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Additional Support for Math99 Students

By:

Dilshad Akrayee

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Summary

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

3(X+Y)= 3x+3Y

5232)53(2

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Example

3

1

3

4

3

2

xxx xx 2

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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.

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Examples

+6 X +9 = +54-6 -8 +48X =

+7 -8X = -56

-5 +7 -35X =

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Multiplying Fractions

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

db

ca

d

c

*

**

ba

15

8

5*3

4*2

5

4*

32 EX)

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Division with Fractions

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

c

d

b

a

d

c*

ba

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Example Divide

7

5

3

2

5

7*

3

215

14

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RuleIf a,b,c,and d are real numbers. b

and d are not equal to zero then

5*210*110

521

cbdad

c**

ba

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Ex) simplify

| 4 7 |

4 7

| 3 |

3

3

3

1

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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 #

=

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Write the prime factorization of 24

242

122

62

331

3*2243

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Addition of Fractions

• If a, b, and c are integers and c is not equal to 0, then

c

ba

c

b

c

a

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Example: Simplify the following

5

3

5

12

5

152

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Subtraction of Fractions

• If a, b, and c are integers and c is not equal to 0, then

c

ba

c

b

c

a

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Write the prime factorization of 24

242

122

62

331

3*2243

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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

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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.

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Note

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

Y

X

B

ABXAY

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Example

10

5

2

1 5*210*1

10 = 10

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Using the Means-Extremes Property

• If you know three parts of a proportion you can find the fourth

4

320

x 3 * 20 = 4 * x

60 = 4x60 = 4x4 4

X = 15

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Chart of

is

A number

Multiply •

equals =

x

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Chart

4 more than x

4 times x

4 less than x

x + 4

4x

x – 4

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Chart

At most it means less or equal which is <

At least it means greater or equal which is>

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Term Example Variable UsingConsecutive Integers 4,5,6,7 X, X+1, X+2, X+3Consecutive Even Integers 2,4,6,8 X, X+2, X+4, X+6Consecutive Odd Integers 3,5,7,9 X, X+2, X+4, X+6

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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 -12X = 14X = 7

X+1 = 7 +1 = 8

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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 -22X = 26X = 13

X+2 = 13 +2 = 15

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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 -22X = 104X = 52

X+2 = 52 +2 = 54

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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

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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

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EX) Find the x-and y-intercepts for

To find x-intercept, let y=0

2x +y= 2

(1, 0)x-intercept

2x+0 = 2x=1

To find y-intercept, let x=0

2(0)+y = 2y=2

y-intercept (0, 2)

(1, 0)

(0, 2)

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Ex) Find the x-intercept and the y-intercept: 3x-y=6

X-intercept (2, 0)

Y-Intercept (0, -6)

The answer should be

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

)()(

)()(

12

12

xx

yym

)3()5(

)6()2(

m8

4

2

1

x1

y1x2

y2

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Exponent Summary Review

r r rab a br

r ss

aa

a

r r

r

a a

b b

Properties

sr aa sra sra sra

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Exponents’ Properties

1) If a is any real number and

r and s are integers then

ar

as

* asr

=

To multiply with the same base, add exponents and

use the common base

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Examples of Property 1

23 xx 523 xx

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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.

sra

sra

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Example of Property 2

One base, two exponents… multiply the exponents.

32x 6x

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Exponents’ Properties

3) If a and b are any real number and r is an integer, then

Distribute the exponent.

rab rr ba

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Examples of Property 3

25x 2 25 x 225x

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EX) Complete the following

Xx

2

x3

x4

42 8 16

3 9 27 81)2( 4 8 16

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Exponents’ Properties4) If a is any real number and

r and s are integers then

ar

as a

sr=

To divide with the same base, subtract exponents and

use the common base

0) ( a

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Example

3

2

a

aaa 12-3a=

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EX) Complete the following table

B

x6

x2

A*A B BA

x4

x2

x125

x23

x248

x62

x26 x2

4x4

10 x 2

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Exponent Summary Review

10r

ra a

a

1a a

0 1 0a a

Definitions

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Examples of Foil

A) (m + 4)(m - 3)=

B) (y + 7)(y + 2)=

C) (r - 8)(r - 5)=

m2 + m - 12

y2 + 9y + 14

r2 - 13r + 40

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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

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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.

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Find the GCF of 6m4, 9m2, 12m5

6m4 = 2 · 3 · m2 · m2

9m2 = 3 · 3 · m2

12m5 = 2 · 2 · 3 · m2 · m3

GCF = 3m2

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Factor

56152 xxFirst list the factors of 56.

1 562 28

4 19

7 8

Now add the factors.

57

30

23

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Notice that 7 and 8 sum to the middle term.

)8)(7( xx

2x x7 x8 56Check with Multiplication.

)568()7( 2 xxx)7(8)7( xxx

)8)(7( xx

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Factor

24142 xxFirst list the factors of 24.

1 242 12

3 8

4 6

Now add the factors.

25

14

11

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Notice that 2 and 12 sum to the middle term.

)12)(2( xx

2x x2 x12 24Check with Multiplication.

)2412()2( 2 xxx)2(12)2( xxx

)12)(2( xx

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Zero-Factor PropertyIf a and b are real numbers

and if ab =0, then

a = 0 or

b = 0.

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Ex) Solve the equation (x + 2)(2x - 1)=0By the zero factor property we know...

Since the product is equal to zero then one of the factors must be zero.

0)2( x

2x

OR (2 1) 0x 12 x

2

1

2

2x

2

1x

}2

1,2{ issolution xthe

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Solve.

1892 xx01892 xx

0)3)(6( xx

}3,6{x

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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

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Absolute Value Example

|5 – 7| – |3 – 8|

= |-2| – |-5|

= 2 – 5

= -3

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Definition:

Two numbers whose product is 1 are called reciprocals

For example: the reciprocals of is

ba

ab

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Example Simplify

3

2

5

2

4

xyxy

2 3

2

x y

x

y

y

x

4

2 5

2

3

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Memorize the First 10 Perfect Cubesn n2 n3

1 1 1

2 4 8

3 9 27

4 16 64

5 25 125

6 36 216

7 49 343

8 64 512

9 81 729

10 100 1000

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What is the Root?

64

3 64

6 64

64444 4

6488 8

64222222 2

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Examples

16

81

7

i4

i9

7i

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If you square a radical you get the radicand

1 i 25 5

12i

2 2

Whenever you have i2 the next turn you will have -1 and no

i.

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Subtract

9 i13

)204()75( ii

ii 20475

First distribute the negative sign.

Now collect like terms.

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Powers of i0i11i2i3i

Anything other than 0 raised to the 0 is 1.

Anything raised to the 1 is itself.i12 i1iii 23 i)1( ii

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The Quadratic Formula

:are solutions two the0 a where

02

cbxax

a

acbbx

2

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The Quadratic Theorem: For any quadratic equation in the form

a

acbbxand

2

4

2

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Ex) Use the quadratic formula tosolve the following:

0263 2 xx

3

33 isanswer

xThe

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Ex. Solve. x2 = 64

642 x

642 x

Take the square root of both sides.

Do not forget the ±.

8x}8,8{x

The solution set has two answers.

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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)