1. Real Numbers and Algebraic Expressions
Chapter 1
Real Numbers and Algebraic Expressions

2. Tips for Success in Mathematics
§ 1.1
Tips for Success in Mathematics

3. Getting Ready for This Course
Positive Attitude
Believe you can succeed.
Scheduling
Make sure you have time for your classes.
Be Prepared
Have all the materials you need, like a lab manual, calculator, or other supplies.
Objective A

4. General Tips for Success
Details
Get a contact person.
Exchange names, phone numbers or addresses with at least one other person in class.
Attend all class periods.
Sit near the front of the classroom to make hearing the presentation, and participating easier.
Do you homework.
The more time you spend solving mathematics, the easier the process becomes.
Check your work.
Review your steps, fix errors, and compare answers with the selected answers in the back of the book.
Learn from your mistakes.
Find and understand your errors. Use them to become a better math student.
Objective B
Continued

5. General Tips for Success
Details
Get help if you need it.
Ask for help when you don’t understand something. Know when your instructors office hours are, and whether tutoring services are available.
Organize class materials.
Organize your assignments, quizzes, tests, and notes for use as reference material throughout your course.
Read your textbook.
Review your section before class to help you understand its ideas more clearly.
Ask questions.
Speak up when you have a question. Other students may have the same one.
Hand in assignments on time.
Don’t lose points for being late. Show every step of a problem on your assignment.
Objective B Continued

6. Using This Text Resource Details Continued Practice Problems.
Try each Practice Problem after you’ve finished its corresponding example.
Chapter Test Prep Video CD.
Chapter Test exercises are worked out by the author, these are available off of the CD this book contains.
Lecture Video CDs.
Exercises marked with a CD symbol are worked out by the author on a video CD. Check with your instructor to see if these are available.
Symbols before an exercise set.
Symbols listed at the beginning of each exercise set will remind you of the available supplements.
Objectives.
The main section of exercises in an exercise set is referenced by an objective. Use these if you are having trouble with an assigned problem.
Objective C
Continued

7. Using This Text Resource Details
Icons (Symbols).
A CD symbol tells you the corresponding exercise may be viewed on a video segment. A pencil symbol means you should answer using complete sentences.
Integrated Reviews.
Reviews found in the middle of each chapter can be used to practice the previously learned concepts.
End of Chapter Opportunities.
Use Chapter Highlights, Chapter Reviews, Chapter Tests, and Cumulative Reviews to help you understand chapter concepts.
Study Skills Builder.
Read and answer questions in the Study Skills Builder to increase your chance of success in this course.
The Bigger Picture.
This can help you make the transition from thinking “section by section” to thinking about how everything corresponds in the bigger picture.
Objective C Continued

8. Get help as soon as you need it.
Getting Help
Tip
Details
Get help as soon as you need it.
Material presented in one section builds on your understanding of the previous section. If you don’t understand a concept covered during a class period, there is a good chance you won’t understand the concepts covered in the next period.
For help try your instructor, a tutoring center, or a math lab. A study group can also help increase your understanding of covered materials.
Objective D

9. Preparing for and Taking an Exam Steps for Preparing for a Test
Review previous homework assignments.
Review notes from class and section-level quizzes you have taken.
Read the Highlights at the end of each chapter to review concepts and definitions.
Complete the Chapter Review at the end of each chapter to practice the exercises.
Take a sample test in conditions similar to your test conditions.
Set aside plenty of time to arrive where you will be taking the exam.
Objective E
Continued

10. Preparing for and Taking an Exam Steps for Taking Your Test
Read the directions on the test carefully.
Read each problem carefully to make sure that you answer the question asked.
Pace yourself so that you have enough time to attempt each problem on the test.
Use extra time checking your work and answers.
Don’t turn in your test early. Use extra time to double check your work.
Objective E Continued

11. Tips for Making a Schedule
Managing Your Time
Tips for Making a Schedule
Make a list of all of your weekly commitments for the term.
Estimate the time needed and how often it will be performed, for each item.
Block out a typical week on a schedule grid, start with items with fixed time slots.
Next, fill in items with flexible time slots.
Remember to leave time for eating, sleeping, and relaxing.
Make changes to your workload, classload, or other areas to fit your needs.
Objective F

12. Algebraic Expressions and Sets of Numbers
§ 1.2
Algebraic Expressions and Sets of Numbers 12

13. Algebraic Expressions
A variable is a letter used to represent any number.
A constant is either a fixed number or a letter that represents a fixed number.
An algebraic expression is formed by numbers and variables connected by the operation of addition, subtraction, multiplication, division, raising to powers, and/or taking roots.

14. Evaluating Algebraic Expressions
To evaluate an algebraic expression, substitute the numerical value for each variable into the expression and simplify the result.
Example:
Evaluate each expression for the given value.
(a) 5x – 2 for x = 8
5(8) – 2 = 40 – 2 = 38
(b) 3a2 + 2a + 4 for a = – 4
3(– 4)2 + 2(– 4) + 4
= 3(16) + (– 8) + 4 = 44

15. Set of Numbers Natural numbers – {1, 2, 3, 4, 5, 6 . . .}
Whole numbers – {0, 1, 2, 3, }
Integers – {. . . –3, -2, -1, 0, 1, 2, }
Rational numbers – the set of all numbers that can be expressed as a quotient of integers, with denominator  0
Irrational numbers – the set of all numbers that can NOT be expressed as a quotient of integers
Real numbers – the set of all rational and irrational numbers combined

16. The Number Line
A number line is a line on which each point is associated with a number. 2
– 2 1 3 4 5
– 1
– 3
– 4
– 5
– 4.8
1.5
Negative numbers
Positive numbers

17. Set Builder Notation
A set can also be written in set builder notation. This notation describes the members of a set, but does not list them.
Example:
{ x | x is an even natural number less than 10}
x is an even natural number less than 10
The set of all x
such that

18. Vocabulary Variable is a symbol used to represent a number.
Algebraic expressions are a collection of numbers, variables, operations, grouping symbols, but NO = or inequalities.
In describing some of the previous sets, we used a symbol, called an ellipsis. It means to continue in the same pattern.
The members of a set are called its elements.
When we list (or attempt to list with an ellipsis) the elements of a set, the set is written in roster form.

19. Absolute Value
The absolute value of a real number a, denoted by |a|, is the distance between a and 0 on the number line.
| – 4| = 4
|5| = 5
Symbol for absolute value
Distance of 4
Distance of 5 2
– 2 1 3 4 5
– 1
– 3
– 4
– 5

20. Translating Phrases Addition (+) Subtraction (–) Multiplication (·)
Division
()
sum
plus
added to
more than
increased by
total
difference
minus
subtract
less than
decreased by
less
product
times
multiply
multiplied by of
double/triple
quotient
divide
shared equally
among
divided by
divided into
Objective A

21. Translating Phrases Example:
Write as an algebraic expression. Use x to represent “a number.”
a.) 5 decreased by a number
b.) The quotient of a number and 12
a.)
In words: 5
decreased by
a number –
Translate: 5 x
Objective A
The quotient of
b.)
In words:
a number
and 12
Translate: x  12

22. Translating Phrases Example:
Translate the following phrases into algebraic expressions.
The sum of 4 and twice y
4 + 2y
x less than the product of y and z
yz - x

23. Operations on Real Numbers
§ 1.3
Operations on Real Numbers 23

24. Adding Real Numbers Example: –2 Adding two numbers with the same sign
Add their absolute values.
Use common sign as sign of sum.
Adding two numbers with different signs
Take difference of absolute values (smaller subtracted from larger).
Use the sign of larger absolute value as sign of sum.
Example:
Add the following numbers.
(–3) (–5) = –2

25. Subtracting Real Numbers
Substitute the opposite of the number being subtracted
Add.
a – b = a + (– b)
Example:
Subtract the following numbers.
(– 5) – 6 – (– 3) =
(– 5) + (– 6) + 3 =
– 8

26. Multiplying or Dividing Real Numbers
Multiplying or dividing two real numbers with same sign
Result is a positive number
Multiplying or dividing two real numbers with different signs
Result is a negative number

27. Multiplying or Dividing Real Numbers
Example:
Find each of the following products.
4 · (–2) · 3 =
–24
(–4) · (–5) = 20

28. Reciprocals If b is a real number, 0 · b = b · 0 = 0.
Reciprocals are two numbers whose product is 1.
The quotient of any real number and 0 is undefined.
The quotient of 0 and any real number = 0.
a  0

29. Simplifying Real Numbers
If a and b are real numbers, and b  0,
Example:
Simplify the following.

30. Exponents
Exponents that are natural numbers are shorthand notation for repeating factors.
34 = 3 · 3 · 3 · 3
3 is the base
4 is the exponent (also called power)
Note by the order of operations that exponents are calculated before other operations.

31. Evaluating Exponential Expressions
Example:
Evaluate each of the following expressions. 34
= 3 · 3 · 3 · 3
= 81
(–5)2
= (– 5)(–5)
= 25
–62
= – (6)(6)
= –36
(2 · 4)3
= (2 · 4)(2 · 4)(2 · 4)
= 8 · 8 · 8
= 512
3 · 42
= 3 · 4 · 4
= 48

32. Evaluating Exponential Expressions
Example:
Evaluate each of the following expressions.
a.) Find 3x2 when x = 5.
3x2 = 3(5)2
= 3(5 · 5)
= 3 · 25
= 75
b.) Find –2x2 when x = –1.
–2x2 = –2(–1)2
= –2(–1)(–1)
= –2(1)
= –2

33. Using Exponential Notation
We may use exponential notation to write products in a more compact form.
can be written as
“three to the fourth power”
“three to the third power” or “three cubed”
“three to the second power” or “three squared.”
In Words
Expression
Objective A
Example:
Evaluate 26.

34. Square Roots
Opposite of squaring a number is taking the square root of a number.
A number b is a square root of a number a if b2 = a.
In order to find a square root of a, you need a # that, when squared, equals a.

35. Principal Square Roots
The principal (positive) square root is noted as
The negative square root is noted as

36. Square Roots
Example:

37. The Order of Operations
Simplify expressions using the order that follows. If grouping symbols such as parentheses are present, simplify expression within those first, starting with the innermost set. If fraction bars are present, simplify the numerator and denominator separately.
1. Evaluate exponential expressions, roots, or absolute values in order from left to right.
2. Multiply or divide in order from left to right.
3. Add or subtract in order from left to right.
Objective D

38. Using the Order of Operations
Example:
Evaluate:
Write 32 as 9.
Divide 9 by 3.
Objective D Continued
Add 3 to 6.
Divide 9 by 9.

39. Evaluating Algebraic Expressions
To evaluate an algebraic expression, substitute the numerical value for each variable into the expression and simplify the result.
Example:
Evaluate each expression for the given value.
(a) 5x – 2 for x = 8
5(8) – 2 = 40 – 2 = 38
(b) 3a2 + 2a + 4 for a = – 4
3(– 4)2 + 2(– 4) + 4
= 3(16) + (– 8) + 4 = 44

40. Properties of Real Numbers
§ 1.4
Properties of Real Numbers 40

41. Algebraic Equations
Algebraic equation is a statement that two expressions have equal value.
Equality is denoted by the phrases
equals
gives
is/was/should be
yields
amounts to
represents
is the same as

42. Translating Phrases Example:
Translate the following sentences into algebraic symbols.
The difference of 7 and a number is less than 42.
7 – x < 42
The quotient of y and twice x is the same as the product of 4 and z

43. Equality and Inequality Symbols
Meaning
a = b
a  b
a < b
a > b
a  b
a  b
a is equal to b.
a is not equal to b.
a is less than b.
a is greater than b.
a is less then or equal to b.
a is greater than or equal to b.

44. Order Property for Real Numbers
For any two real numbers a and b, a is less than b if a is to the left of b on the number line.
a < b means a is to the left of b on a number line.
a > b means a is to the right of b on a number line.
Example:
Insert < or > between the following pair of numbers to make a true statement.

45. Identities and Inverses
for addition: 0 is the identity since a + 0 = a and 0 + a = a.
for multiplication: 1 is the identity since a · 1 = a and 1 · a = a.
Inverses
for addition: a and –a are inverses since a + (–a) = 0.
for multiplication: b and are inverses (b  0) since b · = 1.

46. Commutative Properties
For real numbers a and b,
Addition: a + b = b + a
Multiplication: a · b = b · a
Example: Evaluate the expression: (– 24)
(– 24) = 24 + (– 24) + 7
Use the commutative property.
= 7
Add.

47. Associative Properties
For real numbers a, b, and c,
Addition: a + (b + c) = (a + b) + c = a + b + c
Multiplication: a · (b · c) = (a · b) · c = a · b · c
Example: Evaluate the expression: (– 245)
(– 245)
= 70 + [245 + (– 245)]
Group using the Associative Property.
= 70
Add.

48. The Distributive Property
For real numbers a, b, and c,
a (b + c) = ab + ac
Example:
Use the Distributive Property to remove the parentheses.
7(4 + 2) =
7(4 + 2)
= (7)(4) +
(7)(2) =
= 42