Chapter 1 - The Language of Algebra Section 1-1 - Variables and Expressions A. Write mathematical expressions 1. Definitions

a.) Variables - Symbols used to represent numbers. ie: x, y, z, a, π b.) Algebraic expression - One or more numbers and variables along with one or more algebraic expressions.

ie: 2x, 3 7y − , 5xy

+

1.) a dot can mean multiply. 2 x⋅ c.) The quantities being multiplied are called the factors. d.) The answer is called the product.

Ex 1: Write an algebraic expression for each verbal expression. a.) five less than a number c.

5c − b.) the sum of 9 and two times the number d. 2 9d +

e.) An expression like 3x is called a power. f.) The variable x is called the base. g.) The 3 is called the exponent.

Ex 2: Write each expression algebraically.

a.) the product of ¾ and y to the third power.

334

y

Ex 3: Evaluate each expression. a.) 33 =27 b.) 29 =81

B. Write verbal expressions Ex 4: Write a verbal expression for each algebraic expression.

a.) 36x =The product of 6 and x to the third power. b.) 42r s+ =The sum of s to the fourth and 2 times r.

HW: Algebra 1 1-1 p. 8-9 11-43 odd, 44, 48-57 Section 1-2 - Order of Operations

A. Evaluate rational expressions 1. The rule for what order to do a numerical expression is called order of operations.

a. Please Excuse My Dear Aunt Sally b. Parenthesis, Exponents, (Multiply and Divide), (Add and Subtract). Ex 1: Evaluate each expression. a.) 6 4 2 3+ ⋅ − b.) 348 2 3 5÷ ⋅ + 6 8 3+ − 48 8 3 5÷ ⋅ + 14 3− 6 3 5⋅ + 11 18 5+ 23

Ex 2: Evaluate each expression a.) (8 3) 3(3 2)− ⋅ + b.) 24[12 (6 2)]÷ − (5) 3(5)⋅ 24[12 4]÷ 75 24[3] 4[9] 36

Ex 3: Evaluate 5

3

2 6 23 5 3 2

− ⋅− ⋅ −

.

=32 6 2

27 5 3 2− ⋅

− ⋅ −

=32 12

27 15 2−

− −

=2010

=2

B. Evaluate Algebraic Expressions

Ex 4: Evaluate 2 22( )x y z− + if x = 4, y = 3 and z = 2.

= 2 22(4 3) 2− + = 22(16 3) 2− + = 22(13) 2+ = 2(13) 4+ = 26 4+ =30

HW: Algebra 1-2 p. 14-15 15-29 odd, 30-32, 33-39 odd, 40, 46-48, 51-71 odd Section 1-3 - Open Sentences

A. Solve Equations 1. Definitions

a. A mathematical statement with one or more variables is called an open sentence.

b. Finding a replacement for the variable that results in a true statement is called solving the open sentence.

c. This replacement is called a solution of the open sentence.

Ex 1: Replace b in 8 8 40b − = with the values 4, 5, 6, and 7. Then see whether each replacement results in a true or false sentence.

Replace b with: 8 8 40b − = True or false? 4 8(4) 8 40− = false

5 8(5) 8 40− = false

6 8(6) 8 40− = true

7 8(7) 8 40− = false

The solution set is {6}.

d. A set of numbers from which replacements for a variable may be chosen is called a replacement set.

e. The solution set of an open sentence is the set of all the replacements for the variable that make the sentence true.

Ex 2: Find the solution set for 8 5x− < if the replacement set is {0, 2, 4, 6, 8}.

Replace y with: 8 5x− < True or false?

0 8 0 5− < false

2 8 2 5− < false

4 8 4 5− < true

6 8 6 5− < true

8 8 8 5− < true

The solution set is {4, 6, 8}.

f. A sentence that contains an equal sign =, is an

equation. g. A sentence having the symbols , , ,or≤ ≥ > < is

called an inequality. HW: Algebra 1-3 p. 19-20 15-25 odd, 26-28, 29-43 odd, 47-48, 51-52, 53-67odd

Section 1-4 - Identity and Equality Properties A. Identity and Equality Properties

Additive Identity Prop

For any number a, 0 0a a a+ = + =

Multiplicative Identity Prop

For any number a, 1 1a a a⋅ = ⋅ =

Multiplicative Prop of Zero

For any number a, 0 0 0a a⋅ = ⋅ =

Multiplicative Inverse Prop For every nonzero number

ab

,

where a, b≠0, there is exactly one

number ba

such that 1a bb a

⋅ =

Reflexive Prop of Equality

For any number a, a = a.

Symmetric Prop of Eq.

For any numbers a and b, if a = b, then b = a.

Transitive Prop of Equality

For any numbers a, b and c, if a = b, b = c, then a = c.

Substitution Prop of Equality

If a = b, then a may be replaced by b in any expression.

Ex 1: Name the property used in each equation, then find the value of n.

a.) 12 0n ⋅ = b.) 1

15

n ⋅ =

multiplicative prop of zero multiplicative inverse n = 0 n = 5 c.) 0 8n+ = additive identity n = 8

Ex 2: Evaluate 1

(12 8) 3(15 5 2)4

− + ÷ − , then name

the property in each step.

1

(12 8) 3(15 5 2)4

− + ÷ − Given

1(4) 3(15 5 2)

4+ ÷ − substitution

1(4) 3(3 2)

4+ − substitution

1(4) 3(1)

4+ substitution

1(4) 3

4+ multiplicative identity

1 3+ multiplicative inverse 4 substitution

HW: Algebra 1-4 p. 23-25 13-29 odd, 30-33, 37-40, 45-61 odd

Section 1-5 - The Distributive Property A. Evaluate Expressions

1. Take the formula for perimeter, 2( )P l w= + -plug in the numbers for my white board. (1 meter x 5 meters)

2. Now take the other formula for perimeter 2 2P l w= + .

-plug in the same numbers.

3. This is the distributive property. For any numbers a, b, and c, ( )a b c ab ac+ = + .

Ex. 1: Use the distributive property to find each product.

a.) 16(101)= 16(100) + 16(1) = 1600 + 16 = 1616

b.) 9(10.6)= 9(10) + 9(0.6) = 90 + 5.4 = 95.4

Ex 2: Rewrite each expression using the distributive property, then evaluate. a.) 5(7 2)+ b.) (12 7)3− =35 + 10 =36 - 21 =45 =15

B. Simplify expressions 1. A term is a number, variable, or a product or

quotient of numbers and variables.

a. ie: 2 1, , and 6

4x a b

2. Like terms are terms that have the same variables, and those variables have the same powers.

3. An expression in simplest form is when it has no like terms and no parenthesis.

4. The coefficient is the numerical factor before the variable.

Ex. 3: Simplify 2 2 21 112

4 4x x x+ +

= 2 2 21 112

4 4x x x+ +

= 2 2122

4x x+

= 2 23 2x x+ = 25x

Ex. 4: Name the coefficient in each term. a.) 145xy =145 b.) 2ab =1

c.) 24

5a

45

=

Ex. 5: Simplify each expression

a.) 4 4 2 24 3 2w w w w+ + − b.) 3

324a

a+

4 25w w+ 312

4a

HW: Algebra 1-5 p. 30-31 15-27 odd, 29-30, 31-35 odd, 37-38, 43-53 odd, 55-57, 59-71 odd

Section 1-6 - Commutative and Associative Properties A. Definitions

1. Commutative Property says the order in which you add or multiply two numbers does not change their sum or product. (think commute)

2. Associative Property says the way you group

three (or more) numbers when adding or multiplying does not change their sum or product. (think who you associate or group with)

Ex 1: Name the property used in each step.

8(2 4) 7b b+ + Given

16 32 7b b= + + Distributive 16 7 32b b= + + Associative 23 32b= + Substitution

Ex 2: Evaluate without using a calculator. a.) 5 3 6 4⋅ ⋅ ⋅ b.) 14 18 26 2+ + + 5 4 3 6= ⋅ ⋅ ⋅ 14 26 18 2= + + + 20 18= ⋅ 40 20= + 360= 60=

HW: Algebra 1-6 p. 34-36 (no calculators) 17-27 odd, 28-29, 33-47 odd, 50-51, 53-65 odd

Section 1-7 Logical Reasoning A. Conditional Statements

1. Conditional statements can be written in the form “If A, then B”. 2. A conditional statement is also known as an “if-then statement”. If A, then B. 3. If you don’t eat your meat, then you can’t have any pudding.

Ex 1: Identify the hypothesis and conclusion.

a.) If you love me, then you will obey my commands. b.) If 7 5 26y + ≤ , then 3y ≤ Ex 2: Rewrite each expression in if-then form. a.) I go to youth group on Wednesdays.

If it is Wednesday, then I go to youth group.

The part following the if is the hypothesis.

The part following the then is the conclusion.

hypothesis conclusion

hypothesis conclusion

B. Deductive Reasoning and Counterexamples

Ex 3: 1.) Determine if a valid conclusion can be reached from the statement. 2.) If a valid conclusion does not follow, write no valid conclusion and explain why. a.) If one number is odd and another is even,

the sum is odd. 7 + 20 = 27 ◊ valid conclusion

1. A counterexample is a specific case which a

statement is false. It takes only one counterexample to show a statement is false.

Ex 4: Provide a counterexample for the statement. a.) If you wear a helmet, then you play football. You could ride a motorcycle and wear a helmet.

HW: Algebra 1-7 p. 40-42 19-43 odd, 44-45, 52-53, 57-77 odd