Linear Equations:.

14. Solving Linear Equations Simplify each side separately –Get rid of parentheses –Multiply by LCD to get rid of fractions and decimals –Combine like terms Get the variable by itself on one side by adding or subtracting the same terms on both sides If the coefficient of the variable term is not 1, then divide both sides by the coefficient

15. Determine if the equation is linear. If it is, solve it:

16. Problem 4 Solve: Answer:

17. Linear Equations with No Solution or All Real Numbers as Solutions Many linear equations only have one number as a solution, but some have no solution and others have all numbers as solutions In trying to solve a linear equation, if the variable disappears (same variable & coefficient on both sides) and the constants that are left make a statement that is: –false, the equation has “no solution” (no number can replace the variable to make a true statement) –true, the equation has “all real numbers” as solutions (every real number can replace the variable to make a true statement)

18. Solve the Linear Equation

20. Problem 5 Solve: Answer:

21. Problem 6 Solve: Answer:

22. Problem 7 Solve: Answer:

23. Formulas A “formula” is an equation containing more than one variable Familiar Examples:

24. Solving Formulas To solve a formula for a specific variable means that we need to isolate that variable so that it appears only on one side of the equal sign and all other variables are on the other side If the formula is “linear” for the variable for which we wish to solve, we pretend other variables are just numbers and solve as other linear equations (Be sure to always perform the same operation on both sides of the equal sign)

25. Example Solve the formula for

26. Problem 8 Solve for n: Answer:

27. Steps in Solving Application Problems Read the problem carefully trying to understand what the unknowns are (take notes, draw pictures, don’t try to write equation until all other steps below are done ) Make word list that describes each unknown Assign a variable name to the unknown you know the least about (the most basic unknown) Write expressions containing the variable for all the other unknowns Read the problem one last time to see what information hasn’t been used, and write an equation about that Solve the equation (make sure that your answer makes sense, and specifically state the answer)

28. Example of Solving an Application Problem With Multiple Unknowns A mother’s age is 4 years more than twice her daughter’s age. The sum of their ages is 76. What is the mother’s age? List of unknowns –Mother’s age –Daughter’s age What else does the problem tell us that we haven’t used? Sum of their ages is 76 What equation says this?

29. Example Continued Solve the equation: Answer to question? Mother’s age is 2x + 4:

30. Example of Solving an Application Involving a Geometric Figure The length of a rectangle is 4 inches less than 3 times its width and the perimeter of the rectangle is 32 inches. What is the length of the rectangle? Draw a picture & make notes: What is the rectangle formula that applies for this problem?

31. Geometric Example Continued List of unknowns: –Length of rectangle: –Width of rectangle: What other information is given that hasn’t been used? Use perimeter formula with given perimeter and algebra names for unknowns:

32. Geometric Example Continued Solve the equation: What is the answer to the problem? The length of the rectangle is:

33. Problem 9 The perimeter of a certain rectangle is 16 times the width. The length is 12 cm more than the width. Find the width. Answer:

34. Inequalities An “inequality” is a comparison between expressions involving these symbols: <“is less than” “is less than or equal to” >“is greater than” “is greater than or equal to”

35. Inequalities Involving Variables Inequalities involving variables may be true or false depending on the number that replaces the variable Numbers that can replace a variable in an inequality to make a true statement are called “solutions” to the inequality Example: What numbers are solutions to: All numbers smaller than 5 Solutions are often shown in graph form:

36. Graphing Solutions to Inequalities Graph solutions to:

37. Linear Inequalities A linear inequality looks like a linear equation except the = has been replaced by: Examples: Our goal is to learn to solve linear inequalities

38. Solving Linear Inequalities Linear inequalities are solved just like linear equations with the following exceptions: –Isolate the variable on the left side of the inequality symbol –When multiplying or dividing by a negative, reverse the sense of inequality –Graph the solution on a number line

39. Example of Solving Linear Inequality

40. Problem 10 Solve and graph solution: Answer:

41. Three Part Linear Inequalities Consist of three algebraic expressions compared with two inequality symbols Both inequality symbols MUST have the same sense (point the same direction) AND must make a true statement when the middle expression is ignored Good Example: Not Legitimate:.

42. Expressing Solutions to Three Part Inequalities “Standard notation” - variable appears alone in the middle part of the three expressions being compared with two inequality symbols: “Graphical notation” – same as with two part inequalities: “Interval notation” – same as with two part inequalities:

43. Solving Three Part Linear Inequalities Solved exactly like two part linear inequalities except that solution is achieved when variable is isolated in the middle

44. Example of Solving Three Part Linear Inequalities

45. Problem 11 Solve: Answer:

46. Exponential Expression An exponential expression is: where is called the base and is called the exponent An exponent applies only to what it is immediately adjacent to (what it touches) Example:

47. Meaning of Exponent The meaning of an exponent depends on the type of number it is An exponent that is a natural number (1, 2, 3,…) tells how many times to multiply the base by itself Examples:

48. Rules of Exponents Product Rule: When two exponential expressions with the same base are multiplied, the result is an exponential expression with the same base having an exponent equal to the sum of the two exponents Examples:

49. Rules of Exponents Power of a Power Rule: When an exponential expression is raised to a power, the result is an exponential expression with the same base having an exponent equal to the product of the two exponents Examples:

50. Rules of Exponents Power of a Product Rule: When a product is raised to a power, the result is the product of each factor raised to the power Examples:

51. Rules of Exponents Power of a Quotient Rule: When a quotient is raised to a power, the result is the quotient of the numerator to the power and the denominator to the power Example:

52. Using Combinations of Rules to Simplify Expression with Exponents Examples:

53. Integer Exponents Thus far we have discussed the meaning of an exponent when it is a natural (counting) number: 1, 2, 3, … An exponent of this type tells us how many times to multiply the base by itself Next we will learn the meaning of zero and negative integer exponents Examples:

54. Definition of Integer Exponents The following definitions are used for zero and negative integer exponents: These definitions work for any base,, that is not zero:

55. Quotient Rule for Exponential Expressions When exponential expressions with the same base are divided, the result is an exponential expression with the same base and an exponent equal to the numerator exponent minus the denominator exponent Examples:.

56. “Slide Rule” for Exponential Expressions When both the numerator and denominator of a fraction are factored then any factor may slide from the top to bottom, or vice versa, by changing the sign on the exponent Example: Use rule to slide all factors to other part of the fraction: This rule applies to all types of exponents Often used to make all exponents positive

57. Simplify the Expression: (Show answer with positive exponents)

58. Problem 12 Evaluate: Answer:

59. Problem 13 Evaluate: Answer:

60. Problem 14 Use rules of exponents to simplify and use only positive exponents in answer: Answer:

61. Polynomial Polynomial – a finite sum of terms Examples:

62. Special Names for Certain Polynomials Number of Terms One term: Two terms: Three terms: Special Name

63. Adding and Subtracting Polynomials To add or subtract polynomials horizontally: –Distribute to get rid of parentheses –Combine like terms Example:

64. Multiplying Polynomials To multiply polynomials: –Get rid of parentheses by multiplying every term of the first by every term of the second using the rules of exponents –Combine like terms Examples:

65. Problem 15 Multiply and simplify: Answer:

66. Squaring a Binomial To square a binomial means to multiply it by itself (the result is always a trinomial) Although a binomial can be squared by foiling it by itself, it is best to memorize a shortcut for squaring a binomial:

67. Problem 16 Multiply and simplify: Answer:

68. Dividing a Polynomial by a Monomial Write problem so that each term of the polynomial is individually placed over the monomial in “fraction form” Simplify each fraction by dividing out common factors

69. Problem 17 Divide: Answer:

70. Dividing a Polynomial by a Polynomial First write each polynomial in descending powers If a term of some power is missing, write that term with a zero coefficient Complete the problem exactly like a long division problem in basic math

71. Example

72. Problem 18 Divide: Answer: