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1-1 Variables and Expressions
You should have the following words defined in your notebook:
variables, algebraic expression, factors, product, power, base, exponent, evaluate.
Ex 1: The sum of a number and five
“The sum” means to ADD, “of a number and five” means that you will ADD x to five.
ANSWER: x + 5
Ex 2: The quotient of six and a number
“The quotient” means to DIVIDE, “of six and a number” means that six is being
DIVIDED by a number.
6
ANSWER: 6 ÷ x or
x
Ex 3: A number squared
“Squared” indicates an exponent of two
ANSWER: x2
Ex 4: A number cubed
“Cubed” indicates an exponent of three
ANSWER: x3
Note the difference between the following:
Ex 5: A) Twice the sum of a number and nine
2(x + 9)
B) The sum of twice a number and nine
2x + 9
In A, you would first determine the sum of the number and three, then double it (twice means
double or times two.)
In B, only “the number” is being doubled.
Ex 6: A)
B)
The difference of a number and four, squared
The difference of a number squared and four
(x – 4)2
x2 – 4
In A, the difference is being squared, so you must put the difference in
parenthesis and then put the square outside.
In B, just “the number” is being squared.
Ex 7: A)
B)
Seven less than a number
Seven less a number
x–7
7–x
In A, because it says THAN, you switch the order of the terms. So the 7 is after the minus
sign and the x is first.
In B, there is no THAN, so it is in the order that it appears– 7 before the minus and x after.
Algebra One Cp2 – Chapter 1 Notes for the Web
1-3 Order of Operations
There are four steps for the order of operations:
(1)
P- Parenthesis and Grouping Symbols
(2)
E- Exponents
(3)
M/D- Multiplication and Division from left to right
(4)
A/S- Addition and Subtraction from left to right
Use the order of operations to find the answer. Show EVERY step doing ONE simplification (please
circle it) at a time. Only work DOWN! Answers can be whole #’s, mixed #’s or reduced fractions. No
decimals!
Ex 1:
95 – 32 + 20 ÷ 2 ● 5
Are there Parenthesis or Grouping Symbols? NO
Are there Exponents? YES
95 – 9 + 20 ÷ 2 ● 5
Is there any Multiplication or Division? YES
Which one comes first (left to right)? DIVISION
95 – 9 + 10 ● 5
Is there any more Multiplication or Division? YES
Which one comes next (left to right)? MULT.
95 – 9 + 50
Is there any Addition or subtraction? YES
Which one comes first (left to right)? SUBTR.
86 + 50
Is there any more Addition or subtraction? YES
Which one comes first (left to right)? ADDITION
136
Tips for using your calculator:
On your TI 83+ or TI 84+, you can enter in a whole expression at once. The calculator uses the order of
operations to evaluate that expression. However, if you do not enter the expression in correctly, you will
not get the correct answer.
Tip #1:
In order to do exponents, use the carrot button. It is located in the last column of buttons on the right
under the CLEAR button. It looks like ^ .
Ex 2: 54
You would make the following keystrokes: 5
^
4 ENTER .
The answer that would appear on your screen is 625
This is what you would write on your paper for me:
2)
54
Copy the problem
5 ^ 4 = 625
Show the work and box the final answer
Tip #2:
When doing fractions, put the numerator and denominator in parenthesis.
18  4 and enter in accordingly.
18  4
Ex 3:
Think of this as
76
7  6
Make the following keystrokes: ( 1 8 – 4 ) / (
The answer that would appear on your screen is 1.076923077
7
+ 6
)
ENTER
This is what you would write on your paper for me:
3)
(18 – 4)/(7 + 6) = 1.08 [answer should be rounded to the nearest hundredth]
Algebra One Cp2 – Chapter 1 Notes for the Web
1-5 Open Sentences
You should have the following words defined in your notebook:
Open Sentence, Solution, Replacement Set, Element, Solution Set, Equation, and Inequality
Ex 1: State whether 4x2 + 5 = 149 is true or false using the given value, x = 3
4( )2 + 5 = 149
Rewrite using an open parenthesis for each variable
4( 3 )2 + 5 = 149
Substitute the value of the variable into the parenthesis
4( 9 ) + 5 = 149
Exponents first, 32 = 3 ● 3 = 9
36 + 5 = 149
Multiplication next, 4 ● 9 = 36
41 = 149
Addition last, 36 + 5 = 41
FALSE
41 ≠ 149
Ex 2: Find the solution set for 48  x ● 2 < 8 + x2 given the replacement set x = {3, 4, 8}.
x=3
48  ( ) ● 2 < 8 + ( )2
48  (3) ● 2 < 8 + (3)2
Label work with the element you are using
Rewrite using an open parenthesis for each variable
Substitute the value of the variable into the parenthesis
Concentrate on one side of the statement and then do the other side. Therefore you will do one step of the order of operations per side.
Mult and Div from Left to Right: 48  3 = 16
Now do the multiplication: 16 ● 2 = 32
48  (3) ● 2 < 8 + (3)2
16 ● 2 < 8 + 9
32 < 17
Exponents: 32 = 3 ● 3 = 9
Now do the addition: 8 + 9 = 17
FALSE because 32 is bigger than 17, so x = 3 is NOT a solution
(Ex 2 is continued on the next page.)
Algebra One Cp2 – Chapter 1 Notes for the Web
x=4
48  ( ) ● 2 < 8 + ( )2
48  (4) ● 2 < 8 + (4)2
Label work with the element you are using
Rewrite using an open parenthesis for each variable
Substitute the value of the variable into the parenthesis
Concentrate on one side of the statement and then do the other side. Therefore you will do one step of the order of operations per side.
Mult and Div from Left to Right: 48  4 = 12
Now do the multiplication: 12 ● 2 = 24
48  (4) ● 2 < 8 + (4)2
12 ● 2 < 8 + 16
24 < 24
Exponents: 42 = 4 ● 4 = 16
Now do the addition: 8 + 16 = 24
TRUE because 24 is less than or equal to 24, so x = 4 is a solution.
x=8
48  ( ) ● 2 < 8 + ( )2
48  (8) ● 2 < 8 + (8)2
Label work with the element you are using
Rewrite using an open parenthesis for each variable
Substitute the value of the variable into the parenthesis
Concentrate on one side of the statement and then do the other side. Therefore you will do one step of the order of operations per side.
Mult and Div from Left to Right: 48  8 = 6
Now do the multiplication: 6 ● 2 = 12
48  (8) ● 2 < 8 + (8)2
6 ● 2 < 8 + 64
12 < 72
Exponents: 82 = 8 ● 8 = 64
Now do the addition: 8 + 64 =724
TRUE because 12 is less than or equal to 72, so x = 8 is a solution.
Since x = 4 and x = 8 make the statement true, the solution set will have these two elements. Therefore the answer is
{4, 8}
Algebra One Cp2 – Chapter 1 Notes for the Web
1-6 Identity and Equality Properties
Name the property illustrated.
Ex 1: 32 + 0 = 32
ADDITIVE IDENTITY PROPERTY
(When you add zero to a number, the result is the number.)
Ex 2: 1● 5 = 5
MULTIPLICATIVE IDENTITY PROPERTY
(When you multiply one and a number, the result is the number)
Ex 3: 13 ● 0 = 0
MULTIPLICATIVE PROPERTY OF ZERO
(When you multiply zero and a number, the result is zero.)
Ex 4:
7
2

 1
2
7
MULTIPLICATIVE INVERSE PROPERTY
(When you multiply reciprocals, the result is one.)
Ex 5: 8 + 0 = 8 + 0
REFLEXIVE PROPERTY OF EQUALITY
(There is no difference between the left and right sides of the equation.)
Ex 6: If 5 + 4 = 9, then 9 = 5 + 4
SYMMETRIC PROPERTY OF EQUALITY :
(The expressions on both sides of the equation are the same, they have just
switched which side of the equal sign they are on.)
Ex 7: If 2 ● 1 = 2 and 2 = 10  5, then 2 ● 1= 10  5
TRANSITIVE PROPERTY OF EQUALITY
(There are three parts to this statement. Each part is equal to the others.)
Ex 8: 12 + 4 – 2 = 16 – 2
SUBSTITUTION PROPERTY OF EQUALITY
(Since 12 + 4 = 16, you can substitute 12 + 4 with 16.)
Algebra One Cp2 – Chapter 1 Notes for the Web
1-7 The Distributive Property
You should have the following words defined in your notebook:
term, like terms, equivalent expressions, simplest form, and coefficient
An expression is simplified if there are NO like terms and NO parentheses.
Distributive Property:
Ex 1: 3(5x + 22)
3(5x) + 3(22)
15x + 66
x(y + z) = xy + xz
x(y – z) = xy – xz
Distribute the 3 to all terms in the parentheses. (You must show this step)
Multiply (only when doing the distributive property in this chapter, you
can do both multiplications in one step)
Since 15x and 66 are not like terms, 15x + 66 is the final answer.
Ex. 2: (8x – 3y)7
(8x)7 – (3y)7
56x – 21y
Distribute the 7 to all terms in the parentheses. (It doesn’t matter that the
7 is on the right side of the parentheses.
Mulitply.
Since 56x and 21y are not like terms, 56x – 21y is the final answer.
Ex. 3: 7x + x + 9y – 8y
8x + 9y – 8y
8x + y
7x and x are like terms. Remember x = 1x, so you are doing (7 + 1)x = 8x
9y and 8y are like terms.(9 – 8)y = 1y. Because of the Mult.ID Prop,1y = y
Since 8x and y are not like terms, 8x + y is the final answer.
Ex. 4: 6(4x + 11) – 20
6(4x) + 6(11) – 20
24x + 66 – 20
24x + 46
Distribute the 6 to get rid of the parentheses.
Mulitply
Since 66 and 20 are like terms, combine them.
Since 24x and 6 are not like terms, 24x + 46 is the final answer.
Ex. 5: 5(2x + 7) + 4(10 + 9y)
5(2x) + 5(7) + 4(10 + 9y)
10x + 35 + 4(10 + 9y)
10x + 35 + 4(10) + 4(9y)
10x + 35 + 40 + 36y
10x + 75 + 36y
Distribute the 5 to every terms in the first parentheses.
Multiply
Distribute the 4 to every terms in the second parentheses.
Multiply
Since 35 and 40 are like terms, combine them.
Since 10x, 75 and 36y are not like terms, 10x + 75 + 36y is the final answer.
Algebra One Cp2 – Chapter 1 Notes for the Web
1-8 Commutative and Associative Properties
Commutative means ORDER
Associative mean GROUPING
(Associative’s abbreviation is 5 letters – ASSOC!)
Name the property illustrated.
1)
6+7=7+6
The order of the numbers has changed. The operation is addition. The
Commutative Property of Addition says “The order terms are added does
not affect the sum.”
Comm. Prop. (+)
2)
(6 • 10) • 3 = 6 • (10 • 3)
The order of the numbers is the same. For both sides the numbers
are 6, 10, and 3. However, the grouping did change. On the left, 6
and 10 are grouped. On the right, 10 is now grouped with 2. The
operation is multiplication. The Associative Property of
Multiplication says “The grouping of the terms does not affect the
product.”
Assoc. Prop. (*)
3)
(17 • 2) • 9 = (2 • 17) • 9
This is a tricky one. Even though there are grouping symbols the
17 and 2 are grouped together on each side. Since the same
numbers are in the grouping symbols, it can’t be the Assoc. Prop.
The order of the 17 and 2 switched on the right hand side.
Therefore, if the order is different, it is the Commutative Property.
The operation is multiplication so the answer is:
Comm. Prop. (*)
Name the property illustrated in each step.
4)
8+¼•0•4•7
=8+0•¼•4•7
Comm. Prop. (*) – The order of 0 and 4 switched
= 8 + 0 • (¼ • 4) • 7
Assoc. Prop. (*) – ¼ and 4 were grouped together
=8+0•1•7
Mult. Inv. Prop. – ¼ and 4 are recipropcals, their
product is 1.
= 8 + 0 • (1 • 7)
Assoc. Prop. (*) – 1 and 7 were grouped together
=8+0•7
Mult. ID. Prop. – the product of a number and one
is the number.
=8+0
Mult. Prop. Zero – the product of a number and
zero is the zero.
=8
Add ID. Prop. – the sum of a number and zero is
the number.
Algebra One Cp2 – Chapter 1 Notes for the Web