beginning algebra

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Tutorial 1: How to Succeed in a Math Class Tutorial 2: Symbols and Sets of Numbers Tutorial 3: Fractions Tutorial 4: Introduction to Variable Expressions and Equations Tutorial 5: Adding Real Numbers Tutorial 6: Subtracting Real Numbers Tutorial 7: Multiplying and Dividing Real Numbers Tutorial 8: Properties of Real NumbersTutorial 9: Reading Graphs Tutorial 10: Practice Test on Tutorials 2 - 9

Tutorial 11: Simplifying Algebraic Expressions Tutorial 12: The Addition Property of Equality Tutorial 13: The Multiplication Property of Equality Tutorial 14: Solving Linear Equations (Putting it all together) Tutorial 15: Introduction to Problem SolvingTutorial 16: Percent and Problem Solving Tutorial 17: Further Problem Solving Tutorial 18: Solving Linear Inequalities Tutorial 19: Practice Test on Tutorials 11 - 18

Tutorial 20: The Rectangular Coordinate System Tutorial 21: Graphing Linear Equations Tutorial 22: Intercepts Tutorial 23: Slope Tutorial 24: Graphing Linear Inequalities Tutorial 25: Practice Test on Tutorials 20 - 24

Tutorial 26: ExponentsTutorial 27: Adding and Subtracting PolynomialsTutorial 28: Multiplying PolynomialsTutorial 29: Negative Exponents and Scientific NotationTutorial 30: Division of PolynomialsTutorial 31: Practice Test on Tutorials 26 - 30

Tutorial 32: FormulasTutorial 33: Basic GeometryTutorial 34: Central TendenciesTutorial 35: Reasoning SkillsTutorial 36: Practice Test on Tutorials 32 - 35

Tips on How to Succeed in a Math Class Yes, You Can Learn Math!!!

Get a can do attitude: If you can do it in sports, music, dance, etc., you can do it in math! Try not to let fear or negative experiences turn you off to math.

Practice a little math every day: It helps you build up your confidence and move your brain away from the panic button at test time.

Take advantage of your math class: If you are a college or high school student, realize that most colleges and universities require at least college algebra for any bachelor's degree. Some classes, like chemistry, nursing, statistics, etc. will require some algebra skills to succeed in them. If you are getting a bachelor's degree, then chances are you are going for a professional job. Most professional jobs require at least some math. Granted, some more than others, but nonetheless math (problem solving, numbers, etc...) is everywhere. So make sure that you embrace your math experience and make the most of it.

Get help outside the classroom: Go to your instructors office for extra help during office hours or by appointment. Use the WTAMU Virtual Math Lab (http://www.wtamu.edu/mathlab) as a reference as you go through your class. Anytime you need to see some more examples, want to go through some practice problems or want to take a practice test on an algebra topic, it is just a click away. See if your school has any tutors in math.

WTAMU provides the following FREE tutoring services for WT students:1. Educational Services Tutoring EST offers free one-on-one tutoring to all WT students in a variety of subjects including math Located on campus: Student Success Center, 1st floor of Classroom Center

2. SMARTHINKING SMARTHINKING is an online tutoring service that WT has contracted with to provide free live one-on-one and offline web-based tutoring in a variety of subjects including basic math, algebra, trigonometry, geometry, calculus I&II and stats for WT students. Located online: WT students can access this service by logging into and clicking on the SMARTHINKING link found on your WTClass homepage.

Online whiteboards equipped with math symbols and graphs are used to communicate between the math e-structors and students. When posting a math question to SMARTHINKING, make sure that you type in the directions, the problem, how far you have gotten on the problem and your specific questions about it. For general information about SMARTHINKING go to their website at http://www.smarthinking.com/

See if your school has a learning lab for math. Here at WTAMU, we have a Math Lab located in Classroom Center 411. It is a place where WT students can work on math homework and, as problems arise, get help. The workers will be unable to sit with you one on one for long periods of time like a tutor, however they can help you work on specific questions. Remember that they are not there to do your homework, but to answer specific questions that you have. There are also computer programs, internet connections, and videos in there to help you.

Attend class full time: Math is a sequential subject. That means that what you are learning today builds on what you learned yesterday. Even problems based on a new math concept will need some old skills to work them. (Think: Can you work problems with fractions if you dont know the multiplication tables?)

Keep up with the homework: It sounds simple but your time is limited, you have a job to go to, etc.. Think of it this way: No homework, no learning. Homework helps you practice the applications of math concepts. Its like learning how to drive: the longer you practice, the better your driving skills become and the more confidence you will have on the road. If you only read the drivers manual, youll never learn to drive with confidence and skill. We suggest you try some of the unassigned problems, too, for extra practice.

Try to understand the math problems: When you work homework problems, ask yourself what you are looking for and how you are going to get there. Dont just follow the example. Work the problem step-by-step until you know why you are doing what you are and have arrived at the solution. If you follow the what, how, and whys, youll know what to do when you see a similar problem later.

Use index cards to study tests: Heres how you do that: When studying for a test, make sure you can understand the problems on each math concept as well as work them. Then make the index cards with problems on them. Mix the index cards (yes, shuffle the cards to mix them up) and set the timer. Start working the problems in each card as it is dealt to you. Oh, yeah, hide your textbook! This will simulate a math test taking experience.

Ask questions in class: Dont be ashamed to ask questions. The instructor WILL NOT make fun of you. In fact, at least one other person may have the same question.

Ask questions outside of class: OK, so like most people, you dont want to ask questions in class, OR you think of a question too late. Then go to the instructors office and ask away.

Check homework assignments: Make sure that when you get your graded homework back you look over what you got right as well as what you missed.

Pay attention in class: Math snowballs. If you dont stay alert to the instructors presentation, you may miss important steps to learning concepts. Remember, todays information sets the foundation for tomorrows work.

Dont talk in class: If you have questions, please ask the instructor. The information you get from classmates may be mathematically wrong! And if it isnt related to math info for this class, save it for outside the classroom.

Read the math textbook and study guide: Yes, theres a reason why we ask you to spend all that money on them. If you look carefully, you will see that your book contains pages with great examples, explanations and definitions of terms. Take advantage of them.

Practice Problems

In all of the other tutorials at this Beginning Algebra website, we will have practice problems with links to the answers for you to go through. Since this tutorial did not have any math concepts there will be no practice problems for this tutorial only. We do suggest that you go back to the top and reread the tips on how to succeed in a math class and think about which one(s) will help you the most to be successful in your math class.

Beginning Algebra Tutorial 2: Symbols and Sets of NumbersLearning Objectives

After completing this tutorial, you should be able to: 1. Know what a set and an element are.2. Write a mathematical statement with an equal sign or an inequality.3. Identify what numbers belong to the set of natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.4. Use the Order Property for Real Numbers.5. Find the absolute value of a number.

Introduction

Have you ever sat in a math class, and you swear the teacher is speaking some foreign language? Well, algebra does have it's own lingo. This tutorial will go over some key definitions and phrases used when specifically working with sets of numbers as well as absolute values. Even though it may not be the exciting part of math, it is very important that you understand the language spoken in algebra class. It will definitely help you do the math that comes later. Of course, numbers are very important in math. This tutorial helps you to build an understanding of what the different sets of numbers are. You will also learn what set(s) of numbers specific numbers, like -3, 0, 100, and even (pi) belong to. Some of them belong to more than one set. I think you are ready to go forward. Let's make you a numeric set whiz kid (or adult).

Tutorial

Sets and Elements

A set is a collection of objects. Those objects are generally called members or elements of the set.

Roster Form

Roster form just lists out the elements of a set between two set brackets. For example, {January, June, July}

Equal =

To notate that two expressions are equal to each, use the symbol = between them.

Inequalities Not Equal Read left to right a < b : a is less than b a < b : a is less than or equal to b a > b : a is greater than b a > b : a is greater than or equal to b

Mathematical Statement

A mathematical statement uses the equality and inequality symbols shown above. It can be judged either true or false.

Natural (or Counting) Numbers N = {1, 2, 3, 4, 5, ...}

Makes sense, we start counting with the number 1 and continue with 2, 3, 4, 5, and so on.

Whole Numbers {0, 1, 2, 3, 4, 5, ...}

The only difference between this set and the one above is that this set not only contains all the natural numbers, but it also contains 0, where as 0 is not an element of the set of natural numbers.

Integers Z = {..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ...}

This set adds on the negative counterparts to the already existing whole numbers (which, remember, includes the number 0). The natural numbers and the whole numbers are both subsets of integers.

Rational Numbers Q = {| a and b are integers and}

In other words, a rational number is a number that can be written as one integer over another. Be very careful. Remember that a whole number can be written as one integer over another integer. The integer in the denominator is 1 in that case. For example, 5 can be written as 5/1. The natural numbers, whole numbers, and integers are all subsets of rational numbers.

Irrational Numbers I = {x | x is a real number that is not rational}

In other words, an irrational number is a number that can not be written as one integer over another. It is a non-repeating, non-terminating decimal. One big example of irrational numbers is roots of numbers that are not perfect roots - for exampleor. 17 is not a perfect square - the answer is a non-terminating, non-repeating decimal, which CANNOT be written as one integer over another. Similarly, 5 is not a perfect cube. It's answer is also a non-terminating, non-repeating decimal. Another famous irrational number is (pi). Even though it is more commonly known as 3.14, that is a rounded value for pi. Actually it is 3.1415927... It would keep going and going and going without any real repetition or pattern. In other words, it would be a non terminating, non repeating decimal, which again, can not be written as a rational number, 1 integer over another integer.

Real Numbers R = {x | x corresponds to point on the number line}

Any number that belongs to either the rational numbers or irrational numbers would be considered a real number. That would include natural numbers, whole numbers and integers.

Real Number Line

Above is an illustration of a number line. Zero, on the number line, is called the origin. It separates the negative numbers (located to the left of 0) from the positive numbers (located to the right of 0). I feel sorry for 0, it does not belong to either group. It is neither a positive or a negative number.

Order Property for Real Numbers

Given any two real numbers a and b, if a is to the left of b on the number line, then a < b. If a is to the right of b on the number line, then a > b.

Absolute Value

Most people know that when you take the absolute value of ANY number (other than 0) the answer is positive. But, do you know WHY? Well, let me tell you why! The absolute value of x, notated |x|, measures the DISTANCE that x is away from the origin (0) on the real number line. Aha! Distance is always going to be positive (unless it is 0) whether the number you are taking the absolute value of is positive or negative. The following are illustrations of what absolute value means using the numbers 3 and -3:

Example 1: Replace ? with , or = . 3 ? 5

Since 3 is to the left of 5 on the number line, then 3 < 5.

Example 2: Replace ? with , or = . 7.41 ? 7.41

Since 7.41 is the same number as 7.41, then 7.41 = 7.41.

Example 3: Replace ? with , or = . 2.5 ? 1.5

Since 2.5 is to the right of 1.5 on the number line, then 2.5 > 1.5.

Example 4: Is the following mathematical statement true or false? 2 > 7

Since 2 is to the left of 7 on the number line, then 2 < 7. Therefore, the given statement is false.

Example 5: Is the following mathematical statement true or false? 5 > 5

Since 5 is the same number as 5 and the statement includes where the two numbers are equal to each other, then this statement is true.

Example 6: Write the sentence as a mathematical statement. 2 is less than 5.

Reading it left to right we get: 2 is less than 5 2 -1

Example 10: Write the sentence as a mathematical statement. 5 is not equal to 2.

Reading it left to right we get: 5 is not equal to 2

Example 11: List the elements of the following sets that are also elements of the given set {-4, 0, 2.5, ,,, 11/2, 7} Natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.

Natural numbers? The numbers in the given set that are also natural numbers are {, 7}. Note that simplifies to be 5, which is a natural number.

Whole numbers? The numbers in the given set that are also whole numbers are {0,, 7}.

Integers? The numbers in the given set that are also integers are {-4, 0,, 7}.

Rational numbers? The numbers in the given set that are also rational numbers are {-4, 0, 2.5, , 11/2, 7}.

Irrational numbers? The numbers in the given set that are also irrational numbers are {, }. These two numbers CANNOT be written as one integer over another. They are non-repeating, non-terminating decimals.

Real numbers? The numbers in the given set that are also real numbers are {-4, 0, 2.5,,,, 11/2, 7}.

Example 12: Replace ? with , or = . |-2.5| ? |2.5|

Since |-2.5| = 2.5 and |2.5| = 2.5, then the two expressions are equal to each other: |-2.5| = |2.5|

Example 13: Replace ? with , or = . -3 ? |3|

First of all, |3| = 3 .

Since -3 is to the left of 3 on the number line, then -3 1.

Practice Problems

These are practice problems to help bring you to the next level. It will allow you to check and see if you have an understanding of these types of problems. Math works just like anything else, if you want to get good at it, then you need to practice it. Even the best athletes and musicians had help along the way and lots of practice, practice, practice, to get good at their sport or instrument. In fact there is no such thing as too much practice. To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that problem. At the link you will find the answer as well as any steps that went into finding that answer.

Practice Problems 1a - 1c: Replace ? with < , > , or = . 1a. 5 ? 0 (answer/discussion to 1a) 1b. (answer/discussion to 1b)

1c. -2 ? 2(answer/discussion to 1c)

Practice Problems 2a - 2b: Is the following mathematical statement true or false? 2a. -3 4 (answer/discussion to 2b)

Practice Problems 3a - 3c: Write each sentence as a mathematical statement. 3a. - 4 is less than 0. (answer/discussion to 3a) 3b. 3 is not equal to -3. (answer/discussion to 3b)

3c. 5 is greater than or equal to -5. (answer/discussion to 3c)

Practice Problems 4a - 4f: List the elements of the following set that are also elements of the given set: {-1.5, 0, 2, , } 4a. Natural numbers (answer/discussion to 4a)4b. Whole numbers (answer/discussion to 4b)

4c. Integers (answer/discussion to 4c)4d. Rational numbers (answer/discussion to 4d)

4e. Irrational numbers (answer/discussion to 4e)4f. Real numbers (answer/discussion to 4f)

Answer/Discussion to Practice Problems Tutorial 2: Symbols and Sets of Numbers Answer/Discussion to 1a 5 ? 0

Since 5 is to the right of 0 on the number line, then 5 > 0. (return to problem 1a)

Answer/Discussion to 1b

First of all,. Next we have. Since both absolute values equal the same number , then. (return to problem 1b)

Answer/Discussion to 1c -2 ? 2

Since -2 is to the left of 2 on the number line, then -2 < 2. (return to problem 1c)

Answer/Discussion to 2a -3 4

Since 2 is to the left of 4 on the number line, then 2 Virtual Math Lab > Beginning Algebra > Tutorial 3: Fractions

Answer/Discussion to 1a 100

*Rewrite 100 as a product of primes

(return to problem 1a)

Answer/Discussion to 2a

Step 1: Write the numerator and denominator as a product of prime numbers.

*Rewrite 75 as a product of primes *Rewrite 30 as a product of primes

Step 2: Use the Fundamental Principle of Fractions to cancel out the common factors.

*Div. the common factors of 3 and 5 out of both num. and den.



Step 1: Write the numerator and denominator as a product of prime numbers.

*Rewrite 9 as a product of primes

Step 2: Use the Fundamental Principle of Fractions to cancel out the common factors.

*There are no common factors to divide out

(return to problem 2b)


*Write as prod. of num. over prod. of den. *Div. the common factors of 2 and 5 out of both num. and den.



*Rewrite as the mult. of the reciprocal *Write as prod. of num. over prod. of den. *Div. the common factor of 7 out of both num. and den.


Answer/Discussion to 3c

Step 1: Combine the numerators together. AND Step 2: Put the sum or difference found in step 1 over the common denominator.

*Write the sum over the common den.

Step 3: Reduce to lowest terms if necessary.

*Div. the common factor of 7 out of both num. and den.

(return to problem 3c)

Answer/Discussion to 3d

Rewriting the numbers as fractions we get:

*Rewrite whole number 5 as 5/1 *Rewrite mixed number 2 1/4 as 9/4

Step 1: Find the Least Common Denominator (LCD) for all denominators.

The first fraction has a denominator of 1 and the second fraction has a denominator of 4. What is the smallest number that is divisible by both 1 and 4. If you said 4, you are correct? Therefore, the LCD is 4.

Step 2: Rewrite fractions into equivalent fractions with the common denominator.

*What number times 1 will result in 4? *Multiply num. and den. by 4.

The fraction 9/4 already has a denominator of 4, so we do not have to rewrite it.

Step 3: Add and subtract the fractions with common denominators as described above.

*Write the difference over the common den.

Note that this fraction is in simplest form. There are no common factors that we can divide out of the numerator and denominator

(return to problem 3d)

Answer/Discussion to 3e

Step 1: Find the Least Common Denominator (LCD) for all denominators.

The first fraction has a denominator of 4, the second has a denominator of 5, and the third has a denominator of 10. What is the smallest number that is divisible by 4, 5, and 10? If you said 20, you are correct? Therefore, the LCD is 20.

Step 2: Rewrite fractions into equivalent fractions with the common denominator.

Writing an equivalent fraction of 3/4 with the LCD of 20 we get:






Step 3: Add and subtract the fractions with common denominators as described above.

*Write the sum and difference over the common den.

1Tutorial 4: Introduction to Variable Expressions and Equations

WTAMU > Virtual Math Lab > Beginning Algebra Learning Objectives

After completing this tutorial, you should be able to: 1. Evaluate an exponential expression.2. Simplify an expression using the order of operations.3. Evaluate an expression.4. Know when a number is solution to an equation or not.5. Translate an english expression into a math expression.6. Translate an english statement in to a math equation.

Introduction

This tutorial will go over some key definitions and phrases used when specifically working with algebraic expressions as well as evaluating them. We will also touch on the order of operations. It is very IMPORTANT that you understand some of the math lingo that is used in an algebra class, otherwise it may all seem Greek to you. Knowing the terms and concepts on this page will definitely help you build an understanding of what a variable is and get you more comfortable working with them. Variables are a HUGE part of algebra, so it is very important for you to feel at ease around them in order to be successful in algebra. So let's get going and help you get on the road to being variable savvy.

Tutorial

Exponential Notation

An exponent tells you how many times that you write a base in a PRODUCT. In other words, exponents are another way to write MULTIPLICATION. Lets illustrate this concept by rewriting the product (4)(4)(4) using exponential notation:

In this example, 4 represents the base and 3 is the exponent. Since 4 was written three times in a product, then our exponent is 3. We always write our exponent as a smaller script found at the top right corner of the base. You can apply this idea in the other direction. Lets say you have it written in exponential notation and you need to evaluate it. The exponent will tell you how many times you write the base out in a product. For example if you had 7 as your base and 2 as your exponent and you wanted to evaluate out you could write it out like this:

Example 1: Evaluate

In this problem, what is the base? If you said 5, you are correct! What is the exponent? If you said 4, you are right! Lets rewrite it as multiplication and see what we get for an answer:

*Rewrite the base 5, four times in a product *Multiply

Example 2: Evaluate

In this problem, what is the base? If you said 7, you are correct! What is the exponent? If you said 1, you are right! Lets rewrite it as multiplication and see what we get for an answer:

*Rewrite the base 7, one time in a product

Example 3: Evaluate

In this problem, what is the base? If you said 1/3, you are correct! What is the exponent? If you said 2, you are right! Lets rewrite it as multiplication and see what we get for an answer:

*Rewrite the base 1/3, two times in a product *Multiply

Note that when you have a 2 as an exponent, which is also known as squaring the base. In this problem we could say that we are looking for 1/3 squared.

Order of Operations Please Parenthesis or grouping symbols Excuse Exponents (and radicals) My Dear Multiplication/Division left to right Aunt Sally Addition/Subtraction left to right

When you do have more than one mathematical operation, you need to use the order of operations as listed above. You may have already heard of the saying "Please Excuse My Dear Aunt Sally". It is just a way to help you remember the order you need to go in when applying the order of operations.

Example 4: Simplify.

*Multiply *Add *Subtract

Example 5: Simplify

*Inside ( ) *Exponent *Multiply *Add


Note that the absolute value symbol | | is a fancy grouping symbol. In terms of the order of operations, it would be including on the first line with parenthesis. So in this problem, the first thing we need to do is work the inside of the absolute value. And then go from there.

*Inside | | *Exponent *Add in num. and subtract in den.

Variable

A variable is a letter that represents a number. Don't let the fact that it is a letter throw you. Since it represents a number, you treat it just like you do a number when you do various mathematical operations involving variables. x is a very common variable that is used in algebra, but you can use any letter (a, b, c, d, ....) to be a variable.

Algebraic Expressions

An algebraic expression is a number, variable or combination of the two connected by some mathematical operation like addition, subtraction, multiplication, division, exponents, and/or roots. 2x + y, a/5, and 10 - r are all examples of algebraic expressions.

Evaluating an Expression

You evaluate an expression by replacing the variable with the given number and performing the indicated operation.

Value of an Expression

When you are asked to find the value of an expression, that means you are looking for the result that you get when you evaluate the expression.

So keep in mind that vary means to change - a variable allows an expression to take on different values, depending on the situation. For example, the area of a rectangle is length times width. Well, not every rectangle is going to have the same length and width, so we can use an algebraic expression with variables to represent the area and then plug in the appropriate numbers to evaluate it. So if we let the length be the variable l and width be w, we can use the expression lw. If a given rectangle has a length of 4 and width of 3, we would evaluate the expression by replacing l with 4 and w with 3 and multiplying to get a value of 4 times 3 or 12. Lets step through some examples that help illustrate these ideas.

Example 7: Evaluate the expression when x = 4, y = 6, z = 8.

Plugging in the corresponding value for each variable and then evaluating the expression we get:

*Plug in 4 for x, 6 for y, and 8 for z *Exponent *Multiply *Add *Subtract

Example 8: Evaluate the expression when x = 3, y = 5, and z = 7.


*Plug in 3 for x, 5 for y, and 7 for z *Exponent *Multiply *Add

Equation Two expressions set equal to each other.

Solution A value, such that, when you replace the variable with it, it makes the equation true. (the left side comes out equal to the right side)

Solution Set Set of all solutions.

Example 9: Is 2 a solution of?

Replacing x with 2 we get:

*Plug in 2 for x *Evaluate both sides

Is 2 a solution? Since we got a TRUE statement (7 does in fact equal 7), then 2 is a solution to this equation.

Example 10: Is 5 a solution of?



Is 5 a solution? Since we got a FALSE statement (16 does not equal 14), then 5 is not a solution.

Translating an English Phrase Into an Algebraic Expression

Sometimes, you find yourself having to write out your own algebraic expression based on the wording of a problem. In that situation, you want to 1. read the problem carefully,2. pick out key words and phrases and determine their equivalent mathematical meaning,3. replace any unknowns with a variable, and4. put it all together in an algebraic expression.The following are some key words and phrases and their translations:

Addition: sum, plus, add to, more than, increased by, total

Subtraction: difference of, minus, subtracted from, less than, decreased by, less

Multiplication: product, times, multiply, twice, of

Division: quotient divide, into, ratio

Example 11: Write the phrase as an algebraic expression. The sum of a number and 10.

In this example, we are not evaluating an expression, so we will not be coming up with a value. However, we are wanting to rewrite it as an algebraic expression. It looks like the only reference to a mathematical operation is the word sum. So, what operation will we have in this expression? If you said addition, you are correct!!! The phrase 'a number' indicates that it is an unknown number. There was no specific value given to it. So we will replace the phrase 'a number' with the variable x. We want to let our variable represent any number that is unknown Putting everything together, we can translate the given english phrase with the following algebraic expression:

The sum of a number and 10

*'sum' = + *'a number' = variable x

Example 12: Write the phrase as an algebraic expression. The product of 5 and a number.

Again, we are wanting to rewrite this as an algebraic expression, not evaluate it. This time, the phrase that correlates with our operation is 'product' - so what operation will we be doing this time? If you said multiplication, you are right on. Again, we have the phrase 'a number', which again is going to be replaced with a variable, since we do not know what the number is. Lets see what we get for this answer:

The product of 5 and a number

*'product' = multiplication *'a number' = variable x

Translating a Sentence into an Equation

Since an equation is two expressions set equal to each other, we will be using the same mathematical translations we did above. The difference is we will have an equal sign between the two expressions. The following are some key words and phrases that translate into an equal sign (=):

Equal Sign (=) : equals, gives, is, yields, amounts to, is the same as

Example 13: Write the sentence as an equation. Let x represent the unknown number. The quotient of 3 and a number is .

Do you remember what quotient translates into? If you said division, you are doing great. 'Is' will be replaced by the symbol =. Lets put together everything going left to right:

The quotient of 3 and a number is

Example 14: Write the sentence as an equation. Let x represent the unknown number. 7 less than 3 times a number is the same as 0.

Do you remember what less than translates into? If you said subtraction, you are doing great. Do you remember what times translates into? If you said multiplication, you are correct. 'Is the same as' will be replaced by the symbol =. Lets put together everything going left to right:

7 less than 3 times a number is the same as 0.

Practice Problems


Practice Problems 1a - 1b: Evaluate. 1a. (answer/discussion to 1a)1b. (answer/discussion to 1b)

Practice Problems 2a - 2b: Simplify each expression. 2a. (answer/discussion to 2a)2b. (answer/discussion to 2b)

Practice Problem 3a: Evaluate the expression if x = 1, y = 2, and z = 3. 3a. (answer/discussion to 3a)

Practice Problems 4a - 4b: Decide whether the given number is a solution of the given equation. 4a. Is 0 a solution to? (answer/discussion to 4a)4b. Is 8 a solution to ? (answer/discussion to 4b)

Practice Problems 5a - 5b: Write each phrase as an algebraic expression. Let x represent the unknown number. 5a. 9 less than 5 times a number. (answer/discussion to 5a)5b. The product of 12 and a number. (answer/discussion to 5b)

Practice Problems 6a - 6b: Write each sentence as an equation. Let x represent the unknown number. 6a. The sum of 10 and 4 times a number is the same as 18. (answer/discussion to 6a)

6b. The quotient of a number and 9 is 1/3. (answer/discussion to 6b)

Answer/Discussion to Practice Problems Tutorial 4: Introduction to Variable Expressions and Equations

WTAMU > Virtual Math Lab > Beginning Algebra > Tutorial 4: Introduction to Variable Expressions and Equations


In this problem, what is the base? If you said 2, you are correct! What is the exponent? If you said 5, you are right! Let's rewrite it as multiplication and see what we get for an answer:

*Rewrite the base 2, five times in a product *Multiply



In this problem, what is the base? If you said 1/6, you are correct! What is the exponent? If you said 3, you are right! Let's rewrite it as multiplication and see what we get for an answer:

*Rewrite the base 1/6, three times in a product *Multiply



*Inside ( ) *Multiply *Add



*Inside absolute value *Exponent *Add num and subtract den. *Simplify fraction




*Plug in 1 for x, 2 for y, and 3 for z *Inside parenthesis *Exponent in [ ] *Add in [ ] *Multiply





Is 0 a solution? Since we got a FALSE statement (7 does not equal 9), then 0 is not a solution.





Is 8 a solution? Since we got a TRUE statement (6 does equal 6), then 8 is a solution.


Answer/Discussion to 5a 9 less than 5 times a number.

What operation will we replace less than with? If you said subtraction you are correct!!! What operation will we replace times with? If you said multiplication you are correct!!! The phrase 'a number' indicates that it is an unknown number. There was no specific value given to it. So we will replace the phrase 'a number' with the variable x. We want to let our variable represent any number that is unknown Putting everything together we can translate the given english phrase with the following algebraic expression: 9 less than 5 times a number


Answer/Discussion to 5b The product of 12 and a number.

What operation will we replace product with? If you said multiplication you are correct!!! The phrase 'a number' indicates that it is an unknown number. There was no specific value given to it. So we will replace the phrase 'a number' with the variable x. We want to let our variable represent any number that is unknown Putting everything together we can translate the given english phrase with the following algebraic expression: The product of 12 and a number


Answer/Discussion to 6a The sum of 10 and 4 times a number is the same as 18.

Do you remember what sum translates into? If you said addition, you are doing great. Do you remember what times translates into? If you said multiplication, you are doing great. 'Is the same as' will be replaced by the symbol =. Let's put together everything going left to right: The sum of 10 and 4 times a number is the same as 18


Answer/Discussion to 6b The quotient of a number and 9 is 1/3.

Do you remember what quotient translates into? If you said division, you are correct. 'Is' will be replaced by the symbol =. Let's put together everything going left to right: The quotient of a number and 9 is 1/3.

Tutorial 5: Adding Real Numbers


After completing this tutorial, you should be able to: 1. Add real numbers that have the same sign.2. Add real numbers that have different signs.3. Find the additive inverse or the opposite of a number.

Introduction

This tutorial reviews adding real numbers as well as finding the additive inverse or opposite of a number . I have the utmost confidence that you are familiar with addition, but sometimes the rules for negative numbers (yuck!) get a little mixed up from time to time. So, it is good to go over them to make sure you have them down.

Tutorial

Adding Real Numbers

Adding Real Numbers with the Same Sign

Step 1: Add the absolute values. If you need a review of absolute values, go to Tutorial 2: Symbols and Sets of Numbers.

Step 2: Attach their common sign to sum. In other words: If both numbers that you are adding are positive, then you will have a positive answer. If both numbers that you are adding are negative then you will have a negative answer.

Example 1: Add -6 + (-8).

-6 + (-8) = -14 The sum of the absolute values would be 14 and their common sign is -. That is how we get the answer of -14. You can also think of this as money. I know we can all relate to that. Think of the negative as a loss. In this example, you can think of it as having lost 6 dollars and then having lost another 8 dollars for a total loss of 14 dollars.

Example 2: Add -5.5 + (-8.7).

-5.5 + (-8.7) = -14.2 The sum of the absolute values would be 14.2 and their common sign is -. That is how we get the answer of -14.2. You can also think of this as money - I know we can all relate to that. Think of the negative as a loss. In this example, you can think of it as having lost 5.5 dollars and then having lost another 8.7 dollars for a total loss of 14.2 dollars.

Adding Real Numbers with Opposite Signs

Step 1: Take the difference of the absolute values. If you need a review of absolute values, go to Tutorial 2: Symbols and Sets of Numbers.

Step 2: Attach the sign of the number that has the higher absolute value. Which did you have more of, negative or positive? If the number with the larger absolute value is negative, then your sum is negative. In other words you have more negative than positive. If the number with the larger absolute value was positive, then your sum is positive. In other words you have more positive than negative.

Example 3: Add -8 + 6.

-8 + 6 = -2. The difference between 8 and 6 is 2 and the sign of 8 (the larger absolute value) is -. That is how we get the answer of -2. Thinking in terms of money: we lost 8 dollars and got back 6 dollars, so we are still in the hole 2 dollars.

Example 4: Add.

*Mult. top and bottom of first fraction by 2 to get the LCD of 6 *Take the difference of the numerators and write over common denominator 6 *Reduce fraction

The difference between 4/6 and 1/6 is 3/6 = 1/2 and the sign of 4/6 (the larger absolute value) is +. That is how we get the answer of 1/2. Thinking in terms of money: we had 2/3 of a dollar and lost 1/6 of a dollar, so we would come out ahead 1/2 of a dollar. Note that if you need help on fractions, go back to Tutorial 3: Fractions.

Example 5: Add -10 + 7 + (-2) + 5.

In this example, we are needing to combine more than two numbers together, but we will still follow the same thought process we do if there are only two numbers. Im going to go ahead and step us through it going left to right.

* -10 + 7 = -3 *-3 + (-2) = -5

Example 6: Add.

In this addition problem, we have some absolute values thrown into the mix. Remember that we need to do what is inside the absolute values (grouping symbol) first and then add those numbers together. If you need a review on order of operations go to Tutorial 4: Introduction to Variable Expressions and Equations.

*Add inside the absolute values *Evaluate the absolute values *Add

Opposites

Opposites are two numbers that are on opposite sides of the origin (0) on the number line, but have the same absolute value. In other words, opposites are the same distance away from the origin, but in opposite directions. The opposite of x is the number -x. Keep in mind that the opposite of 0 is 0. The following is an illustration of opposites using the numbers 3 and -3:

Double Negative Property For every real number a, -(-a) = a.

When you see a negative sign in front of an expression, you can think of it as taking the opposite of it. For example, if you had -(-2), you can think of it as the opposite of -2. Since a number can only have one of two signs, either a '+' or a '-', then the opposite of a negative would have to be positive. So, -(-2) = 2. Example 7: Write the additive inverse or opposite of 1.5.

The opposite of 1.5 is -1.5, since both of these numbers have the same absolute value but are on opposite sides of the origin on the number line.

Example 8: Write the opposite of -3.

The opposite of -3 is 3, since both of these numbers have the same absolute value but are on opposite sides of the origin on the number line.

Example 9: Simplify -(-10).

When you have a negative in front of a parenthesis like this, it is another way to write that you need to find the additive inverse or opposite. Since the opposite of a negative is a positive, our answer is 10.

Example 10: Simplify -|-5.2|.

-|-5.2| = -(5.2) = -5.2*Evaluate the absolute value *Find the opposite

Practice Problems


Practice Problems 1a - 1d: Add. 1a. -15 + 7 (answer/discussion to 1a)1b. (answer/discussion to 1b)

1c. 3.2 + (-1.3) + (- 4.1) (answer/discussion to 1c)1d. |- 4 + (-3) + 2| (answer/discussion to 1d)

Practice Problems 2a - 2b: Find the additive inverse or opposite. 2a. (answer/discussion to 2a)2b. -20 (answer/discussion to 2b)

Practice Problems 3a - 3b: Simplify. 3a. -(- 4) (answer/discussion to 3a)3b. (answer/discussion to 3b)

Answer/Discussion to Practice Problems Tutorial 5: Adding Real Numbers

WTAMU > Virtual Math Lab > Beginning Algebra > Tutorial 5: Adding Real Numbers

Answer/Discussion to 1a -15 + 7

-15 + 7 = -8 The difference between 15 and 7 is 8 and the sign of 15 (the larger absolute value) is -, that is how we get the answer of -8.



*Mult. top and bottom of first fraction by 2 to get the LCD of 10 *Add the numerators and write over common denominator 10

The sum of the absolute values would be 13/10 and their common sign is -, that is how we get the answer of -13/10.


Answer/Discussion to 1c 3.2 + (-1.3) + (- 4.1)

*3.2 + (-1.3) = 1.9 *1.9 + (-4.1) = -2.2


Answer/Discussion to 1d |- 4 + (-3) + 2|

*- 4 + (-3) = -7 * - 7 + 2 = -5 *Evaluate the absolute value

(return to problem 1d)

Answer/Discussion to 2a 1/2

The opposite of 1/2 is -1/2, because both of these numbers have the same absolute value, but are on opposite sides of the origin on the number line.


Answer/Discussion to 2b -20

The opposite of -20 is 20, because both of these numbers have the same absolute value, but are on opposite sides of the origin on the number line.


Answer/Discussion to 3a -(- 4)

Since the opposite of a negative is a positive, our answer is 4.



*Evaluate the absolute value

Tutorial 6: Subtracting Real Numbers


After completing this tutorial, you should be able to: 1. Subtract real numbers that have the same sign.2. Subtract real numbers that have different signs.3. Simplify an expression that has subtraction in it using the order of operations.

Introduction

This tutorial reviews subtracting real numbers and intertwines that with some order of operation and evaluation problems. I have the utmost confidence that you are familiar with subtraction, but sometimes the rules for negative numbers (yuck!) get a little mixed up from time to time. So, it is good to go over them to make sure you have them down. Even in this day and age of calculators, it is very important to know these basic rules of operations on real numbers. Even if you are using a calculator, you are the one that is putting the information into it, so you need to know things like when you are subtracting versus adding and the order that you need to put it in. Also, if you are using a calculator you should have a rough idea as to what the answer should be. You never know, you may hit a wrong key and get a wrong answer (it happens to the best of us). Also, your batteries in your calculator may run out and you may have to do a problem by hand (scary!!!). You want to be prepared for those Murphy's Law moments.

Tutorial

Subtracting Real Numbers a - b = a + (-b) or a - (-b) = a + b

In other words, to subtract b, you add the opposite of b. Now, you do not have to write it out like this if you are already comfortable with it. This just gives you the thought behind it.

Example 1: Subtract -3 - 5.

-3 - 5 = -3 + (-5) = -8. Subtracting 5 is the same as adding a -5. Once it is written as addition, we just follow the rules for addition, as shown in Tutorial 5: Adding Real Numbers, to complete for an answer of -8.

Example 2: Subtract -3 - (-5).

-3 - (-5) = -3 + 5 = 2. Subtracting -5 is the same as adding 5. Once it is written as addition, we just follow the rules for addition, as shown in Tutorial 5: Adding Real Numbers, to complete for an answer of 2.

Example 3: Subtract.

*Rewrite as addition *Mult. top and bottom of 1st fraction by 2 and 2nd by 3 to get theLCD of 6 *Take the difference of the numerators and write over common denominator 6

The difference between 14/6 and 3/6 is 11/6 and the sign of 14/6 (the larger absolute value) is -. That is how we get the answer -11/6


Since we have several operations going on in this problem, we will have to use the order of operations to make sure that we get the correct answer. If you need to review the order of operations go to Tutorial 4: Operations of Real Numbers.

*Exponent *Multiply *25 - 8 = 17

Example 5: Simplify


*Eval. inside absolute value *Exponent *Multiplication *7 + 6 = 13 *13 - 15 = -2

Example 6: Evaluate the expression if x = -2 and y = 5.

To review evaluating an expression go to Tutorial 4: Introduction to Variable Expressions and Equations. Plugging -2 for x and 5 for y and simplifying we get:

*Plug in -2 for x and 5 for y *Rewrite num. as addition of opposite *Add *Simplify fraction

Example 7: Is -1 a solution of -x + 4 = 6 + x?

If you need a review on what a solution to an equation is go to Tutorial 4: Introduction to Variable Expressions and Equations. Replacing x with -1 we get:

*Plug in -1 for x *Take the opposite of -1 *Add

Is -1 a solution? Since we got a TRUE statement (5 does in fact equal 5), then -1 is a solution to this equation.

Practice Problems


Practice Problems 1a - 1b: Subtract. 1a. -10 - (-2) (answer/discussion to 1a)1b. - 4.1 - 5.3 (answer/discussion to 1b)

Practice Problems 2a - 2b: Simplify. 2a. (answer/discussion to 2a)2b. (answer/discussion to 2b)

Practice Problem 3a: Evaluate the expression when x = 2 and y = -2. 3a. (answer/discussion to 3a)

Practice Problem 4a: Is -2 a solution to the given equation? 4a. (answer/discussion to 4a)

Answer/Discussion to Practice Problems Tutorial 6: Subtracting Real Numbers

WTAMU > Virtual Math Lab > Beginning Algebra > Tutorial 6: Subtracting Real Numbers

Answer/Discussion to 1a -10 - (-2)

-10 - (-2) = -10 + 2 = -8. Subtracting -2 is the same as adding 2. Once it is written as addition, we just following the rules for addition to complete.


Answer/Discussion to 1b - 4.1 - 5.3

- 4.1 - 5.3 = - 4.1 + (-5.3) = -9.4. Subtracting 5.3 is the same as adding -5.3. Once it is written as addition, We just following the rules for addition to complete.




*Exponent *Multiplication *27 + (-2) = 25




*Eval. inside of absolute value *Exponent *Multiplication *16 - 4 = 12



To review evaluating an expression go to Tutorial 4: Introduction to Variable Expressions and Equations. Plugging 2 for x and -2 for y and simplifying we get:

*Plug in 2 for x and -2 for y



If you need a review on what a solution to an equation is go to Tutorial 4: Introduction to Variable Expressions and Equations. Replacing x with -2 we get:

*Plug in -2 for x

Is -2 a solution? Since we got a TRUE statement (7 does in fact equal 7), then -2 is a solution to this equation.

Tutorial 7: Multiplying and Dividing Real Numbers


After completing this tutorial, you should be able to: 1. Find the reciprocal of a number.2. Multiply positive and negative numbers.3. Divide positive and negative numbers.4. Multiply by zero.5. Know that dividing by zero is undefined.

Introduction

This tutorial reviews multiplying and dividing real numbers and intertwines that with some order of operation and evaluation problems. It also reminds you that dividing by 0 results in an undefined answer. In other words, it is a big no, no. I have the utmost confidence that you are familiar with multiplication and division, but sometimes the rules for negative numbers (yuck!) get a little mixed up from time to time. So, it is good to go over them to make sure you have them down.

Tutorial

Multiplicative Inverse (or reciprocal) For each real number a, except 0, there is a unique real number such that

In other words, when you multiply a number by its multiplicative inverse the result is 1. A more common term used to indicate a multiplicative inverse is the reciprocal. A multiplicative inverse or reciprocal of a real number a (except 0) is found by flipping a upside down. The numerator of a becomes the denominator of the reciprocal of a and the denominator of a becomes the numerator of the reciprocal of a.

Example 1: Write the reciprocal (or multiplicative inverse) of -3.

The reciprocal of -3 is -1/3, since -3(-1/3) = 1. When you take the reciprocal, the sign of the original number stays intact. Remember that you need a number that when you multiply times the given number you get 1. If you change the sign when you take the reciprocal, you would get a -1, instead of 1, and that is a no no.

Example 2: Write the reciprocal (or multiplicative inverse) of 1/5.

The reciprocal of 1/5 is 5, since 5(1/5) = 1.

Quotient of Real Numbers If a and b are real numbers and b is not 0, then

Multiplying or Dividing Real Numbers

Since dividing is the same as multiplying by the reciprocal, dividing and multiplying have the same sign rules. Step 1: Multiply or divide their absolute values. Step 2: Put the correct sign. If the two numbers have the same sign, the product or quotient is positive. If they have opposite signs, the product or quotient is negative.

Example 3: Find the product (-4)(3).

(-4)(3) = -12. The product of the absolute values 4 x 3 is 12 and they have opposite signs, so our answer is -12.

Example 4: Find the product.

*Mult. num. together *Mult. den. together *(-)(-) = (+) *Reduce fraction

The product of the absolute values 2/3 x 9/10 is 18/30 = 3/5 and they have the same sign, so that is how we get the answer 3/5. Note that if you need help on fractions go to Tutorial 3: Fractions

Example 5: Find the product

Working this problem left to right we get:

*(3)(-2) = -6 *(-6)(-10) = 60

Example 6: Divide (-10)/(-2).

(-10)/(-2) = 5 The quotient of the absolute values 10/2 is 5 and they have the same signs, so our answer is 5.

Example 7: Divide.

*Div. is the same as mult. by reciprocal *Mult. num. together *Mult. den. together *(+)(-) = - *Reduce fraction

The quotient of the absolute values 4/5 and 8 is 4/40 = 1/10 and they have opposite signs, so our answer is -1/10. Note that if you need help on fractions go to Tutorial 3: Fractions

Multiplying by and Dividing into Zero a(0) = 0 and 0/a = 0 (when a does not equal 0)

In other words, zero (0) times any real number is zero (0) and zero (0) divided by any real number other than zero (0) is zero (0).

Example 8: Multiply 0().

0() = 0. Multiplying any expression by 0 results in an answer of 0.

Example 9: Divide 0/5.

0/5 = 0. Dividing 0 by any expression other than 0 results in an answer of 0.

Dividing by Zero a/0 is undefined

Zero (0) does not go into any number, so whenever you are dividing by zero (0) your answer is undefined. Example 10: Divide 5/0.

5/0 = undefined. Dividing by 0 results in an undefined answer.



*Evaluate inside the absolute values *Subtract *(-)/(-) = +

Example 12: Evaluate the expression if x = -2 and y = - 4.

To review evaluating an expression go to Tutorial 4: Introduction to Variable Expressions and Equations. Plugging -2 for x and - 4 for y and simplifying we get:

*Plug in -2 for x and -4 for y *Exponent *Multiply *Add

Practice Problems


Practice Problems 1a - 1c: Multiply. 1a. (-2)(-25) (answer/discussion to 1a)1b. (0)(-100) (answer/discussion to 1b)

1c. (-2)(3)(5) (answer/discussion to 1c)

Practice Problems 2a - 2c: Divide. 2a. (answer/discussion to 2a)2b. (answer/discussion to 2b)

2c. (answer/discussion to 2c)

Practice Problem 3a: Simplify. 3a. (answer/discussion to 3a)

Practice Problem 4a: Evaluate the expression whenx = 5 and y = -5. 4a. (answer/discussion to 4a)

Answer/Discussion to Practice Problems Tutorial 7: Multiplying and Dividing Real Numbers

WTAMU > Virtual Math Lab > Beginning Algebra > Tutorial 7: Multiplying and Dividing Real Numbers

Answer/Discussion to 1a (-2)(-25)

(-2)(-25) = 50. The product of the absolute values 2 and 25 is 50 and they have the same sign, so that is how we get the answer 50.


Answer/Discussion to 1b (0)(-100)

(0)(-100) = 0.


Answer/Discussion to 1c (-2)(3)(5)

Working it left to right we get:

(-2)(3)(5) = (-6)(5) = -30



(-25)/(5) = -5. The quotient of the absolute values (25)/(5) = 5 and they have opposite signs, so that is how we get the answer -5.



7/0 = undefined.


Answer/Discussion to 2c

*Div. is the same as mult. by reciprocal *Mult. num. together *Mult. den. together *(-)(-) = - *Reduce fraction




*Evaluate inside the absolute values *Multiply *Add *Reduce fraction



To review evaluating an expression go to Tutorial 4: Introduction to Variable Expressions and Equations. Plugging 5 for x and -5 for y and simplifying we get:

*Plug in 5 for x and -5 for y *Exponent *Multiply *Subtract

Tutorial 8: Properties of Real Numbers


After completing this tutorial, you should be able to: 1. Identify and use the addition and multiplication commutative properties.2. Identify and use the addition and multiplication associative properties.3. Identify and use the distributive property.4. Identify and use the addition and multiplication identity properties.5. Identify and use the addition and multiplication inverse properties.

Introduction

It is important to be familiar with the properties in this tutorial. They lay the foundation that you need to work with equations, functions, and formulas all of which are covered in later tutorials, as well as, your algebra class. In some cases, it isn't very helpful to rewrite an expression, but in other cases it helps to write an equivalent expression to be able to continue with a problem and solve it. An equivalent expression is one that is written differently, but has the same value. The properties on this page will get you up to speed as to how you can write expressions in equivalent forms.

Tutorial

The Commutative Properties of Addition and Multiplication a + b = b + a and ab = ba

The Commutative Property, in general, states that changing the ORDER of two numbers either being added or multiplied, does NOT change the value of it. The two sides are called equivalent expressions because they look different but have the same value.

Example 1: Use the commutative property to write an equivalent expression to 2.5x + 3y.

Using the commutative property of addition (where changing the order of a sum does not change the value of it) we get 2.5x + 3y = 3y + 2.5x.

Example 2: Use the commutative property to write an equivalent expression to.

Using the commutative property of multiplication (where changing the order of a product does not change the value of it), we get

The Associative Properties of Addition and Multiplication a + (b + c) = (a + b) + c and a(bc) = (ab)c

The Associative property, in general, states that changing the GROUPING of numbers that are either being added or multiplied does NOT change the value of it. Again, the two sides are equivalent to each other. At this point it is good to remind you that both the commutative and associative properties do NOT work for subtraction or division. Example 3: Use the associative property to write an equivalent expression to (a + 5b) + 2c.

Using the associative property of addition (where changing the grouping of a sum does not change the value of it) we get (a + 5b) + 2c = a + (5b + 2c).

Example 4: Use the associative property to write an equivalent expression to (1.5x)y.

Using the associative property of multiplication (where changing the grouping of a product does not change the value of it) we get (1.5x)y = 1.5(xy)

Distributive Properties a(b + c) = ab + ac or (b + c)a = ba + ca

In other words, when you have a term being multiplied times two or more terms that are being added (or subtracted) in a ( ), multiply the outside term times EVERY term on the inside. Remember terms are separated by + and -. This idea can be extended to more than two terms in the ( ). Example 5: Use the distributive property to write 2(x - y) without parenthesis.

Multiplying every term on the inside of the ( ) by 2 we get:

*Distribute 2 to EVERY term inside ( )

Example 6: Use the distributive property to write - (5x + 7) without parenthesis.

*A - outside a ( ) is the same as times (-1) *Distribute the (-1) to EVERY term inside ( ) *Multiply

Basically, when you have a negative sign in front of a ( ), like this example, you can think of it as taking a -1 times the ( ). What you end up doing in the end is taking the opposite of every term in the ( ).

Example 7: Use the distributive property to find the product 3(2a + 3b + 4c).

As mentioned above, you can extend the distributive property to as many terms as are inside the ( ). The basic idea is that you multiply the outside term times EVERY term on the inside.

*Distribute the 3 to EVERY term *Multiply

Identity Properties

Addition The additive identity is 0 a + 0 = 0 + a = a

In other words, when you add 0 to any number, you end up with that number as a result.

Multiplication Multiplication identity is 1 a(1) = 1(a) = a

And when you multiply any number by 1, you wind up with that number as your answer.

The Inverse Properties

Additive Inverse (or negative) For each real number a, there is a unique real number, denoted -a, such that a + (-a) = 0.

In other words, when you add a number to its additive inverse, the result is 0. Other terms that are synonymous with additive inverse are negative and opposite.

Multiplicative Inverse (or reciprocal) For each real number a, except 0, there is a unique real number such that

In other words, when you multiply a number by its multiplicative inverse the result is 1. A more common term used to indicate a multiplicative inverse is the reciprocal. A multiplicative inverse or reciprocal of a real number a (except 0) is found by "flipping" a upside down. The numerator of a becomes the denominator of the reciprocal of a and the denominator of a becomes the numerator of the reciprocal of a. These two inverses will come in big time handy when you go to solve equations later on. Keep them in your memory bank until that time. Example 8: Write the opposite (or additive inverse) and reciprocal (or multiplicative inverse) of -3.

The opposite of -3 is 3, since -3 + 3 = 0.

The reciprocal of -3 is -1/3, since -3(-1/3) = 1. When you take the reciprocal, the sign of the original number stays intact. Remember that you need a number that when you multiply times the given number you get 1. If you change the sign when you take the reciprocal, you would get a -1, instead of 1, and that is a no no.

Example 9: Write the opposite (or additive inverse) and reciprocal (or multiplicative inverse) of 1/5.

The opposite of 1/5 is -1/5, since 1/5 + (-1/5) = 0.

The reciprocal of 1/5 is 5, since 5(1/5) = 1.

Practice Problems


Practice Problems 1a - 1b: Use a commutative property to write an equivalent expression. 1a. xy (answer/discussion to 1a)1b. .1 + 3x (answer/discussion to 1b)

Practice Problems 2a - 2b: Use an associative property to write an equivalent expression. 2a. (a + b) + 1.5 (answer/discussion to 2a)2b. 5(xy) (answer/discussion to 2b)

Practice Problems 3a - 3b: Use the distributive property to find the product. 3a. -2(x - 5) (answer/discussion to 3a)3b. 7(5a + 4b + 3c) (answer/discussion to 3b)

Practice Problems 4a - 4b: Write the opposite (additive inverse) and the reciprocal (multiplicative inverse) of each number. 4a. -7 (answer/discussion to 4a)4b. 3/5 (answer/discussion to 4b)

Answer/Discussion to Practice Problems Tutorial 8: Properties of Real Numbers

WTAMU > Virtual Math Lab > Beginning Algebra > Tutorial 8: Properties of Real Numbers

Answer/Discussion to 1a xy Using the commutative property of multiplication (where changing the order of a product does not change the value of it), we get xy = yx (return to problem 1a)

Answer/Discussion to 1b .1 + 3x Using the commutative property of addition (where changing the order of a sum does not change the value of it), we get .1 + 3x = 3x + .1 (return to problem 1b)

Answer/Discussion to 2a (a + b) + 1.5 Using the associative property of addition (where changing the grouping of a sum does not change the value of it), we get (a + b) + 1.5 = a + (b + 1.5) (return to problem 2a)

Answer/Discussion to 2b 5(xy) Using the associative property of multiplication (where changing the grouping of a product does not change the value of it), we get 5(xy) = (5x)y (return to problem 2b)

Answer/Discussion to 3a -2(x - 5)

*Distribute -2 to EVERY term *Multiply


Answer/Discussion to 3b 7(5a + 4b + 3c)

*Distribute 7 to EVERY term *Multiply


Answer/Discussion to 4a -7 The opposite of -7 is 7, since -7 + 7 = 0. The reciprocal of -7 is -1/7, since -7(-1/7) = 1. (return to problem 4a)

Answer/Discussion to 4b 3/5 The opposite of 3/5 is -3/5, since 3/5 + (-3/5) = 0. The reciprocal of 3/5 is 5/3, since (3/5)(5/3) = 1.

Tutorial 9:Reading Graphs


After completing this tutorial, you should be able to: 1. Read a bar graph.2. Read a line graph.3. Read a double line graph.4. Draw and read a Venn diagram.

Introduction

In this tutorial we will be reading graphs. Graphs can be used to visually represent the relationship of data. It can help organize and show people statistics, which can be good for some and not so good for others, depending on what the statistics show. Organizing data graphically can come in handy in fields like business, sports, teaching, politics, advertising, etc.. Let's start looking at some graphs.

Tutorial

Bar Graph

A bar graph can be used to give a visual representation of the relationship of data that has been collected. It is made up of a vertical and a horizontal axis and bars that can run vertically or horizontally. Vertical Bar Graph

If the bars are vertical, match the top of the bar with the vertical axis found at the side of the overall graph to find the information the bar associates with on the vertical axis. You will find what the bar associates with on the horizontal axis at the base of the bar. The bar graph below has vertical bars:

The horizontal axis represents years and the vertical axis represents profit in thousands of dollars. The first bar on the left associates with the year 1999 AND the profit of $20,000. The red line shows how the top of the bar lines up with 20 on the vertical axis. The second bar from the left associates with the year 2000 and the profit of $30,000. The blue line shows how the top of the bar lines up with 30 on the vertical axis.

Horizontal Bar Graph

If the bars are horizontal, match the right end of the bar with the horizontal axis found at the bottom of the overall graph to find the information the bar associates with on the horizontal axis. You will find what the the bar associates with on the vertical axis at the left end of the bar. The bar graph below has horizontal bars: (Note that this graph shows the same information the above graph does, just with horizontal bars instead of vertical bars.)

The vertical axis represents years and the horizontal axis represents profit in thousands of dollars. The first bar on the bottom associates with the year 1999 AND the profit of $20,000. The red line shows how the right end of the bar lines up with 20 on the horizontal axis. The second bar from the bottom associates with the year 2000 and the profit of $30,000. The blue line shows how the right end of the bar lines up with 30 on the horizontal axis.

Example 1: The bar graph below shows the number of students in a math class that received the grades shown. Use this graph to answer questions 1a - 1d. 1a. Find the number of students who received an A. 1b. Find the number of students who received an F. 1c. Find the number of students who passed the course (D or higher). 1d. Which grade did the most students receive?

1a. Find the number of students who received an A. (return to bar graph)

The bar that associates with the grade A is the first bar on the left. The top of that bar matches with 6 on the vertical axis. 6 students received an A.

1b. Find the number of students who received an F. (return to bar graph)

The bar that associates with the grade F is the fifth bar from the left. The top of that bar matches with 2 on the vertical axis. 2 students received an F.

1c. Find the number of students who passed the course (D or higher). (return to bar graph)

We will have to do a little calculating here. We will need to find the number of students that received an A, B C, and D and then ad them together. The bar that associates with the grade A is the first bar on the left. The top of that bar matches with 6 on the vertical axis. The bar that associates with the grade B is the second bar from the left. The top of that bar matches with 16 on the vertical axis. The bar that associates with the grade C is the third bar from the left. The top of that bar matches with 12 on the vertical axis. The bar that associates with the grade D is the fourth bar from the left. The top of that bar matches with 4 on the vertical axis. 6 + 16 + 12 + 4 = 38 students passed the course.

1d. Which grade did the most students receive? (return to bar graph)

It looks like more students received a B than any other single grade.

Example 2: The bar graph below shows the number of civilians holding various federal government jobs. Use the graph to answer questions 2a - 2d. 2a. About how many civilians work for Congress? 2b. About how many civilians work for the State Department? 2c. About how many civilians work for the armed forces (Navy, Air Force, and Army)? 2d. Which federal government job listed has the most civilian workers?

2a. About how many civilians work for Congress? (return to bar graph)

The bar that associates with Congress is the fourth bar up. The right of that bar lines up a little to the left of 50 on the horizontal axis. Note how the question asks ABOUT how many. In some cases, if it does not directly line up with a number that is marked you may need to approximate. This is very close to and less than 50. A good approximation is 25. About 25,000 civilians work for Congress.

2b. About how many civilians work for the State Department? (return to bar graph)

The bar that associates with the State Department is the sixth bar up. The right of that bar lines up with 50 on the horizontal axis. About 50,000 civilians work for the State Department.

2c. About how many civilians work for the armed forces (Navy, Air Force, and Army)? (return to bar graph)

We will have to do a little calculating on this one. We will need to find the number of civilians that work for each branch of the armed services and then add them up. The bar that associates with the Navy is the third bar up. The right of that bar ends between 300 and 350 on the horizontal axis. 310 is a good approximation for this number. The bar that associates with the Air Force is the second bar up. The right of that bar ends between 200 and 250 on the horizontal axis. 210 is a good approximation for this number. The bar that associates with the Army is the first bar from the bottom. The right of that bar ends just under 350 on the horizontal axis. 340 is a good approximation for this number. About 310,000 + 210,000 + 340,000 = 860,000 civilians work for the State Department.

2d. Which federal government job listed has the most civilian workers? (return to bar graph)

It looks like the Army has the most civilian workers.

Line Graph

A line graph is another way to give a visual representation of the relationship of data that has been collected. It is made up of a vertical and horizontal axis and a series of points that are connected by a line. Each point on the line matches up with a corresponding vertical axis and horizontal axis value on the graph. In some cases, you are giving a value from the horizontal axis and you need to find its corresponding value from the vertical axis. You find the point on the line that matches the given value from the horizontal axis and then match it up with its corresponding vertical axis value to find the value you are looking for. You would do the same type of process if you were given a vertical axis value and needed to find a horizontal axis value. The graph below is a line graph: (Note that this graph shows the same information the above graphs under vertical and horizontal graphs do, just with a line instead of bars.)

The horizontal axis represents years and the vertical axis represents profit in thousands of dollars. The first point on the left associates with the year 1999 AND the profit of $20,000. The red line shows how it lines up with 20 on the vertical axis and 1999 on the horizontal axis. The second point from the left associates with the year 2000 and the profit of $30,000. The blue line shows how it lines up with 30 on the vertical axis and 2000 on the horizontal axis.

Example 3: The line graph below shows the distance traveled of a vacationer going 70 mph down I-40 from 0 to 4 hours. Use the graph to answer questions 3a - 3b. 3a. How far has the vacationer traveled at 3 hours? 3b. How long does it take the vacationer to travel 140 miles?

3a. How far has the vacationer traveled at 3 hours? (return to line graph)

The point that matches with 3 on the horizontal axis also matches with 210 on the vertical axis. The vacationer has traveled 210 miles.

3b. How long does it take the vacationer to travel 140 miles? (return to line graph)

The point that matches with 140 on the vertical axis also matches with 2 on the horizontal axis. It takes the vacationer 2 hours to travel 140 miles.

Example 4: The line graph below shows the profit a local candy company made over the months of September through December of last year. Use the graph to answer questions 4a - 4c. 4a. About how much was the profit in the month of October? 4b. Which month had the lowest profit? 4c. What is the difference between the profits of November and December?

4a. About how much was the profit in the month of October? (return to line graph)

The point that matches with October on the horizontal axis also matches between 20 and 25 on the vertical axis. It looks to be about 23. The profit for the month of October is about $23,000.

4b. Which month had the lowest profit? (return to line graph)

It looks like September had the lowest profit.

4c. What is the difference between the profits of November and December? (return to line graph)

The point that matches with November on the horizontal axis also matches with 15 on the vertical axis. The point that matches with December on the horizontal axis also matches with 20 on the vertical axis. The difference between the profits of November and December would be 20,000 - 15,000 = $5,000.

Double Line Graph

A double line graph is another way to give a visual representation of the relationship of data that has been collected. It is similar to the line graph mentioned above. The difference is there are two lines of data instead of one. It is made up of a vertical and horizontal axis and two series of points each one connected by a line. The legend will show which line represents what set of points. Most times a solid line and a dashed line are used. But varying colors can also distinguish the two lines apart. Each point on each line matches up with a corresponding vertical axis and horizontal axis value on the graph. In some cases, you are giving a value from the horizontal axis and you need to find its corresponding value from the vertical axis. You find the point on the line that matches the given value from the horizontal axis and then match it up with its corresponding vertical axis value to find the value you are looking for. You would do the same type of process if you were given a vertical axis value and needed to find a horizontal axis value. The graph below is a double line graph:

The horizontal axis represents the year and the vertical axis represents profit in thousands of dollars. The legend towards the top of the graph indicates which line represents which product. The solid line corresponds with Product A and the dashed line goes with Product B. The first point on the solid line on the left associates with the year 1995 AND the profit of $30,000. The second point on the solid line from the left associates with the year 1996 AND the profit of $40,000. The third point on the solid line from the left associates with the year 1997 AND the profit of $40,000. The fourth point on the solid line from the left associates with the year 1998 AND the profit of $30,000. The fifth point on the solid line from the left associates with the year 1999 AND the profit of $60,000. The first point on the dashed line on the left associates with the year 1995 AND the profit of $20,000. The second point on the dashed line from the left associates with the year 1996 AND the profit of $20,000. The third point on the dashed line from the left associates with the year 1997 AND the profit of $15,000. The fourth point on the dashed line from the left associates with the year 1998 AND the profit of $40,000. The fifth point on the dashed line from the left associates with the year 1999 AND the profit of $50,000.

Example 5: The double line graph below shows the total enrollment of students in a local college from 1990 - 1995, broken down into part-time and full-time students. Use the graph to answer questions 5a - 5c. 5a. What was the full-time enrollment in 1992? 5b. For what year shown on the graph did the number of part-time students exceed the previous years number of part-time students by the greatest number? 5c. What was the total enrollment from 1993 to 1995?

5a. What was the full-time enrollment in 1992? (return to double line graph)

Since we are looking for full-time students, are we going to look at the solid or dashed line? According to the legend, we need to look at the dashed line. The point that is on the dashed line and matches with 1992 on the horizontal line also matches with 200 on the vertical line. There were 200 full-time students enrolled in 1992.

5b. For what year shown on the graph did the number of part-time students exceed the previous years number of part-time students by the greatest number? (return to double line graph)

Since we are looking for part-time students, are we going to look at the solid or dashed line? According to the legend, we need to look at the solid line. When looking at the graph, we are only interested in a rise in the number of part-time students. From 1990 to 1991, the number of part-time students went up 100 to 150. From 1991 to 1992, it went down from 150 to 50. From 1992 to 1993, there was increase from 50 to 250. From 1993 to 1994, there was another increase, this time from 250 to 300. The last years, 1994 - 1995, it held steady at 300. So, what year exceeded the previous number of part-time students by the greatest number? Looks like 1993. There were 200 more part-time students in 1993 than there were in 1992.

5c. What was the total enrollment from 1993 to 1995? (return to double line graph)

Lets break this down into part-time and full-time students. Looking at the dashed line to see the number of full-time students we get 250 + 400 + 500 = 1150. Looking at the solid line to see the number of part-time students we get 250 + 300 + 300 = 850. Putting those together, we have 1150 + 850 = 2000 students who were enrolled from 1993 to 1995.

Venn Diagrams

Venn diagrams are a visual way of organizing information. It can be very helpful when you have a problem to solve that categorizes or shows relationships between things. A common use for Venn diagrams is analyzing the results of a survey. For example, you may have a survey of students asking them which classes they like and perhaps you listed math and english. The student could check 0, 1, or 2 of these choices. You would strategically place the results in a Venn diagram. If they only choose math, then they would go in a particular region of the diagram that shows that, if they picked both, they would go into the area of the diagram that depicts that, etc. Of course there are other uses for the Venn diagram, that is one of the more common ones. The graph below is a Venn diagram:

This diagram represents the results of a survey of people who were asked if they liked Coke or Pepsi. They could choose only Coke, only Pepsi, both, or neither. Note that a lot of times you do not see the letter U or the roman numerals on a Venn Diagram (just the box and the circles), I use them as references so you know what area of the diagram I'm talking about in the lesson. The rectangle box represents the universal set U. The universal set is the set of all elements considered in a problem. In this example, the universal set are all the people who took the survey. The circles represent the categories or subsets involving the universal set. In this example, the two categories or choices on the survey were Coke and Pepsi. When you draw a Venn diagram, you want to overlap the circles in case there are some that pick both categories. We need to make sure we accurately place those people and do not count them more than one time. The roman numerals are called region numbers. Region I represents everyone who selected ONLY Coke which was 575 people. Region II is where the two circles intersect or overlap. It represents everyone who selected BOTH Coke and Pepsi which was 100 people. Region III represents everyone who selected ONLY Pepsi which was 225 people. Region IV is inside the rectangle, but outside the circles. It represents everyone who selected NEITHER Coke nor Pepsi which was 15 people.

Example 6: A teacher took a survey on pets in her class of 40 students. 12 students said they had a cat. 9 students said they had a dog. 2 said they had both a cat and a dog. How many students picked neither? How many students had only a cat? How many students had only a dog?

The first thing we need to do is draw a Venn diagram with two adjoining circles - one for cats and one for dogs.

Now we need to fill in numbers into the correct regions based on the information that was given. We need to start with something that only goes with one region and then work our way out from that. The only statement that deals with one region is 2 said they had both a cat and a dog. That would correlate with region II. So in region II, we would put a 2 as shown below:

Next let's look at the statement 12 students said they had a cat. Be careful here. It is very tempting to put a 12 in region I - but region I is reserved for those students who ONLY have a cat, which is different. When it says they had a cat, it means they checked it off on the survey and may or may not have checked off dog also. The cat circle includes regions I and II. Since we already have region II filled in with a 2, we can use that with the fact that I and II have to add up to be 12 to figure out what goes in region I - what do you think??? If we take 12 - 2 we get 10 left that have no other place to go but region I. This puts everybody in the correct spot AND does not count students more than 1 time. We can use the same type of argument when working with the statement "9 students said they had a dog." Again, it did not say ONLY dog - so 9 will have to fit into regions II and III and since we already have II filled in with 2 students that will leave 9

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