Transcript CH1

Sect 1.1
Algebraic Expressions
Variable
Consist of variables and/or numbers, often with operation signs
and grouping symbols.
Any symbol that represents a number…letters or
Constant A value that never changes.
Variable Expression An expression that contains a variable.
Evaluating the Expression To evaluate an expression, we substitute a value in for
each variable in the expression and calculate the result.
Area formula
Perimeter
A = (base)(height) = bh
A = (length)(width) = lw
P = all exterior sides added together.
Rectangle: P = 2l + 2w
Sect 1.1
added to
sum of
plus
more than
increased by
5 pounds was added to the number
The sum of a number and 12
7 plus some number
20 more than the number
The number increased by 3
n+5
x + 12
7+m
r + 20
y+3
subtracted from
difference of
minus
less than
decreased by
2 was subtracted from the number
difference of two numbers
8 minus some number
9 less than the number
The number decreased by 10
w-2
a-b
8-c
d-9
f - 10
Sect 1.1
multiplied by
product of
times
twice
of
the number multiplied by 4
the product of two numbers
13 times some number
twice the number
half of the number
divided by
quotient of
divided into
ratio of
per
3 divided by the number
the quotient of two numbers
8 divided into some number
the ratio of 9 to some number
There were 28 miles per g gallons
4n
xy
13z
2t
1
x
2
3q
mn
h 8
9 r
28 g
Sect 1.1
Four less than Joe’s height in inches.
h–4
Eighteen increased by a number.
18 + n
A day’s pay divided by eight hours.
p/
8
Half of the pallet.
1/ p
2
Seven more than twice a number.
2x + 7
Six less than the product of two numbers.
ab – 6
Nine times the difference of a number and 3.
9(m – 3)
Eighty five percent of the enrollment.
85%(e) = 0.85(e)
Twice the sum of a number and 3.
2(x + 3)
The sum of twice a number and 3.
2x + 3
Sect 1.1
The symbol = (“equals”) indicates that the expressions on either side of the
equal sign represents the same number. An equation is when two algebraic
expression are equal to each other. Equations can be true or false.
4  8  32
True 32 = 32
94  6
False 5 = 6
x  5  13
We don’t know the value of x.
In the last example, replacing the “x” with a value that makes the equation
true is called a solution. Some equations have more than one solution, and
some have no solutions. When all solutions have been found, we have solved
the equation.
Determine whether 8 is a solution of x + 5 = 13.
8 + 5 = 13
13 = 13
True, 8 is a solution
Sect 1.1
When translating phrases into expressions to equations, we need to look for the
phrases “is the same as”, “equal”, “is”, and “are” for the = sign.
Translate.
What number plus 478 is 1019?
x + 478 = 1019
Twice the difference of a number and 4 is 24.
2 ( ________
x
– _________
4
) = 24
“than” makes the terms
switch around the minus sign
Three times a number plus seven is the same as the number less than one.
3x
+ 7
=
x
– 1
The Taipei Financial Center, or Taipei 101, in Taiwan is the world’s tallest building.
At 1666 ft, it is 183 ft taller than the Petronas Twin Towers in Kuala Lumpur. How
tall are the Petronas Twin Towers?
1666 – 183 =
P
1483 = P
+ 183 – 183
Sect 1.2
Equivalent expressions.
4 + 4 + 4, 3 4 , and 3(4)
Laws that keeps expressions equivalent.
Commutative Law
for Addition
switch around the times sign
for Multiplication
ab  ba
a b  b a
a b  b a
switch around the plus sign
Associative Law
for Addition
for Multiplication
abc   abc
a  b  c  a  b  c
Move ( )’s around new plus sign
5 1 3  5  9
10 + 3 + 10 = 23
Move ( )’s around new times sign
5 1 3  9  4
20
27 = 540
Sect 1.2
Use the Commutative Law and Associative Law for Addition.
7  x  3
x  7   3
3  7  x 
x  7  3
3  7  x
Use the Commutative Law and Associative Law for Multiplication.
4xy
x4y
y4 x 
x4 y 
 y4x
Use the Commutative Law for Addition and Multiplication.
7x  3
3 7 x
3  x7
Sect 1.2
Distributive Property ab  c   ab  ac
Separate by place values & add.
4x  3
5237   5200  30  7
1000 + 150 + 35 = 1185
42 x  y  6
4x – 12
2a5b  3c  9d 1
8x – 4y + 24
10ab + 6ac – 18ad – 2a
Factor using the Distributive Property
7x  7 y
7 x 7 y
12 x  8 y  4
4  3 x  4  2 y  4 1
7 x  y 
4 3x  2 y  1
GCF leftovers
Sect 1.2
Terms vs Factors
Term is any number, variable, or quantity being multiplied together. Be
careful of the definition that terms are separated by plus or minus signs.
Only if the ( )’s are simplified away!
One term
two terms
three terms
x
10   7 w
2ab  5
2ab 10a
2
2 x  3 y  4
Multiplying
Factors are the number, variable or quantity being multiplied together.
2ab  5 The factors are 2, a, and (b – 5)
Sect 1.3
Review: Natural Numbers = { 1, 2, 3, 4, 5, 6, …..}
List factors of 18.
The factors are 1, 2, 3, 6, 9, 18
Prime Numbers are Natural numbers that have 2 different factors, 1 and itself.
{2, 3, 5, 7, 11, 13, 17, 19, 23, …}
Composite Numbers are Natural numbers that have 3 or more factors.
{4, 6, 8, 9, 10, 12, 14, 15, …}
Notice that “1” is not in either set!
Sect 1.3
List the prime factorization of 48.
48  2  2  2  2  3
Tree method 48
4
2
Always start with smallest prime numbers
and work up to largest prime number.
12
6
2 2
2
Staircase Method
2
3
Division Rules
2: any even number
3: sum of the digits is divisible by 3
5: ends in 0 or 5
48
2 24
2 12
2 6
3
The prime number outside the upside down
division boxes should be all the prime
numbers.
48  2  2  2  2  3
Sect 1.3
Fraction notation.
a
 numerator
b
denominator
Fraction Properties
Notation for 1
a
1
a
Notation for 0
Undefined
0
0
b
a
 undefined
0
Why do we use the undefined term?
We have to define Multiplication and
Division with the same numbers.
Example
5  3  15
15  3  5
5 0  0
Multiplying by 0 and divide by 0 doesn’t
00  5
We start with 5 and finish with 5 when
we multiply by 3 and divide by 3.
return to the original value, not defined.
Sect 1.3
Fraction multiplication.
Tops together and bottoms together.
3 4
3 4
12
12 1
1





8 15 8 15 120 12 10 10
a c ac
 
b d bd
Another technique is to Simply first.
Multiplicative inverse (Reciprocal)
a b
 1
b a
13
3 4
4 1 1 1 1





8 15 4  2 3  5 2  5 10
We don’t divide by fractions, but Multiply by the reciprocal of
Fraction Division the fraction that we are dividing by.
1 1
a c a d ad
7 35 7 48 7 12  4 4
   

 
 

b d b c bc
12 48 12 35 12 7  5 5
a
a
a c ac
1 
 
Multiplicative Identity
b
b
b c bc
Use this property to get common denominators.
Sect 1.3
Simplify the fraction by multiplication rules.
15 5  3 3


40 5  8 8
36 12  3 3


24 12  2 2
9
9 1 1


72 9  8 8
Canceling errors!
4 1 1

42 2
23 3

2
1
Can’t cancel with addition or subtraction!
Addition and Subtraction of Fractions (same Denominators)
a c ac
 
b b
b
1
5
1
6
1



12 12
12 2
2
a c ac
 
b b
b
3
11 5 6
3
 

8 8 8
4
4
Sect 1.3
Addition and Subtraction of Fractions (with different Denominators)
a c a d c b ad bc ad  bc Rule that works every
     


b d b d d b bd bd
bd
time, however, can
a c a d c b ad bc ad  bc create huge numbers!
     


b d b d d b bd bd
bd
7 11
7 12 11 8 84 88 172 4  43 43

    




8 12
8 12 12 8 96 96
4  24 24
96
We can work with smaller numbers and prior knowledge…staircase method.
7 11 4 8, 12

2, 3
8 12
Multiply the
outsides for the
LCD = 4(2)(3) = 24
Notice cross multiplying = 24
9 5

8 6
2 8, 6
4, 3
Multiply the
outsides for the
LCD = 2(4)(3) = 24
21 22 43
3 7 11 2
  



3 8 12 2
24 24 24
7
3 9 5 4 27 20
   


3 8 6 4
24 24 24
Sect 1.4 Positive and Negative Real Numbers
Review: The Set of Numbers


REAL NUMBERS
Any number on the number line.
IRRATIONAL NUMBERS
Numbers that can’t be written
as a fraction
Examples :  , 3  1.73205...
0
RATIONAL NUMBERS
Numbers that CAN be written
as a fraction
Examples : 9, 34 ,0.3, 4  2
INTEGERS NUMBERS
… -3, -2, -1, 0, 1, 2, 3, …
WHOLE NUMBERS
0, 1, 2, 3, …
NATURAL NUMBERS
1, 2, 3, …
Less Than, Greater Than
<
>
Less Than or
Equal to,
<

2.0310 ___ 2.0309
Greater Than or
Equal to
>
To compare decimal numbers, both numbers need to
have the same number of decimal places. Add a 0 to
the end of the left number and compare place values
until different.
10 > 9
To compare fractions, we need common denominators.
Multiply the other denominators to the numerators and
compare the products.
6 12
7 11 72 ___
 77
___
11 12
12 11
6 7

11 12
Sect 1.4 Positive and Negative Real Numbers
Absolute Value The POSITIVE distance a number is away from zero
on the number line.
5 5
  7   7  7
5 units long
-5 -4 -3 -2 -1 0 1 2
Convert a repeating decimal to fraction.
0.3
Step 1. Set the repeating decimal = x
Step 2. Get the decimal point to the left
of the repeating digits. Already done.
Step 3. Get the decimal point to the right
of the repeating digits. Multiply by 10’s to
both sides of the equation. This moves the
decimal point one place for each 10.
Step 4. Subtract Step 3 – Step 2 and
solve for x.
x  0.3 
1
3
x  0.3
10  x  10  0.33
10x  3.3
x  0.3
9x  3
x
3
9

1
3
Convert a repeating decimal to fraction.
0.63
Step 1. Set the repeating decimal = x
Step 2. Get the decimal point to the left
of the repeating digits. Already done.
x  0.63
7
11
x  0.63
Step 3. Get the decimal point to the right 100  x  100  0.6363
of the repeating digits. Multiply by 10’s to
100 x  63.63
both sides of the equation. This moves the
decimal point one place for each 10.
x  0.63
Step 4. Subtract Step 3 – Step 2 and
solve for x.
99 x  63
x
63
99
7
 11
Convert a repeating decimal to fraction.
0.16
Step 1. Set the repeating decimal = x
Step 2. Get the decimal point to the left
of the repeating digits. Multiply by 10.
Step 3. Get the decimal point to the right
of the repeating digits. Multiply by 10’s to
both sides of the equation. This moves the
decimal point one place for each 10.
Step 4. Subtract Step 3 – Step 2 and
solve for x.
x  0.16 
1
6
10x  1.66
10 10x  10 1.66
100x  16.66
10x  1.66
90 x  15
x  15
90 
1
6
Sect 1.5 and 1.6 Add & Subtract sign numbers
Add & Subtract with number line.
3 Step rule. Any two signed numbers.
1. Remove all double signs.
a–(-b)
a+b
a+(-b)
a–b
2. Keep the sign of the largest number ( absolute value ).
+Large – small = Positive answer
Small – Large = Negative answer
3. a. Same Signs Sum
b. Different Signs Difference (subtract)
+Large – small = + (Large – small)
– Large + small = – ( Large – small)
- a – b = - (a + b)
+ a + b = + (a + b)
Sect 1.5 and 1.6 Add & Subtract sign numbers
-12 + (-7)
1. Double signs
-15 + 9
2 Sign of Largest number
1. Double signs
NONE
-12 – 7 = – 19
-16
– 18
1. Double signs
NONE
-32 – (-4)
23 + (-11) 1. Double signs
2 Sign of Largest number
2 Sign of Largest number
23 – 11 = + 12
= – 34
3 Same signs
SUM
1. Double signs 2 Sign of Largest number
19 – (-7)
3 Different signs
Difference LG - sm
3 Different signs
Difference LG - sm
1. Double signs
2 Sign of Largest number
19 + 7 = + 26
3 Same signs
SUM
-9 + (-7) – (-4) + 3 – 8 – (-12)
1. Double signs
= –6
3 Different signs
Difference LG - sm
3 Same signs
SUM
-32 + 4 = – 28
2 Sign of Largest number
Add all positive numbers 1st and negative numbers 2nd.
-9 – 7 + 4 + 3 – 8 + 12
19
– 24 = – 5
2 Sign of Largest number
Law of Opposites: a + (-a) = 0
Good to use this property when adding a
long list of sign numbers…canceling is good!
3 Different signs
Difference LG - sm
Sect 1.5 and 1.6 Add & Subtract sign numbers
Combine Like Terms
Defn. 1. Must have the same variables in the individual terms.
2. The exponents on each variable must be the same.
Identify the like terms. 7x + 3y – 5 + 2x – 9y – 8x + 10
x – terms
y – terms
Now Combine them. 7x + 2x – 8x
constants
+ 3y – 9y – 5 + 10
x – 6y + 5
Combine Like Terms
2a + (- 3b) + (-5a) + 9b
2xy + 3x – 7y + 5 – 8x – 2 + y
2a – 3b – 5a + 9b
2xy + 3x – 7y + 5 – 8x – 2 + y
– 3a + 6b
2xy – 5x – 6y + 3
Sect 1.7 Mult and Division of sign numbers
2 steps
1. Determine the sign.
Even number of Negatives being multiplied or divided = Positive answer.
Odd number of Negatives being multiplied or divided = Negative answers.
2. Multiply or divide the values.
3 2 5 14 3  360
1
2
10
3
4
1
3 2 7  1

20
4 514 3
36
Multiply by 0 rule. 310 15 110 372
Sign on the fraction rule.
a
a
a


b
b
b
0
Sect 1.8 Exponential Notation & Order of Operations
Exponential notation is a short cut to writing out repetitive multiplication.
Negative quantities are
6
aaaaaa  a
 a  1 a defined as a -1 multiplied to
the positive quantity.
Simplify.
3  3 3 3 3
4
 99
 81
2x 
 34
  3  3  3  3
3
 2 x  2 x  2 x 
 8x 3
 99
 81
2x 3
 2 x  x  x
 2x 3
 34  1 34
 1 3  3  3  3
 1 9  9
 81
Sect 1.8 Exponential Notation & Order of Operations
Order of Operation P.E.MD.AS
1. P = ( )’s which means all grouping symbols. ( ), { }, [ ], | |, numerators,
denominators, square roots, etc.
2. E = Exponents. All exponential expressions must be simplified.
3. MD = Multiply or Divide in order from Left to Right
4. AS = Add or Subtract in order from Left to Right
15  2  5  3
15 10  3


2  9  2 2 
129  7   4  5
34  2 3
8  4  9  23  5
3
3
5 3
2  9  2 8
8
2  9 16
27
5
Top
Bottom
129  7  4  5
34  2 3
122  20
81 8
24  20
89
44
44
89
Sect 1.8 Exponential Notation & Order of Operations
Simplify
When variables are present, remove ( )’s by the Distributive
Property and Combine Like Terms.
Include the sign

5x  9  24x  5

7 x 2  3 x 2  2 x  5x
7 x 2  3x 2  6 x  5 x
5x  9  8x  10
13x  1
10x 2  x
2, STO> button, X, enter
Original Expression
Our Answer
 x  7 y  5
1 x  7 y  5
 x  7y 5
7  3x  2
7  3x  6
3x  4 x  2
3x  4x  2
 x2
5t 2  2t  4t 2  9t
9t 2  11t
  
7 x  2  5x  5  8
7 x  2  5 x  3
7 x3  2  5 x3  1  8
3
3
3
1 3x

5t 2  2t   4t 2  9t
3
7 x3  2  5x3  3
2 x3 1
