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Sect 1.1 Algebraic Expressions Variable Constant Variable Expression Evaluating the Expression Area...
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Transcript of Sect 1.1 Algebraic Expressions Variable Constant Variable Expression Evaluating the Expression Area...
Sect 1.1Algebraic Expressions
Variable
Constant
Variable Expression
Evaluating the Expression
Area formula Perimeter
Consist of variables and/or numbers, often with operation signs and grouping symbols.
Any symbol that represents a number…letters or
A value that never changes.
An expression that contains a variable.
To evaluate an expression, we substitute a value in for each variable in the expression and calculate the result.
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
subtracted from
difference of
minus
less than
decreased 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
2 was subtracted from the number
difference of two numbers
8 minus some number
9 less than the number
The number decreased by 10
n + 5
x + 12
7 + m
r + 20
y + 3
w - 2
a - b
8 - c
d - 9
f - 10
Sect 1.1
multiplied by
product of
times
twice
of
divided by
quotient of
divided into
ratio of
per
the number multiplied by 4
the product of two numbers
13 times some number
twice the number
half of the number
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
g
r
h
nm
q
28
9
8
3
x2
1
Sect 1.1
Four less than Joe’s height in inches.
Eighteen increased by a number.
A day’s pay divided by eight hours.
Half of the pallet.
Seven more than twice a number.
Six less than the product of two numbers.
Nine times the difference of a number and 3.
Eighty five percent of the enrollment.
Twice the sum of a number and 3.
The sum of twice a number and 3.
h – 4 ab – 6
18 + n 9(m – 3)
p/8 85%(e) = 0.85(e)
1/2 p 2(x + 3)
2x + 7 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.
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.
3284 135 x649 True 32 = 32 False 5 = 6 We don’t know the value of
x.
8 + 5 = 13
13 = 13
True, 8 is a solution
Sect 1.1When 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?
Twice the difference of a number and 4 is 24.
Three times a number plus seven is the same as the number less than one.
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?
x + 478 = 1019
2 ( ________ – _________ )
x 4 = 24
3x + 7 = x – 1
“than” makes the terms switch around the minus sign
1666 = P + 183 – 183 – 183
1483 = P
Sect 1.2
Equivalent expressions. 4 + 4 + 4, , and 3(4)
Laws that keeps expressions equivalent.Commutative Law for Addition for Multiplication
Associative Law for Addition for Multiplication
43
abba baab
cbacba cabbca
95315 49315
switch around the plus sign
switch around the times sign
abba
Move ( )’s around new plus sign Move ( )’s around new times sign
10 + 10 =3 + 23 20 27 = 540
Sect 1.2
Use the Commutative Law and Associative Law for Addition.
Use the Commutative Law and Associative Law for Multiplication.
Use the Commutative Law for Addition and Multiplication.
37 x
yx4
37 x
37 x 37 x
x 73 x73
yx4 yx 4
xy 4 xy4
x73 73 x
Sect 1.2
Distributive Property
Factor using the Distributive Property
acabcba
2375 34 x
624 yx 19352 dcba
yx 77 4812 yx
Separate by place values & add.
1000 +
7302005
150 + 35 = 11854x – 12
8x – 4y + 24 10ab + 6ac – 18ad – 2a
yx 77
7 yx GCF leftovers
142434 yx
4 123 yx
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
Factors are the number, variable or quantity being multiplied together.
52 ba aab 102 432 yx
wx
72
10
Multiplying
52 ba The factors are 2, a, and (b – 5)
Sect 1.3
Review: Natural Numbers = { 1, 2, 3, 4, 5, 6, …..}
List factors of 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!
The factors are 1, 2, 3, 6, 9, 18
Sect 1.3
List the prime factorization of 48.
Tree method Staircase Method
Division Rules
2: any even number
3: sum of the digits is divisible by 3
5: ends in 0 or 5
48
4 12
2 2 2 6
2 3
3222248
24482
Always start with smallest prime numbers and work up to largest prime number.
2
122
62
3
3222248
The prime number outside the upside down division boxes should be all the prime numbers.
Sect 1.3Fraction notation.
Fraction Properties
Notation for 1 Notation for 0 Undefined
b
a numeratordenominator
1a
a 00
b
undefineda
0
Why do we use the undefined term?We have to define Multiplication and Division with the same numbers.
Example1535
5315 We start with 5 and finish with 5 when we multiply by 3 and divide by 3.
005 500 Multiplying by 0 and divide by 0 doesn’t
return to the original value, not defined.
Sect 1.3
Fraction multiplication.
Multiplicative inverse (Reciprocal)
Fraction Division
Multiplicative Identity
bd
ac
d
c
b
a
1a
b
b
a
bc
ad
c
d
b
a
d
c
b
a
Tops together and bottoms together.
15
4
8
3
10
1
1012
112
120
12
158
43
Another technique is to Simply first.
15
4
8
3
10
1
52
11
53
4
24
3
11
We don’t divide by fractions, but Multiply by the reciprocal of the fraction that we are dividing by.
48
35
12
7
35
48
12
7
57
412
12
7
11
5
4
b
a
b
a1
bc
ac
c
c
b
a
Use this property to get common denominators.
Sect 1.3
Simplify the fraction by multiplication rules.
Canceling errors!
Addition and Subtraction of Fractions (same Denominators)
40
1524
36
2
32
72
9
24
14
b
ca
b
c
b
a
b
ca
b
c
b
a
85
35
8
3 212
312
2
3
89
19
8
1
Can’t cancel with addition or subtraction!1
3 2
1
12
1
12
5
8
5
8
11
12
6
2
1
1
28
6
4
3
3
4
Sect 1.3Addition and Subtraction of Fractions (with different Denominators)
bd
bcad
bd
bc
bd
ad
b
b
d
c
d
d
b
a
d
c
b
a
bd
bcad
bd
bc
bd
ad
b
b
d
c
d
d
b
a
d
c
b
a
12
11
8
7
6
5
8
9
Rule that works every time, however, can create huge numbers!
8
8
12
11
12
12
8
7
96
88
96
84
96
172
244
434
24
43
We can work with smaller numbers and prior knowledge…staircase method.
12
11
8
7
,8 124,2 3
Multiply the outsides for the LCD = 4(2)(3) = 24 12
11
8
7
3
3
242424
2
2
21 22 43
,8 62,4 3
Multiply the outsides for the LCD = 2(4)(3) = 24 6
5
8
9
3
3
242424
4
4
27 20 7Notice cross multiplying = 24
Sect 1.4 Positive and Negative Real NumbersReview: The Set of Numbers
0
REAL NUMBERSAny number on the number line.
IRRATIONAL NUMBERSNumbers that can’t be written
as a fraction
...73205.13,: Examples
RATIONAL NUMBERSNumbers that CAN be written
as a fraction24,3.0,,9: 4
3 Examples
INTEGERS NUMBERS… -3, -2, -1, 0, 1, 2, 3, …
WHOLE NUMBERS0, 1, 2, 3, …
NATURAL NUMBERS1, 2, 3, …
Less Than, Greater Than
< >
Less Than or Greater Than or
Equal to, 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.
0309.2___0310.210 > 9
To compare fractions, we need common denominators. Multiply the other denominators to the numerators and compare the products.
___1211
126
1112
117
77___72
12
7
11
6
Sect 1.4 Positive and Negative Real Numbers
Absolute Value The POSITIVE distance a number is away from zero on the number line.
5 75 77
-5 -4 -3 -2 -1 0 1 2
5 units long
Convert a repeating decimal to fraction.
3.0 Step 1. Set the repeating decimal = x 3.0x
Step 2. Get the decimal point to the left of the repeating digits. Already done.
3.0x
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.
33.01010 x
3.310 x
Step 4. Subtract Step 3 – Step 2 and solve for x.
3.0x39 x
31
93 x
31
Convert a repeating decimal to fraction.
63.0 Step 1. Set the repeating decimal = x 63.0x
Step 2. Get the decimal point to the left of the repeating digits. Already done.
63.0x
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.
6363.0100100 x
63.63100 x
Step 4. Subtract Step 3 – Step 2 and solve for x.
63.0x6399 x
117
9963 x
117
Convert a repeating decimal to fraction.
61.0 Step 1. Set the repeating decimal = x 61.0x
Step 2. Get the decimal point to the left of the repeating digits. Multiply by 10.
66.110 x
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.
66.1101010 x
66.16100 x
Step 4. Subtract Step 3 – Step 2 and solve for x.
66.110 x1590 x
61
9015 x
61
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.
2. Keep the sign of the largest number ( absolute value ).
3. a. Same Signs Sum
b. Different Signs Difference (subtract)
a + ( - b )
a – b
a – ( - b )
a + b
+Large – small = Positive answer Small – Large = Negative answer
- a – b = - (a + b)
+ a + b = + (a + b) +Large – small = + (Large – small)
– Large + small = – ( Large – small)
Sect 1.5 and 1.6 Add & Subtract sign numbers
-12 + (-7) -15 + 9
-16 – 18 23 + (-11)
-32 – (-4) 19 – (-7)
-9 + (-7) – (-4) + 3 – 8 – (-12)
Law of Opposites: a + (-a) = 0
3 Same signs
SUM
2 Sign of Largest number1. Double signs
-12 – 7 = – 19
Add all positive numbers 1st and negative numbers 2nd.
1. Double signs
NONE
3 Different signs
Difference LG - sm
2 Sign of Largest number
1. Double signs
-9 – 7 + 4 + 3 – 8 + 12
= – 6
1. Double signs
NONE
3 Same signs
SUM
2 Sign of Largest number
= – 34
2 Sign of Largest number1. Double signs
23 – 11 = + 12 3 Different signs
Difference LG - sm
1. Double signs
-32 + 4 = – 28 2 Sign of Largest number
3 Different signs
Difference LG - sm 3 Same signs
SUM
2 Sign of Largest number1. Double signs
19 + 7 = + 26
19 – 24 2 Sign of Largest number
= – 5
3 Different signs
Difference LG - smGood to use this property when adding a long list of sign numbers…canceling is good!
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
Now Combine them.
Combine Like Terms
2a + (- 3b) + (-5a) + 9b 2xy + 3x – 7y + 5 – 8x – 2 + y
x – terms
7x + 2x – 8xy – terms
+ 3y – 9yconstants
– 5 + 10
x – 6y + 5
2a – 3b – 5a + 9b
– 3a + 6b
2xy + 3x – 7y + 5 – 8x – 2 + y
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.
Multiply by 0 rule.
Sign on the fraction rule.
341523 31454
1723
102 3 4 1
36
360 20
1
27301115103 0
b
a
b
a
b
a
Sect 1.8 Exponential Notation & Order of Operations
Exponential notation is a short cut to writing out repetitive multiplication.
Simplify.
6aaaaaaa
43 4343
32x32x
3333
81
99
3333
81
99
43133331
81
991
xxx 222 38x
xxx 232x
aa 1Negative quantities are defined as a -1 multiplied to the positive quantity.
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
35215 3532948 34 23
547912
31015
35
8
32292
8292
1692
72
5
Top Bottom
547912 34 23
20212 881
2024 89
44
89
44
Sect 1.8 Exponential Notation & Order of OperationsSimplify
54295 xx xxxx 5237 22
When variables are present, remove ( )’s by the Distributive Property and Combine Like Terms.
10895 xx
x13 1
Include the sign
xxxx 5637 22
210x x
Original Expression
Our Answer
2, STO> button, X, enter