2. 1 Additional Support for Math99 Students By: Dilshad Akrayee

3. 2 Summary  Distributive a*(b + c) = a*b + a*c 3(X+Y)= 3x+3Y

4. 3 Example

5. 4 Multiplication of Real Numbers (+)(+) = (+) (+)(-) = (-) (-)(+) = (-) (-)(-) = (+) When something good happens to somebody good… that’s good. When something good happens to somebody bad...that’s bad. When something bad happens to somebody good...that’s bad. When something bad happens to somebody bad...that’s good.

6. 5 Examples +6+6 X +9+9 = +54 -6-6-8-8+48 X = +7+7-8-8 X = -56 -5-5 +7+7-35 X =

7. 6 Multiplying Fractions If a, b, c, and d are real numbers then EX)

8. 7 Division with Fractions If a,b,c,and d are real numbers. b,c, and d are not equal to zero then

9. 8 Example Divide

10. 9 Rule If a,b,c,and d are real numbers. b and d are not equal to zero then

11. 10 Ex) simplify

12. 11 Real Number System Natural # {1, 2, 3, 4,…} = {0,1, 2, 3, 4,…} = Whole # Integers # {…-3,-2,-1,0,1, 2, 3,…} Natural #Whole #Integers # =

13. 12 Write the prime factorization of 24 24 2 122 6 2 33 1

14. 13 Addition of Fractions If a, b, and c are integers and c is not equal to 0, then

15. 14 Example: Simplify the following

16. 15 Subtraction of Fractions If a, b, and c are integers and c is not equal to 0, then

17. 16 Write the prime factorization of 24 24 2 122 6 2 33 1

18. 17 Definition LCD  The least common denominator (LCD) for a set of denominators is the smallest number that is exactly divisible by each denominator  Sometimes called the least common multiple

19. 18 Find the LCD of 12 and 18 12 = (2)(2)(3) 18 = (2)(3)(3) The LCD will contain each factor the most number of times it was used. (2)(2)(3)(3) = 36 So the LCD of 12 and 18 is 36.

20. 19 Note For any algebraic expressions A,B, X, and Y. A,B,X,Y do not equal zero

21. 20 Example 10 = 10

22. 21 Using the Means-Extremes Property If you know three parts of a proportion you can find the fourth 3 * 20= 4 * x 60 = 4x 60 = 4x 44 X = 15

23. 22 Chart  of  is  A number Multiply equals = x

24. 23 Chart  4 more than x  4 times x  4 less than x x + 4 4x x – 4

25. 24 Chart At most it means less or equal which is < At least it means greater or equal which is >

26. 25

27. 26 Ex)The sum of two consecutive integers is 15. Find the numbers Let X and X+1 represent the two numbers. Then the equation is: X + X + 1 = 15 2X + 1 = 15 2X = 15 -1 2X = 14 X = 7 X+1 = 7 +1 = 8

28. 27 Ex)The sum of two consecutive odd integers is 28. Find the numbers Let X and X+2 represent the two numbers. Then the equation is: X + X + 2 = 28 2X + 2 = 28 2X = 28 -2 2X = 26 X = 13 X+2 = 13 +2 = 15

29. 28 Ex)The sum of two consecutive even integers is 106. Find the numbers Let X and X+2 represent the two numbers. Then the equation is: X + X + 2 = 106 2X + 2 = 106 2X = 106 -2 2X = 104 X = 52 X+2 = 52 +2 = 54

30. 29 Definition - Intercepts  The x-intercept of a straight line is the x- coordinate of the point where the graph crosses the x-axis  The y-intercept of a straight line is the y-coordinate of the point where the graph crosses the y-axis. x-intercept y-intercept

31. 30 2 Ex) Find the x-intercept and the y- intercept of 3x – 2y = 6 and graph. The x-intercept occurs when y = 0 (, 0) The y-intercept occurs when x = 0 (0, ) -3

32. 31 EX) Find the x-and y-intercepts for To find x-intercept, let y=0 2x +y= 2 (1, 0) x-intercept 2x+0 = 2 x=1 To find y-intercept, let x=0 2(0)+y = 2 y=2 y-intercept (0, 2) (1, 0) (0, 2)

33. 32 Ex) Find the x-intercept and the y-intercept: 3x-y=6 X-intercept(2, 0) Y-Intercept(0, -6) The answer should be

34. 33 Find the slope between (-3, 6) and (5, 2) x1 y1 x2 y2

35. 34 Exponent Summary Review Properties

36. 35 Exponents’ Properties 1) If a is any real number and r and s are integers then * = To multiply with the same base, add exponents and use the common base

37. 36 Examples of Property 1

38. 37 Exponents’ Properties 2) If a is any real number and r and s are integers, then A power raised to another power is the base raised to the product of the powers.

39. 38 Example of Property 2 One base, two exponents… multiply the exponents.

40. 39 Exponents’ Properties 3) If a and b are any real number and r is an integer, then Distribute the exponent.

41. 40 Examples of Property 3

42. 41 EX) Complete the following X

43. 42 Exponents’ Properties 4) If a is any real number and r and s are integers then = To divide with the same base, subtract exponents and use the common base

44. 43 Example =

45. 44 EX) Complete the following table *

46. 45 Exponent Summary Review Definitions

47. 46 Examples of Foil A) (m + 4)(m - 3)= B) (y + 7)(y + 2)= C) (r - 8)(r - 5)= m 2 + m - 12 y 2 + 9y + 14 r 2 - 13r + 40

48. 47 Finding the Greatest Common Factor for Numbers Write each number in prime factored form. Use each factor the least number of times that it occurs in all of the prime factored forms. Usually multiply for final answer. Find GCF of 36 and 48 36 = 2 ·2 ·3 ·3 48 = 2 ·2 ·2 ·2 ·3 2 occurs twice in 36 and four times in 48 3 occurs twice in 36 and once in 48. GCF = 2 ·2 ·3 =12

49. 48 Find the GCF of 30, 20, 15 30 = 2 · 3 · 5 20 = 2 · 2 · 5 15 = 3 · 5 Since 5 is the only common factor it is also the greatest common factor GCF.

50. 49 Find the GCF of 6m 4, 9m 2, 12m 5 6m 4 = 2 · 3 · m 2 · m 2 9m 2 = 3 · 3 · m 2 12m 5 = 2 · 2 · 3 · m 2 · m 3 GCF = 3m 2

51. 50 Factor First list the factors of 56. 1 56 2 28 4 19 7 8 Now add the factors. 57 30 23 15 Notice that 7 and 8 sum to the middle term. Check with Multiplication.

52. 51 Factor First list the factors of 24. 1 24 2 12 3 8 4 6 Now add the factors. 25 14 11 10 Notice that 2 and 12 sum to the middle term. Check with Multiplication.

53. 52 Zero-Factor Property If a and b are real numbers and if ab =0, then a = 0 or b = 0.

54. 53 Ex) Solve the equation (x + 2)(2x - 1)=0 By the zero factor property we know... Since the product is equal to zero then one of the factors must be zero. OR

55. 54 Solve.

56. 55 Fun Facts About Opposites Each negative number is the opposite of some positive number. Each positive number is the opposite of some negative number. -(-a) = a When you add any two opposites the result is always zero. a + (-a) = 0

57. 56 Absolute Value Example |5 – 7| – |3 – 8| = |-2| – |-5| = 2 – 5 = -3

58. 57 Definition: Two numbers whose product is 1 are called reciprocals For example: the reciprocals of is

59. 58 Example Simplify

60. 59 Memorize the First 10 Perfect Cubes nn2n2 n3n3 111 248 3927 41664 525125 636216 749343 864512 981729 101001000

61. 60 What is the Root?

62. 61 Examples

63. 62 If you square a radical you get the radicand 2 2 Whenever you have i 2 the next turn you will have -1 and no i.

64. 63 Subtract First distribute the negative sign. Now collect like terms.

65. 64 Powers of i Anything other than 0 raised to the 0 is 1. Anything raised to the 1 is itself.

66. 65 The Quadratic Formula The Quadratic Theorem: For any quadratic equation in the form

67. 66 Ex) Use the quadratic formula to solve the following:

68. 67 Ex. Solve. x 2 = 64 Take the square root of both sides. Do not forget the ±. The solution set has two answers.

69. 68 Identify the Vertex y = a(x - a) 2 + b y = -3(x - 3) 2 + 48 y = 5(x + 16) 2 - 1 (a, b) (3, 48) (-16, -1)