Additional Support for Math 99 Students By Dilshad

Additional Support for Math 99 Students By: Dilshad Akrayee 1

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

Example 3

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. 4

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

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

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

Example Divide 8

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

Ex) simplify 10

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

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

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

Example: Simplify the following 14

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

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

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

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. 18

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

Example 10 = 10 20

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

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

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

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

25

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 + 1 = 15 2 X = 15 -1 2 X = 14 X=7 X+1 = 7 +1 = 8 26

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 + 2 = 28 2 X = 28 -2 2 X = 26 X = 13 X+2 = 13 +2 = 15 27

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 + 2 = 106 2 X = 106 -2 2 X = 104 X = 52 X+2 = 52 +2 = 54 28

Definition - Intercepts q The x-intercept of a straight line is the xcoordinate 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. y-intercept q x-intercept 29

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

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

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

Find the slope between (-3, 6) and (5, 2) y 2 x 1 y 1 x 2 33

Exponent Summary Review Properties 34

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

Examples of Property 1 36

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

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

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

Examples of Property 3 40

EX) Complete the following X 41

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

Example = 43

EX) Complete the following table * 44

Exponent Summary Review Definitions 45

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

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 · 3 · 3 48 = 2 · 2 · 3 2 occurs twice in 36 and four times in 48 3 occurs twice in 36 and once in 48. GCF = 2 · 3 =12 47

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

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

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

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

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

Ex) Solve the equation (x + 2)(2 x - 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 53

Solve. 54

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 55