1. Solve for x. 30° 2x + 10 2x + 10 = 60 – 10 – 10 2x = 50 2 2 x = 25

3.  The exam will be 40 multiple choice questions with 2 extra credit questions.  You will have 1 hour to complete the exam.  No extra time will be given.  You may bring ONE sheet of notes to use on the exam.

4. The final exam will cover: 1. Inequalities 2. Probability 3. Area & Perimeter of Polygons & Circles 4. Angles & Lines 5. Exponents 6. Radicals 7. Polynomials

6.  = Open Circle  = Closed Circle  < or < = Shade to the left.  > or > = Shade to the right.... Greater Than  Less Than

7. x < 3 Open or Closed? Right or Left? x > -4 Open or Closed? Right or Left?.

8. -5 + x < -1 + 5 +5 x < 4 Now graph it!

9. 4x + 1 > -11 – 1 – 1 4x > -12 4 4 x > -3

10. multiplydivide  Whenever we multiply or divide by a negative number, we must REVERSE the inequality sign. -2x < 6 -2 -2 x > -3 We have to divide by -2. So we have to reverse the sign.

11. -3x + 1 < 10 – 1 – 1 -3x < 9 -3 -3 x > -3 Reverse the Sign!

12. “At least” means greater than or equal to (>) “No more than” means less than or equal to (<) “More than” means greater than (>) “Less than” means less than (<)

13. Chris has $200 in his bank account. He makes $10 an hour at his job. He wants to save at least $400 to buy some chickens. What is the minimum number of hours Chris will have to work? 200 + 10h > 400 – 200 – 200 10h > 200 10 10 h > 20 hours Has already! Wants more than this amount! Mo’ Money, mo’ money, mo’ money!

14. Tom calls a cab which charges $2.50 plus $0.50 a mile. If Tom has no more than $20.00 in his pocket, how far can he go? $2.50 + $0.50m < $20 -2.50 -2.50 0.50m < 17.50 0.50 0.50 m < 35 miles That’s a lie. I got big bank!

16. Event – This is the selected outcome. Ex. If event A is the probability of rolling a 5 or higher, the probability is 2/7, so P(A) = 2/7. Complement – This is the probability of everything other than the event. Ex. In the example above, the complement is rolling 4 or lower, so the complement of event A is 5/7, or P(A) = 5/7. Probability of “A Bar”

17.  If you toss a coin twice, what are the possible outcomes? HH, TT, HT, TH  What is the probability of two heads? HH, TT, HT, TH =  What is the probability of at least one head? HH, TT, HT, TH = It’s complement would be 3/4! It’s complement would be 1/4! 1/4 3/4

18.  To find the probability of two independent events occurring together, multiply their probabilities!  Ex. Find the probability of tossing a coin twice and having heads occur twice. 1212 1212. 1414 = Probability of Toss #1 coming up heads. Probability of Toss #2 coming up heads. Probability of two heads!

19. Ex. A coin is tossed and a card is drawn from a standard deck. a. What is the probability of tossing heads and drawing an ace? b. What is the probability of tossing tails and drawing a face card? 1212 1 13. 1 26 = 1212 3 13. 3 26 =

20.  4! + 3! =  3! – 2! =  4! 2! =  = (4321) + (321) = 30 (321) – (21) = 4 (4321) (21) = 48 6! 4! (654321) (4321) = 30

21. (Order doesn’t matter! AB is the same as BA) n C r = Where: n = number of things you can choose from r = number you are choosing n! r! (n – r)!

22.  There are 6 pairs of shoes in the store. Your mother says you can buy any 2 pairs. How many combination of shoes can you choose? So n = 6 and r = 2 6 C 2 = = 6! 2! (6 – 2)! 654321 21(4321) = 30 2 =15 combinations!

23. (Order does matter! AB is different from BA) n P r = Where: n = number of things you can choose from r = number you are choosing n! (n – r)!

24.  In a 7 horse race, how many different ways can 1 st, 2 nd, and 3 rd place be awarded? So n = 7 and r = 3 7 P 3 = = 7! (7 – 3)! 7654321 (4321) =210 permutations!

25.  You have a choice of 3 meats, 4 cheeses, and 2 breads. How many different types of sandwiches could you make? Multiply the choices! 3 4 2 = 24 different sandwiches

27.  Perimeter – The distance around an polygon.  Area – The amount of space inside a two dimensional shape.

28.  Estimate or calculate the length of a line segment based on other lengths given on a geometric figure. x 17 in 8 in 8 + x = 17 - 8 - 8 x = 9 in Easy!

29.  Compute the perimeter of polygons when all side lengths are given 8 in 7 in 8 in Add all the sides: 8 + 7 + 7 + 8 + 7 + 7 = 44 in Even I can do this!

30.  Compute the area of rectangles when whole number dimensions are given. 25 in 6 in Area of Rectangle = Length Width A = L W A = 25 6 A = 150 in 2

31.  Compute the area and perimeter of triangles and rectangles in simple problems. Area of Triangles = 1/2 Base Height Base Height

32.  Kyla mows lawns for $1.20 per square feet. How much did she charge to cut the lawn below? 23 ft 11 ft A = L W A = 23 11 A = 253 ft 2 Price = 253$1.20 Price = $303.60

33.  Find the missing value, x. x 28 in x + 2x + x = 28 4x = 28 4 4 x = 7 inches I get it. Add up the bottom sides to equal the top! x 2x

34.  Find the perimeter. 17 in Find x: x + 5 = 17 – 5 – 5 x = 12 If x = 12, then x – 1 is 11! x – 1 x + 5 Use x to find perimeter: 17 + 17 + 11 + 11 Perimeter = 56 inches

35. I can find the area by cutting it!  Find the area. 5 ft 15 ft Area of ‘A’ A = 10 7 A = 70 ft 2 Area of ‘B’ A = 5 8 A = 40 ft 2 A = 70 + 40 A = 110 ft 2 8 ft 7 ft 5 ft 10 ft A B

36.  Definition – Quadrilaterals with at least one pair of parallel sides. Area of a Trapezoid = (Find the average of the bases and multiply by the height!) b1b1 b2b2 h b 1 & b 2 are the top and bottom bases. h is the height. (b 1 + b 2 ) 2 h

37. = 9  Find the area of the trapezoid below. 12 9 (b 1 + b 2 ) 2 A = h 16 (12 + 16) 2 = 126 ft 2

38.  Circumference – The distance around a circle. (Perimeter of a circle.)  Radius – The distance from the center of a circle to any point on its circumference.  Diameter – The distance from one side of a circle, passing through the center, to the other side of the circle.. Radius Circumference Diameter

39.  The diameter of a circle is equal to twice the radius, or d = 2r  Circumference of a circle is equal to the diameter multiplied by pi, or C = 2 π r or C = π d

40. C = 2 π r C = 2 π 5 C = 10 π or 31.42 5

41. C = d π C = 25.5 π or 80.11 25.5 Diameter is 25.5!

42.  The area of a circle is equal to: A = π r 2. 6 A = π 6 2 A = 36 π A = 113.10

43.  Pizza World offers two types of pizzas: rectangles and circles. If each pizza cost $12.50, which is the better buy? 12 in 12 in diameter A = 12 12 = 144 in 2 A = π 6 2 = 113.10 in 2 This is the better buy!

45.  You can identify parallel lines by their equations! y = 3x + 7 y = 3x – 9 These two lines are parallel. Their slopes are the same! (Notice that they have different y-intercepts!)

46.  Lines that intersect at right angles (90 0 ) are perpendicular.  Perpendicular lines have slopes that are negative reciprocals.  The product of their slopes = -1. These two lines are perpendicular. They intersect at a right angle.

47. Negative reciprocals 1. What is the reciprocal of ? 2. What is the reciprocal of 3? 2323 3232 So the negative reciprocal is – ! 3232 1313 1313

48. These equations are perpendicular: y = 2x + 8 y = - x – 5 y = - x – 7 y = x + 5 1212 4545 3232 5454

49. Y X  Complementary Angles – Two angles that add up to 90° Angles X and Angle Y are complementary and add up to 90 °.

50.  Find the missing angle. 36° x°x° 90 – 36 = 54°

51.  Supplementary angles - Two angles that add up to 180° Angles X and Angle Y are supplementary and add up to 180 °. YX

52.  Solve for x. x138° 180 – 138 = 42°

53.  Vertical Angles - A pair of opposite angles formed by the intersection of two lines. Vertical angles are always equal. A B Angle A and Angle B are vertical angles. They are equal!

54. n Line n is a transversal. A B Corresponding angles – Two congruent angles that lie on the same side of the transversal.  A and  B are corresponding angles.

55. n Line n is a transversal. A C B D  A =  C  B =  D They are alternate interior angles. (Interior = Inside!)

56. n Line n is a transversal. A C B D  A =  C  B =  D They are alternate exterior angles. (Exterior = Outside!)

57.  Find the missing angles. 42°A A = 138° B = 138° C = 42° BC F E D G D = 42° E = 138° F = 138° G = 42°

58. b° d °65 ° 70 ° Hint: The 3 angles in a triangle sum to 180°. Find the missing angles. 70 + 70 + b = 180 140 + b = 180 b = 40° 40 ° 40 + 65 + d = 180 105 + d = 180 d = 75°

59.  In a RIGHT Triangle, if sides “a” and “b” are the legs and side “c” is the hypotenuse, then a 2 + b 2 = c 2 a b c

60.  Find the length of the hypotenuse. 12 16 C a 2 + b 2 = c 2 12 2 + 16 2 = c 2 144 + 256 = c 2 400 = c 2 √400 = c 20 = c

61.  Find the length of the missing leg. 29 20 a a 2 + b 2 = c 2 a 2 + 21 2 = 29 2 a 2 + 400 = 841 a 2 = 441 a = √441 a = 21

63. Zero Exponent Property – Any number raised to the zero power is 1. x 0 = 1 3 0 = 112 0 = 1 Negative Exponent Property – Any number raised to a negative exponent is the reciprocal of the number. x -4 = 5 -1 = 5x -3 y 5 = 1x41x4 5y 5 x 3 1515

64. Product of Powers – When multiplying numbers with the same bases, ADD the exponents. Quotient of Powers – When dividing numbers with the same bases, SUBTRACT the exponents. x2x8x2x8 = x 10 3x 4 x -2 = 3x 2 x6x2x6x2 = x 4 10x 4 y 3 2x 7 y 5y 2 x 3 =

65. (9 5 ) 3 = 9 15 Power of a Power - When you have an exponent raised to an exponent, multiply the exponents! (x 4 y 5 ) 2 = x 8 y 10 Power of a Product - Distribute the exponent on the outside of the parentheses to all of the terms inside of the parentheses. (xy) 3 = x 3 y 3 (2bc) 2 = 4b 2 c 2

66. Power of a Quotient – Distribute the exponent on the outside of the parentheses to the numerator and the denominator of the fraction. 2x2x = ( ) 3 23x323x3 () = 8x38x3 3 x 2 y = ( ) 4 3 4 x 8 y 4 ( ) = 81 x 8 y 4

68. √8 =√4 ∙ √2 =2√2 8 is not a perfect square, so we will simplify it! 8 is made up of 4 ∙ 2. Look! 4 is a perfect square! √4 = 2 We can’t simplify √2, so we leave him alone.

69. √75 = √25 ∙ √3 = 5√3 Find factors of 75! 25 and 3 are factors of 75! Find the square root of 25! √72 =√36 ∙ √2 =6√2

70. √27 √32 √20 √75 = √9 ∙ √3 = √16 ∙ √2 = √4 ∙ √5 = √25 ∙ √3 = 3√3 = 4√2 = 2√5 = 5√3

71. To combine square roots, combine the coefficients of like square roots. 4√3 + 5 √3= 9√3 7√5 – 4√2 = Combine the coefficients, keep the radical! You cannot combine unlike radicals! 7√5 – 4√2

72. √20 + √5 = √4 ∙ √5 + √5 = √12 + √27 = 2√3 +√3 ∙ √4 3√3 = √9 ∙ √3 = 5√3 2√5 + √5 = 3√5 +

73. When multiplying radicals, you can multiply the two numbers and put the answer under one radical. Simplify! √3 ∙ √2 =√6 √3 ∙ √3 =√9= 3 √3 ∙ √6 =√18= √9 ∙ √2= 3√2

74. 2√5 ∙ 3√5 1. Multiply the coefficients. 2 ∙ 3 = 6 2. Multiply the radicals. √5 ∙ √5 = √25 3. Solve. 6√25 = 6 ∙ 5 = 30

75. 3√7 ∙ 2√5 = 6√35 2√3 ∙ 5√3 =10√9 =10 ∙ 3 4√2 ∙ 3√8 = 12√16 = 12 ∙ 4 = 48 2√5 ∙ 3√2 =6√10 = 30

77. Add: (x 2 + 3x + 1) + (4x 2 +5) Step 1: Identify like terms: Step 2: Add the coefficients of like terms, do not change the powers of the variables: (x 2 + 3x + 1) + (4x 2 +5) Notice: ‘3x’ doesn’t have a like term. (x 2 + 4x 2 ) + 3x + (1 + 5) 5x 2 + 3x + 6

78. Subtract: (3x 2 + 2x + 7) – (x 2 + x + 4) Step 1: Change subtraction to addition. Step 2: Add like terms. (3x 2 + 2x + 7) + (- x 2 + - x + - 4) 2x 2 + x + 3 Change signs of all terms after subtraction sign.

79. Distributive Property – Review!!! 3(x + 5) = 3x + 15 We will need that same concept to multiply polynomials. 3(2x 2 + 4x + 3)= 6x 2 We will distribute the outside term to everything on the inside. + 12x+ 9

80. 4(3x 2 – 2x + 1)= 12x 2 – 8x + 4 2x(2x 2 + x + 5)= 4x 3 + 2x 2 + 10x 4 ∙ 3x 2 4 ∙ –2x 4 ∙ 1 2x ∙ 2x 2 2x ∙ x 2x ∙ 5 12x 2 –8x +4 4x 3 +2x 2 +10x

81. (2x + 1)(4x – 3) Combine: 8x 2 – 6x + 4x – 3 8x 2 – 2x – 3 Mult. 4x – 3 2x + 1 8x 2 – 6x + 4x– 3 First: +2x ∙ +4x = +8x 2 Outer: +2x ∙ -3 = -6x Inner: +1 ∙ +4x = +4x Last: +1 ∙ -3 = -3 Combine: 8x 2 – 6x + 4x – 3 8x 2 – 2x – 3 Same Answer!