1 additional support for math99 students by: dilshad akrayee
Post on 27-Dec-2015
222 Views
Preview:
TRANSCRIPT
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
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)
7
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
9
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
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 #
=
15
Subtraction of Fractions
• If a, b, and c are integers and c is not equal to 0, then
c
ba
c
b
c
a
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.
21
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
25
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
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 -12X = 14X = 7
X+1 = 7 +1 = 8
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 -22X = 26X = 13
X+2 = 13 +2 = 15
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 -22X = 104X = 52
X+2 = 52 +2 = 54
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
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
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 = 2x=1
To find y-intercept, let x=0
2(0)+y = 2y=2
y-intercept (0, 2)
(1, 0)
(0, 2)
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
33
Find the slope between(-3, 6) and (5, 2)
)()(
)()(
12
12
xx
yym
)3()5(
)6()2(
m8
4
2
1
x1
y1x2
y2
35
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
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.
sra
sra
39
Exponents’ Properties
3) If a and b are any real number and r is an integer, then
Distribute the exponent.
rab rr ba
42
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
46
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
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
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.
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
56152 xxFirst list the factors of 56.
1 562 28
4 19
7 8
Now add the factors.
57
30
23
15
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
51
Factor
24142 xxFirst list the factors of 24.
1 242 12
3 8
4 6
Now add the factors.
25
14
11
10
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
53
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
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
Definition:
Two numbers whose product is 1 are called reciprocals
For example: the reciprocals of is
ba
ab
59
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
62
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.
64
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
65
The Quadratic Formula
:are solutions two the0 a where
02
cbxax
a
acbbx
2
42
The Quadratic Theorem: For any quadratic equation in the form
a
acbbxand
2
4
2
67
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.
top related