Additional Support for Math99 Students By: Dilshad Akrayee

1

Summary

 Distributive

a*(b + c) = a*b + a*c

3(X+Y)= 3x+3Y 2 ( 3  5 )  2 3  2 5 2

Example

x

3 2  

x

3 4 

x

3 1   

x

2 

x

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

+ 6 6 + 7 5 X X X X Examples + 9

=

8

=

8 + 7

= =

+ 54 + 48 56 35 5

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

c

*

d



a

*

c b

*

d

EX) 2 3 * 4 5  2 * 4 3 * 5  8 15 6

Division with Fractions

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

c d



a b

*

d c

7

Example Divide

2 3  5 7  2 3 * 7 5  14 15 8

Rule If a,b,c,and d are real numbers. b and d are not equal to zero then a b 1 2 

c



a

*

d



b

*

c d

 5 10  1 * 10  2 * 5 9

Ex) simplify   3  3  3   1 10

Real Number System

Natural #

= {1, 2, 3, 4,…}

Whole # Integers #

= {0,1, 2, 3, 4,…} = {…-3,-2,-1,0,1, 2, 3,…}

Natural #



Whole #



Integers #

11

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

Addition of Fractions

• If

a

,

b

, and

c

are integers and is not equal to 0, then

c a c



b c



a



b c

13

Example: Simplify the following 2 5  1 5  2  1  5 3 5 14

Subtraction of Fractions

• If

a

,

b

, and

c

are integers and is not equal to 0, then

c a c



b c



a



b c

15

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

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 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 A  B

X Y AY



BX

19

1 2  5 10 Example 1 * 10  2 * 5

10 = 10

20

Using the Means-Extremes Property • If you know three parts of a proportion you can find the fourth 3 4 

x

20 3 * 20 = 4 * x 60 = 4x 60 = 4x 4 4 X = 15 21

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

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

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

Term Example Variable Using

Consecutive Integers 4,5,6,7 X, X+1, X+2, X+3 Consecutive Even Integers 2,4,6,8 X, X+2, X+4, X+6 Consecutive Odd Integers 3,5,7,9 X, X+2, X+4, X+6 25

Ex)The sum integers

is

of two consecutive 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 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 + X + 2

=

28 2X + 2 = 28 2X = 28 -2 2X = 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 + X + 2

=

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

Definition - Intercepts

 The

x-intercept

of a straight line is the x coordinate of the point where the graph crosses the x-axis

y-intercept

 The

y-intercept

straight line is the of a y-coordinate of the point where the graph crosses the y-axis.

x-intercept

29

Ex) Find the x-intercept and the y intercept of 3x – 2y = 6 and graph.

• The x-intercept occurs when y = 0 • The y-intercept occurs when x = 0 30

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

) (

0

,

2

) (

1 , 0

) 31

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

y1 x1

m

Find the slope between  ( (-3, 6) and (5, 2)

y

2 )  (

y

1 ) (

x

2 )  (

x

1 ) x2 y2

m

 ( 2 )  ( 6 ) ( 5 )  (  3 )   4 8   2 1 33

a r

Exponent Summary Review 

a s



a r



s

 

s

 Properties

a r



s a r a s



a

 

r



r a b r a

 

a r b r

34

Exponents’ Properties 1) If

a

is any real number and

r

and

s

are integers then

a r

*

a s

=

a r



s

To multiply with the same base, add exponents and use the common base 35

Examples of Property 1

x

3 

x

2 

x

3  2 

x

5 36

Exponents’ Properties 2) If

a

is any real number and

r s

and are integers, then  

s



a r



s

A power raised to another power is the base raised to the product of the powers.

37

Example of Property 2   3 

x

6 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  

r



a r



b r

Distribute the exponent.

39

Examples of Property 3   2  5 2

x

2  25

x

2 40

EX) Complete the following X 2 3 (  2 ) 4 9

x

2 4

x

8 27  8 3

x

4 16 81 16 41

Exponents’ Properties 4) If

a

is any real number and

r

and

s

a r a s

are integers then =

a r



s

(

a

To divide with the same  0) base, subtract exponents and use the common base 42

Example

a

3

a

2

=

a 3 2 

a

1 

a

43

EX) Complete the following table

A x

4 12

x

5  2

x

6

B x

2 2

x

3 2

x

4

A

*

B x

6 24

x

8  4

x

10

A B x

2 6

x

2 

x

2 44

Exponent Summary Review

Definitions

a



r

 1

a r a

1 

a a

 0

a

0  1

a

 0 45

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 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 ·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 47

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.

48

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 49

x

2  15

x

 56 Factor  (

x

 7 )(

x

 8 ) First list the factors of 56.

Check with Multiplication.

Now add the factors.

1 56 2 28 4 19 7 8 57 30 23 15     ( (

x x x

(

x

2 2   7 7

x x

)  8 (

x

8

x

  7 7 ) )(

x

  8 ( 8 )

x



x

  56 56 7 ) 50 ) Notice that 7 and 8 sum to the middle term.

x

2 Factor  14

x

 24  (

x

 2 )(

x

 12 ) First list the factors of 24.

Check with Multiplication.

Now add the factors.

1 24 2 12 3 8 4 6 25 14 11 10    (

x x

2 2  

x

( 

x

( 

x

2 2

x x

)   ( 12

x

 12

x

 2 )   2 )( 

x

12 ( 

x

12 )   24 2 ) 24 ) 51 Notice that 2 and 12 sum to the middle term.

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)(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.

(

x



x

2  )   2 0 OR (2 2

x

2

x

2

x

    1) 1 1 2 0

x



the

solution is

x

 {  2 , 1 2 } 53 1 2

x

2

x

2  Solve.

9

x

 18  9

x

 18  0 (

x



x

6 )(

x

  3 ) { 6 , 3 }  0 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

Absolute Value Example

|5 – 7| – |3 – 8| = |-2| – |-5| = 2 – 5 = -3

56

Definition: Two numbers whose product is 1 are called b reciprocals is a b a 57

Example Simplify 2

x

3

y

2 4

x y

5  2

x

3

y

2 

y

5 4

x

 2

x y

3 2 58

Memorize the First 10 Perfect Cubes 7 8 9 10 4 5 6 n 1 2 3 n 2 1 4 9 16 25 36 49 64 81 100 n 3 1 8 27 64 125 216 343 512 729 1000 59

What is the Root?

64  8 3 64  4 6 64  2 8  8  64 4  4  4  64 2  2  2  2  2  2  64 60

Examples  16  4

i

  81   9

i

 7 

i

7 61

If you square a radical you get the radicand   2  5 Whenever you have

i

2 the next turn you will have -1 and no

i

.

 

i

2 2    2   1 62

First distribute the negative sign.

Subtract Now collect like terms.

( 5  7

i

)  (  4  20

i

)  5  7

i

 4  20

i

 9  13

i

63

Powers of

i

1  

i i

1    

i i i i

1 2 3 0 Anything other than 0 raised to the 0 is 1.

Anything raised to the 1 is itself.

i

3

i

2 

i

 2

i

 1  (  1 )

i

 

i

64

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

ax

2 

bx



c

 0 where a  0 the two solutions are :

x

 

b



b

2  4

ac

2

a and x

 

b



b

2  4

ac

2

a

65

Ex) Use the quadratic formula to solve the following: 3

x

2  6

x

 2  0

The

answer is

x

  3  3 66 3

Ex. Solve. x 2 = 64

x

2  64 Take the square root of both sides.

x

2   64

x x

   8 {  8 , 8 } Do not forget the ±.

The solution set has two answers.

67

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) 68