This chapter reviews basic algebra concepts including the real number system, arithmetic operations, exponents, equations, and solving different types of problems. It emphasizes understanding key vocabulary, memorizing processes and properties, using provided formulas, asking questions when unsure, and checking work.

1. Intermediate Algebra
by Gustafson and Frisk
Chapter 1
A Review of Basic Algebra

2. Section 1.1: The Real Number
System
SETS: collections of objects.
 Natural Numbers  Integers
 Whole Numbers  Positive Numbers
 Rational Numbers  Negative
 Irrational Numbers Numbers
 Real Numbers  Even Numbers
 Odd Numbers
Use { } {x | x > 5}
is read “the set of all x such that
x is greater than 5”

3. Section 1.1: The Real Number
System
GRAPHS: plot on the number line.
Individual numbers are dots
-3 -2 -1 0 1 2 3 4

4. Section 1.1: The Real Number
System
GRAPHS: plot on the number line.
Intervals including end points
-3 -2 -1 0 1 2 3 4
-3 -2 -1 0 1 2 3 4

5. Section 1.1: The Real Number
System
GRAPHS: plot on the number line.
Intervals not including end points
-3 -2 -1 0 1 2 3 4
-3 -2 -1 0 1 2 3 4

6. Section 1.2: Arithmetic & Properties of Real
Numbers
OPERATIONS:
 Addition
 Subtraction (the same as adding a
number with the opposite sign)
 Multiplication
 Division (the same as multiplying by
the reciprocal)

7. Section 1.2: Arithmetic & Properties of Real
Numbers
ADDITION:
Addends that have the same signs
 Add absolute values
 Keep the sign of the addends
Addends that have opposite signs
 Subtract absolute values
 Keep the sign of the addend with the
largest absolute value

8. Section 1.2: Arithmetic & Properties of Real
Numbers
MULTIPLICATION:
 Multiply absolute values
 If the factors have the same signs,
the product is positive
 If the factors have opposite signs,
the product is negative

9. Section 1.2: Arithmetic & Properties of Real
Numbers
STATISTICS: measures of central tendency
 Mean
 Median
 Mode

10. Section 1.2: Arithmetic & Properties of Real
Numbers
Properties:
 Associative – addition, multiplication
 Commutative – addition, multiplication
 Distributive – multiplication is
distributed over addition
a (b + c) = ab + ac

11. Section 1.2: Arithmetic & Properties of Real
Numbers
Identities:
 Addition – zero
 Multiplication – one
Inverses:
 Addition – opposites
 Multiplication – reciprocals

12. Section 1.3: Definition of Exponents
EXPONENTS: repeated multiplication
In the expression: an
a is the base and n is the exponent
 Exponents are NOT factors
 Means to multiply “a” n times

13. Section 1.3: Definition of Exponents
ORDER OF OPERATIONS:
If an algebraic expression has more than one
operation, the following order applies:
1. Clear up any grouping.
2. Evaluate exponents.
3. Do multiplication and division from left to
right.
4. Do addition and subtraction from left to right.

14. Section 1.5: Solving Equations
Algebraic Expression vs. Equation
 Expressions: a combination of
numbers and operations
 Equation: a statement that two
expressions are equal

15. Section 1.5: Solving Equations
EXPRESSIONS:
 Terms
 Like terms
 When multiplying, the terms do not
need to be alike
 Can only add like terms!

16. Section 1.5: Solving Equations
TO SOLVE AN EQUATION IN ONE VARIABLE:
 If you see fractions, multiply both sides by the LCD.
This will eliminate the fractions.
 Simplify the algebraic expressions on each side of the
equal sign (eliminate parentheses and combine like
terms).
 Use the addition property of equality to isolate the
variable terms from the constant terms on opposite
sides of the equal sign.
 Use the multiplication property to make the coefficient
of the variable equal to one.
 Check your results by evaluating.

17. Section 1.5: Solving Equations
TYPES OF EQUATIONS:
 CONDITIONAL: if x equals this, then y
equals that.
 IDENTITY: always true no matter what
numbers you use.
 CONTRADICTION: never true no matter
what numbers you use.
 FORMULAS: conditional equations that
model a relationship between the variables.

18. Section 1.6 & 1.7: Solving Problems, Applications
TYPES OF PROBLEMS:
 Geometry
 Percent
 Physics (forces)
 Uniform motion
 Mixtures
 Good ‘ole common sense analysis

19. Chapter 1: Basic Algebra Review
SUMMARY:
 KNOW YOUR VOCABULARY! You can’t
follow directions if you don’t know what the
words in the instructions mean.
 Memorize the processes and the properties.
 I will provide formulas for your reference.
 Ask questions if you are unsure.
 Always check your work to make sure that
you answered the question, and that your
answer is reasonable.