Introduction to Variables and Properties of Algebra

Home , Expression (mathematics), Quantity, Variable (mathematics)

Chapter 2

CHAPTER 2: INTRODUCTION TO VARIABLES AND PROPERTIES OF ALGEBRA Chapter Objectives By the end of this chapter, students should be able to:  Introduction to variables  Different meanings of variables  Evaluate algebraic expressions  Writing algebraic expressions  Properties of Algebra • Commutative • Associative • Identity • Inverse • Distributive Contents CHAPTER 2: INTRODUCTION TO VARIABLES AND PROPERTIES OF ALGEBRA ...... 87 SECTION 2.1 INTRODUCTION TO VARIABLES ...... 88 A. WHAT IS A VARIABLE? ...... 88 B. MEANING OF A VARIABLE IN MATHEMATICS ...... 90 C. VARIABLE EXPRESSIONS ...... 91 D. WRITING ALGEBRAIC EXPRESSIONS ...... 92 E. EVALUATE ALGEBRAIC EXPRESSIONS ...... 96 F. LIKE TERMS AND COMBINE LIKE TERMS ...... 98 EXERCISE ...... 99 SECTION 2.2 PROPERTIES OF ALGEBRA ...... 101 EXERCISE ...... 104 CHAPTER REVIEW ...... 106

87 Chapter 2 SECTION 2.1 INTRODUCTION TO VARIABLES A. WHAT IS A VARIABLE? When someone is having trouble with algebra, they may say, “I don’t speak math!” While this may seem weird to you, it is a true statement. Math, like English, French, Spanish, or Arabic, is a like a language that you must learn in order to be successful. In order to understand math, you must practice the language.

Action words, like run, jump, or drive, are called verbs. In mathematics, operations are like verbs because they involve doing something. Some operations are familiar, such as addition, multiplication, subtraction, or division.

Operations can also be much more complex like a raising to an exponent or taking a square root.

Variable is one of the most important concepts of mathematics, without variables we would not get far in this study. Variable is a symbol, usually an English letter, written to replace an unknown or changing quantity.

Definitions

A variable, usually represented by a letter or symbol, can be defined as: • A quantity that may change within the context of a mathematical problem. • A placeholder for a specific value.

An algebraic expression is a mathematical statement that can contain numbers, variables, and operations (addition, subtraction, multiplication, division, etc…).

MEDIA LESSON Introduction to variables and variable expressions (Duration 7:54)

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Definition: • A ______is a ______that represents an ______. a) How many hours will you study tomorrow? ______b) How much will you pay to have your car repaired? ______

• A______consists of ______, ______, ______and ______like ______, ______, ______and ______.

• Often ______and ______are used ______. The difference is a ______contain a variable while the ______. 88 Chapter 2 Mathematical Expressions Equations ______• Writing variable expressions a) It costs $9 per adult and $5 per child to go to the movies. Variable expression for the cost of a group go to the movies: ______b) It costs $30 per day to rent a car plus $0.10 per mile. Variable expression for the total rental cost: ______

Evaluating variable expressions c) Evaluate 5 + 7 when = 6 d) Evaluate 4 3 when = 9 and = 7

𝑥𝑥 𝑥𝑥 𝑚𝑚 − 𝑛𝑛 𝑚𝑚 𝑛𝑛

e) Evaluate 3 + 7 when = 11, = 8 f) Evaluate +7 9 when = 9, = 3, 2 𝑝𝑝 − 𝑞𝑞 𝑝𝑝 𝑞𝑞 and = 136 𝑐𝑐 − 𝑓𝑓 𝑑𝑑 𝑒𝑒 𝑑𝑑 𝑓𝑓

MEDIA LESSON Why all the letters in algebra? (Duration 3:03)

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What question students ask a lot when they start Algebra?

______

What are other things besides letters that can be used as a variable?

______

89 Chapter 2 B. MEANING OF A VARIABLE IN MATHEMATICS MEDIA LESSON Meaning of a variable (Duration 4:13)

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Definitions of a variable

1) Define variable as a changing value A variable is a letter that represents ______or ______that may ______within the context of a mathematical problem. Example: ______2) Define variable as a placeholder A variable is a letter that represents ______or ______that will remain ______based on the confines of the mathematical problem. Example: ______

Example: Determine if the description describes a changing value (CV) or a placeholder (P) then determine a possible variable. Changing Value (CV) or Scenario Variable Placeholder (P) The elevation of Mount Humphrey’s

The water level of a pool as it is being filled

The number of calories consumed throughout the day

The monthly car payment with a fixed interest loan

The amount of gas consumed by your car as you drive The cost of a new textbook for a specific class from the bookstore

at the beginning of the semester

YOU TRY

Changing Value (CV) or Scenario Variable Placeholder (P) The number of cars in the parking lot at this moment

The number of cars pass by an intersection throughout the day

The altitude of an airplane during a trip The temperature in the Dead Valley on at midnight on January

1st in 1905 The balance of your checking account today

90 Chapter 2 MEDIA LESSON Why aren't we using the multiplication sign? (Duration 3:08)

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Why the multiplication sign “x” is not being used much when we get to algebra? ______

MEDIA LESSON Why is 'x' the symbol for an unknown? (Duration 3:57)

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Why is it that is the unknown? ______𝑥𝑥 C. VARIABLE EXPRESSIONS In mathematics, especially in algebra, we look for patterns in the numbers that we see. Using mathematical verbs and variables studied in previous lessons, expressions can be written to describe a pattern.

Definition An algebraic expression is a mathematical phrase combining numbers and/or variables using mathematical operations.

An algebraic expression consists of coefficients, variables, and terms. Given an algebraic expression, a • coefficient is the number in front of the variable. • variable is a letter representing any number. • term is a product of a coefficient and variable(s). • constant: a number with no variable attached. A term whose value never changes. For example, , , , , are all examples of terms because each is a product of a coefficient and variable(s). 𝟐𝟐 𝟑𝟑 𝒕𝒕 𝟐𝟐𝒙𝒙 𝟑𝟑𝒔𝒔𝒔𝒔 𝟕𝟕𝒙𝒙 𝟓𝟓𝒂𝒂𝒃𝒃 𝒄𝒄

𝟐𝟐

𝟓𝟓𝟓𝟓𝒚𝒚 −�𝟐𝟐𝟐𝟐 −�𝟑𝟑 �� 𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕 𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄 𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕 𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕 MEDIA LESSON Algebraic expression vocabulary (Duration 5:52)

View the video lesson, take notes and complete the problems below Terms: ______. Constant Term: ______. Example 1: Consider the algebraic expression 4 + 3 22 + 17 5 4 2 a) List the terms:______𝑥𝑥 𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥 b) Identify the constant term: ______Factors: ______. Coefficient: ______. 91 Chapter 2 Example 2: Complete the table below. 1 2 4 2 5 List the 𝑟𝑟 − 𝑚𝑚 −𝑥𝑥 𝑏𝑏ℎ Factors Identify the

Coefficient

Example 3: Consider the algebraic expression 5 8 + 7 4 3 2 𝑦𝑦 a) How many terms are there? ______𝑦𝑦 − 𝑦𝑦 𝑦𝑦 − 4 − b) Identify the constant term. ______c) What is the coefficient of the first term? ______d) What is the coefficient of the second term? ______e) What is the coefficient of the third term? ______f) List the factors of the fourth term.______

YOU TRY

Consider the algebraic expression 2 + 2 8 3 2 a) How many terms are there?𝑚𝑚 ______𝑚𝑚 − 𝑚𝑚 − b) Identify the constant term. ______c) What is the coefficient of the first term? ______d) What is the coefficient of the second term? ______e) List the factors of the third term. ______

D. WRITING ALGEBRAIC EXPRESSIONS

MEDIA LESSON Write algebraic expressions (Duration 6:18)

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Example 1: Juan is 6 inches taller than Niko. Let represent Niko’s height in inches. Write an algebraic expression to represent Juan’s height. 𝑁𝑁 Niko’s height: ______Juan’s height: ______

Example 2: Juan is 6 inches taller than Niko. Let represent Juan’s height in inches. Write an algebraic expression to represent Niko’s height. 𝐽𝐽 Niko’s height: ______

Juan’s height: ______

92 Chapter 2 Example 3: Suppose sales tax in your town is currently 9.8%. Write an algebraic expression representing the sales tax for an item that costs dollars.

Cost: ______𝐷𝐷 Sales tax: ______Example 4: You started this year with $362 saved and you continue to save an additional $30 per month. Write an algebraic expression to represent the total amount saved after months.

Number of months: ______𝑚𝑚 Total amount saved: ______

Example 5: Movie tickets cost $8 for adults and $5.50 for children. Write an algebraic expression to represent the total cost for adults and children to go to a movie.

Number of adults: ______𝐴𝐴 𝐶𝐶 ______Number of children: ______Total Cost: ______

YOU TRY

Complete the following problems. Show all steps as in the media examples. a) There are about 80 calories in one chocolate chip cookie. If we let be the number of chocolate chip cookies eaten, write an algebraic expression for the number of calories consumed. 𝒏𝒏 Number of cookies: Number of calories consumed:

b) Brendan recently hired a contractor to do some necessary repair work. The contractor gave a quote of $450 for materials and supplies plus $38 an hour for labor. Write an algebraic expression to represent the total cost for the repairs if the contractor works for hours.

Number of hours worked: 𝒉𝒉 Total cost:

c) A concession stand charges $3.50 for a slice of pizza and $1.50 for a soda. Write an algebraic expression to represent the total cost for slices of pizza and sodas.

𝑷𝑷 𝑺𝑺 Number of slices of pizza: Number of sodas: Total cost:

93 Chapter 2 MEDIA LESSON The story of (Duration 7:09)

𝑥𝑥 View the video lesson, take notes and complete the problems below

Example 1: Tell the story of in each of the following expressions. a) 5 𝑥𝑥 b) 5

𝑥𝑥 − − 𝑥𝑥

c) 2 d) 2 𝑥𝑥 𝑥𝑥

Example 2: Tell the story of in each of the following expressions. a) 2 + 4 𝑥𝑥 b) 2( + 4)

𝑥𝑥 𝑥𝑥

c) 5( 3) 2 2 𝑥𝑥 − −

Example 3: Write an algebraic expression that summarizes the stories below.

a) Step 1: Add 3 to b) Step 1: Divide by 2 Step 2: Divide by 2 Step 2: Add 3 𝑥𝑥 𝑥𝑥

Example 4: Write an algebraic expression that summarizes the stories below.

Step 1: Subtract from 7 Step 2: Raise to the third power Step 3: Multiply by𝑥𝑥 3 Step 4: Add 1

94 Chapter 2 Below is a table of common English words converted into a mathematical expression. You can use this table to assist in translating expressions.

Operation Words Example Translation Added to 4 added to + 4

More than 2 more than𝑛𝑛 𝑛𝑛 + 2 The sum of The sum of 𝑦𝑦 and 𝑦𝑦 + Addition Increased by increased𝑟𝑟 by 6 𝑠𝑠 𝑟𝑟 +𝑠𝑠6 The total of 𝑚𝑚The total of 8 and 𝑚𝑚8 + Plus plus 2 𝑥𝑥 + 2𝑥𝑥 Minus 𝑐𝑐 minus 1 𝑐𝑐 1 Less than 𝑥𝑥5 less than 𝑥𝑥 − 5 Less 4 less 𝑦𝑦 𝑦𝑦4 − Subtraction Subtracted from 3 subtracted𝑟𝑟 from −3𝑟𝑟 Decreased by decreased by 10𝑡𝑡 𝑡𝑡 − 10 The difference between 𝑚𝑚The difference between and 𝑚𝑚 − Times 12 times 𝑥𝑥 𝑦𝑦 12𝑥𝑥 − 𝑦𝑦 1 Of One-third𝑥𝑥 of ∙ 𝑥𝑥 3 Multiplication The product of The product of𝑣𝑣 and 𝑣𝑣

Multiplied by multiplied by 𝑛𝑛3 𝑘𝑘 𝑛𝑛𝑛𝑛 𝑜𝑜3𝑜𝑜 𝑛𝑛 ∙ 𝑘𝑘 Twice 𝑦𝑦Twice 2 𝑦𝑦 2 Divided by divided𝑑𝑑 by 4 𝑑𝑑 𝑜𝑜𝑜𝑜 ∙ 𝑑𝑑 4 Division 𝑛𝑛 𝑛𝑛 The quotient of The quotient of and 𝑡𝑡 𝑡𝑡 𝑥𝑥 The ratio of The ratio of to 𝑥𝑥 Division 𝑥𝑥 𝑥𝑥 𝑝𝑝 2 Per 2 per 𝑝𝑝

The square of The square𝑏𝑏 of 𝑏𝑏 Power 2 The cube of The cube of 𝑦𝑦 𝑦𝑦 3 Is Are 𝑘𝑘 Equal 𝑘𝑘 Equals Gives Is equal to Is equivalent to Yields Results in was

95 Chapter 2 YOU TRY Complete the following problems.

a) Tell the story of in the expression 𝑥𝑥−3 𝑥𝑥 5

b) Write an algebraic expression that summarizes the story below:

Step 1: Multiply by 2 Step 2: Add 5 Step 3: Raise to the𝑥𝑥 second power.

MEDIA LESSON The beauty of algebra (Duration 10:06)

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What algebraic expression did Sal start with when he discussed why the abstraction of mathematics is so fundamental? ______

E. EVALUATE ALGEBRAIC EXPRESSIONS MEDIA LESSON Evaluate algebraic expressions (Duration 9:33)

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To evaluate a algebraic or variable expression:

1. ______

2. ______

PEMDAS

MD:

AS:

96 Chapter 2

Example 1: Find the value of each expression when w = 2. Simplify your answers. a) – 6 b) 6 – c) 5 – 3

𝑤𝑤 𝑤𝑤 𝑤𝑤

d) w e) 3 f) (3 ) 3 2 2 𝑤𝑤 𝑤𝑤

i) 3 g) h) 𝑤𝑤 4 5𝑤𝑤 5𝑤𝑤 4

Example 2: Evaluate + given = – 5, = 7, and = – 3

𝑎𝑎𝑎𝑎 𝑐𝑐 𝑎𝑎 𝑏𝑏 𝑐𝑐

Example 3: Evaluate given = – 5 and = – 3 2 2 𝑎𝑎 − 𝑏𝑏 𝑎𝑎 𝑏𝑏

Example 4: A local window washing company charges $11.92 for each window plus a reservation fee of $7. a) Write an algebraic expression to represent the total cost from the window washing company for washing windows.

𝑤𝑤 b) Use this expression to determine the total cost for washing 17 windows.

YOU TRY

Given = 5, = – 1, = 2, evaluate the expressions below. Show all steps.

Evaluate𝑎𝑎 𝑏𝑏4 𝑐𝑐 Evaluate 2 5 + 7 2 𝑏𝑏 − 𝑎𝑎𝑎𝑎 𝑎𝑎 − 𝑏𝑏 𝑐𝑐

97 Chapter 2 F. LIKE TERMS AND COMBINE LIKE TERMS Definition

Two terms are like terms if the base variable(s) and exponent on each variable are identical.

For example, 3 and 7 are like terms because they both contain the same base variables, and , and the exponents2 on2 (the is squared on both terms) and are the same. 𝑥𝑥 𝑦𝑦 − 𝑥𝑥 𝑦𝑦 𝑥𝑥 Combining𝑦𝑦 like terms: If two terms𝑥𝑥 are𝑥𝑥 like terms, we add (or subtract)𝑦𝑦 the coefficients, then keep the variables (and exponents on the corresponding variable) the same.

MEDIA LESSON Like terms and combine like terms (Duration 6:30)

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Example 1: Identify the like terms in each of the following expressions

3 6 + 10 5 10 + 6 3 7 + 3 2 + 8 + 2 3 2 3 𝑎𝑎 − 𝑎𝑎 𝑎𝑎 − 𝑎𝑎 𝑥𝑥 − 𝑦𝑦 𝑧𝑧 − 𝑥𝑥 𝑛𝑛 𝑛𝑛 − 𝑛𝑛 𝑛𝑛 𝑛𝑛 − 𝑛𝑛

Example 2: Combine the like terms 3 6 + 10 5 10 + 6 3 7 + 3 2 + 8 + 2 3 2 3 𝑎𝑎 − 𝑎𝑎 𝑎𝑎 − 𝑎𝑎 𝑥𝑥 − 𝑦𝑦 𝑧𝑧 − 𝑥𝑥 𝑛𝑛 𝑛𝑛 − 𝑛𝑛 𝑛𝑛 𝑛𝑛 − 𝑛𝑛

YOU TRY Combine the like terms.

a) 3 4 + 8 b) 5 + 2 4 + + 7 2 2 𝑥𝑥 − 𝑥𝑥 𝑥𝑥 − 𝑥𝑥 − 𝑎𝑎 − 𝑎𝑎 𝑎𝑎

98 Chapter 2 EXERCISE Tell the story of in each of the following expressions.

1) 11 𝒙𝒙 2) + 5 3) 5 𝑥𝑥 − 𝑥𝑥 𝑥𝑥 4) 5) 6) 2 5 3 𝑥𝑥 𝑥𝑥 − 𝑥𝑥 7) 2 3 8) 8 9) (2 ) 2 2 𝑥𝑥 − 𝑥𝑥 𝑥𝑥 10) 7 2 11) 5(7 ) 3 12) 3 − 𝑥𝑥 − 𝑥𝑥 3𝑥𝑥−3 Write an algebraic expression that summarizes the stories below. � 5 �

13) Step 1: Add 8 to 14) Step 1: Divide by 8 Step 2: Raise to the third power Step 2: Subtract 5 𝑥𝑥 𝑥𝑥

15) Step 1: Subtract 3 from 16) Step 1: Multiply by 10 Step 2: Multiply by 7 Step 2: Raise to the 3rd power 𝑥𝑥 Step 3: Multiply 𝑥𝑥by 2

17) Step 1: Add 5 to 18) Step 1: Raise to the second power Step 2: Divide by 2 Step 2: Multiply by 5 Step 3: Raise to the𝑥𝑥 second power Step 3: Subtract𝑥𝑥 from 9 Step 4: Add 8

19) Step 1: Subtract from 2 20) Step 1: Multiply by 4 Step 2: Multiply by 8 Step 2: Add 9 Step 3: Raise to the𝑥𝑥 third power Step 3: Divide by𝑥𝑥 2 − Step 4: Add 1 − Step 4: Raise to the fifth power Step 5: Divide by 3

Find the value of each expression when = – . Simplify your answers.

21) 11 22) 𝒃𝒃+ 5 𝟖𝟖 23) 5

𝑏𝑏 − 𝑏𝑏 𝑏𝑏 24) 25) 26) 2 2 3 𝑏𝑏 𝑏𝑏 − 𝑏𝑏

Evaluate each of the following given = .

27) 2 3 28)𝒒𝒒 8 𝟏𝟏𝟏𝟏 29) (2 ) 2 2 𝑞𝑞 − 31) 7𝑞𝑞 2 32) 2 𝑞𝑞 30) 𝑞𝑞 4 − 𝑞𝑞 7𝑞𝑞

99 Chapter 2 Evaluate the following expressions for the given values. Simplify your answers.

33) for = 6, = 4 34) for = 3 −𝑏𝑏 4𝑥𝑥−8 2𝑎𝑎 𝑎𝑎 𝑏𝑏 5+𝑥𝑥 𝑥𝑥 35) for = 8, = 1 36) 3 + 2 1 for = 1 3 2 2 5 𝑎𝑎𝑎𝑎 𝑎𝑎 𝑏𝑏 3 𝑥𝑥 𝑥𝑥 − 𝑥𝑥 − 37) for = 3, = 2 38) 2 7 for = 5, = 3 2 2 𝑥𝑥 − 𝑦𝑦 𝑥𝑥 − 𝑦𝑦 − 𝑥𝑥 − 𝑦𝑦 𝑥𝑥 𝑦𝑦 39) Shea bought candy bars for $1.50 each. Write an algebraic expression for the total amount Shea spent. 𝑐𝑐 40) Suppose sales tax in your town is currently 9%. Write an algebraic expression representing the sales tax for an item that costs dollars.

41) Ben bought movie tickets𝑑𝑑 for $8.50 each and bags of popcorn for $3.50 each. a) Write an algebraic expression for the total amount Ben spent. b) Use this 𝑚𝑚expression to determine the amount𝑝𝑝 Ben will spend if he buys 6 movie tickets and 4 bags of popcorn.

42) Noelle is 5 inches shorter than Amy. Amy is inches tall. Write an algebraic expression for Noelle's height. 𝐴𝐴 43) Jamaal studied hours for a big test. Karla studied one fourth as long. Write an algebraic expression for the length of time that Karla studied. ℎ 44) A caterer charges a delivery fee of $45 plus $6.50 per guest. a) Write an algebraic expression to represent the total catering cost if guests attend the reception. b) Use the expression to determine the amount of a company luncheon𝑔𝑔 of 50 guests.

45) Tickets to the museum cost $18 for adults and $12.50 for children. a) Write an algebraic expression to represent the cost for adults and children to visit the museum. b) Use this expression to determine the cost for 4 adults and𝑎𝑎 6 children𝑐𝑐 to attend the museum.

46) Consider the algebraic expression 5 + + 2 𝑛𝑛 a) How many terms are there? 8 5 2 𝑛𝑛 − 𝑛𝑛 𝑛𝑛 8 − b) Identify the constant term. c) What is the coefficient of the first term? d) What is the coefficient of the second term? e) What is the coefficient of the third term? f) What is the coefficient of the fourth term?

Combine the like terms. 47) 3 5 + 7 48) 3 + 3 9 + 2 3 2 3 49) 𝑑𝑑 −2 𝑑𝑑+ 4𝑑𝑑 +− 𝑑𝑑 ( 2 ) 50) 3𝑥𝑥 7 𝑥𝑥+ −9 𝑥𝑥 5 𝑥𝑥 −7 𝑥𝑥

𝑎𝑎 − 𝑏𝑏 𝑎𝑎 𝑏𝑏 − − 𝑏𝑏 𝑥𝑥 − 𝑦𝑦 𝑥𝑥 − 𝑦𝑦 −

100 Chapter 2 SECTION 2.2 PROPERTIES OF ALGEBRA

Properties of real numbers are the basic rules when we work with expressions in Algebra.

Addition Multiplication + = =

Identity Any number𝒂𝒂 plus𝟎𝟎 zero𝒂𝒂 is the same Any number 𝒂𝒂multiplies∙ 𝟏𝟏 𝒂𝒂 by one is the Hint: the number number. same number. stays true to its 7 + 0 = 7 7 1 = 7 “identity”. + 0 = 1 =  + 0 =   ∙ 1 =  𝑥𝑥 𝑥𝑥 𝑥𝑥 ∙ 𝑥𝑥 ∙

Multiplicative inverse

Additive Inverse = + ( ) = 𝟏𝟏 if is not zero 𝒂𝒂 ∙ 𝟏𝟏 Inverse Hint: “Inverse”  Any number𝒂𝒂 plus− its𝒂𝒂 opposite𝟎𝟎 equals 𝒂𝒂 Any number multiplies𝑎𝑎 by its reverse  undo zero. 2 + ( 2) = 0 reciprocal is one. 1 + ( ) = 0 3 = 1 − 3 1 𝑥𝑥 −𝑥𝑥 ⋅ = 1

𝑥𝑥 ⋅ + = + 𝑥𝑥 =

Commutative You can𝒂𝒂 add𝒃𝒃 in𝒃𝒃 any 𝒂𝒂order. Hint: “Commute”  You can 𝒂𝒂multiply∙ 𝒃𝒃 𝒃𝒃in∙ any𝒂𝒂 order. 2 + 3 = 3 + 2  2 3 = 3 2 move switch + 5 = 5 + places 2 = 2 +  =  + ∙ ∙ ☺𝑥𝑥 ☺𝑥𝑥 ·  =  · ☺𝑥𝑥 ∙ ∙ 𝑥𝑥☺

( + ) + = + ( + ) ( ) = ( ) Associative Hint: “associate”  When𝒂𝒂 you𝒃𝒃 add, 𝒄𝒄you can𝒂𝒂 group𝒃𝒃 in𝒄𝒄 any When you𝒂𝒂 ⋅ 𝒃𝒃multiply,⋅ 𝒄𝒄 you𝒂𝒂 ⋅ can𝒃𝒃 ⋅ group𝒄𝒄 in different groups  combination. any combination. parenthesis (3 + 5) + 2 = 3 + (5 + 2) (6 2) 7 = 6 (2 7) ( + ) + = + ( + ) ( ) = ( ) ⋅ ⋅ ⋅ ⋅ 𝑥𝑥 𝑦𝑦 𝑧𝑧 𝑥𝑥 𝑦𝑦 𝑧𝑧 𝑥𝑥 ⋅ 𝑦𝑦 ⋅ 𝑧𝑧 𝑥𝑥 ⋅ 𝑦𝑦 ⋅ 𝑧𝑧

Distributive  Hint: “Distribute” Multiplying a number by a group of numbers added together or subtracted give out to each other is the same as doing each multiplication separately. 5(100 + 20) = 5 100 + 5 20 2( + ) = 2 + 2 ⋅ ⋅ 𝑥𝑥 𝑦𝑦 𝑥𝑥 𝑦𝑦 101 Chapter 2 MEDIA LESSON Properties of Real Numbers (Duration 9:59)

View the video lesson, take notes and complete the problems below

The commutative properties

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The associative properties

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Identity properties

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Inverse properties

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102 Chapter 2 Distributive property

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 NOTE: Subtraction and division do not have the associative and communicative properties.

103 Chapter 2 EXERCISE Identify the property that justifies each problem below. 1) + 0 = Name: Name: 2) 𝑤𝑤 + =𝑤𝑤0 Name: 3) −3(𝑥𝑥 +𝑥𝑥 ) = 3 + 3 Name: 4) 5 𝑢𝑢(6 𝑣𝑣) = (6 𝑢𝑢)5 𝑣𝑣 Name: 5) (5𝑤𝑤) 𝑧𝑧 = 1 𝑧𝑧 𝑤𝑤 1 Name: 6) 2 (54𝑥𝑥) =𝑥𝑥(2 4) Name: 7) (5∙ +𝑎𝑎5𝑏𝑏) + 4 =∙ 5 𝑎𝑎+𝑎𝑎 (5 + 4) Name: 8) 2 𝑥𝑥(3 + 7 ) = 2𝑥𝑥(7 + 3 )

𝑥𝑥 𝑦𝑦 𝑧𝑧 𝑥𝑥 𝑧𝑧 𝑦𝑦 Find the additive inverse and the multiplicative inverse for each problem below

Additive inverse Multiplicative inverse

9) 6

10) 1

11) 2 2.5

− 12) (given 0)

𝑚𝑚 𝑚𝑚 ≠ Complete the following table by performing the indicated operations by computing the result in the parentheses first as the order of operations requires.

Problem 1 Problem 2 Are the Results the Same? (5 + 7) + 3 5 + (7 + 3)

13) Addition ______

(10 5) 4 10 (5 4)

14) Subtraction ______− − − −

(2 3) 4 2 (3 4)

15) Multiplication ______∙ ∙ ∙ ∙

(600 ÷ 30) ÷ 5 600 ÷ (30 ÷ 5)

16) Division ______

104 Chapter 2 Use the commutative or associative properties to perform the operations in the order you find most simple. Show all your work.

17) (13 + 29) + 7 18) 5 (6 8)

19) (15 + 4) + 6 20) 4 ∙ 13∙ 5

∙ ∙ Apply distributive property and combine like terms if possible.

21) 8 (5 ) 22) 7 ( 6 + 6)

𝑛𝑛 − 𝑚𝑚 𝑘𝑘 − 𝑥𝑥 23) 6(1 + 6 ) 24) 8 (5 + 10 )

− 𝑥𝑥 − 𝑥𝑥 𝑛𝑛 25) ( 5 + 9 ) 26) ( + 1)2

− − 𝑎𝑎 𝑛𝑛 27) 4( + 7) + 8( + 4) 28) 8( + 6) 8( + 8)

𝑥𝑥 𝑥𝑥 − 𝑛𝑛 − 𝑛𝑛 29) 8 + 9( 9 + 9) 30) 4 7(1 8 )

− 𝑥𝑥 − 𝑥𝑥 𝑣𝑣 − − 𝑣𝑣 31) 10( 2) 3 32) 7(7 + 3 ) + 10(3 10 )

− 𝑥𝑥 − − 𝑣𝑣 − 𝑣𝑣 33) ( 8) (2 7) 34) 9( + 10) + 5 2 2 𝑏𝑏 𝑏𝑏 35) 2𝑥𝑥 (−10 −+ 5)𝑥𝑥 −7(6 10 3 )

𝑛𝑛 − 𝑛𝑛 − − 𝑛𝑛 − 𝑚𝑚

105 Chapter 2 CHAPTER REVIEW KEY TERMS AND CONCEPTS Look for the following terms and concepts as you work through the workbook. In the space below, explain the meaning of each of these concepts and terms in your own words. Provide examples that are not identical to those in the text or in the media lesson.

Variable

Algebraic expression

Coefficient

Term

Constant

Like terms

Identity property of addition

Identity property of multiplication

Additive Inverse property

Multiplicative Inverse property

Associative property of addition

Associative property of multiplication

Commutative property of addition

Commutative property of multiplication

Distributive property

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