Additional Support for Math99 Students

1. Additional Support for Math99 Students By: Dilshad Akrayee

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

3. Example

4. (+)(+) = (+) (+)(-) = (-) (-)(+) = (-) (-)(-) = (+) 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. Multiplication of Real Numbers

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

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

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

8. Example Divide

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

10. Ex) simplify

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

12. Write the prime factorization of 24 24 2 2 12 2 6 3 3 1

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

14. Example: Simplify the following

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

16. Write the prime factorization of 24 24 2 2 12 2 6 3 3 1

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

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.

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

20. Example 10 = 10

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 4 4 X = 15

22. of is A number Multiply • equals = x Chart

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

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

26. Ex)The sum of two consecutive integers is15. 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

27. Ex)The sum of two consecutive odd integers is28. 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

28. Ex)The sum of two consecutive even integers is106. 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

29. Definition - Intercepts y-intercept • 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

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

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

32. Ex) Find the x-intercept and the y-intercept: 3x-y=6 The answer should be

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

34. Properties Exponent Summary Review

35. Exponents’ Properties • 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

36. Examples of Property 1

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.

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

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

40. Examples of Property 3

41. EX) Complete the following

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

43. Example =

44. EX) Complete the following table *

45. Exponent Summary Review Definitions

46. A) (m + 4)(m - 3)= B) (y + 7)(y + 2)= C) (r - 8)(r - 5)= m2 + m - 12 y2 + 9y + 14 r2 - 13r + 40 Examples of Foil

47. 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 Finding the Greatest Common Factor for Numbers

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

49. Find the GCF of 6m4, 9m2, 12m5 6m4 = 2 · 3 · m2 · m2 9m2 = 3 · 3 · m2 12m5 = 2 · 2 · 3 · m2 · m3 GCF = 3m2

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