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
Whole Numbers
Rational Numbers
Irrational Numbers
Real Numbers
Integers
Positive Numbers
Negative 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 -1 3 1 4 2 -2 -3
X = 1

4. [ [ ] Intervals including end points -1 3 1 4 2 -2 -3 -1 3 1 4 2 -2 -3
Section 1.1: The Real Number System
GRAPHS: plot on the number line.
Intervals including end points [ -1 3 1 4 2 -2 -3
{ x | x > 2} and { x | 0 < x < 2} [ ] -1 3 1 4 2 -2 -3

5. ( ( ) Intervals not including end points -1 3 1 4 2 -2 -3 -1 3 1 4 2
Section 1.1: The Real Number System
GRAPHS: plot on the number line.
Intervals not including end points ( -1 3 1 4 2 -2 -3
{ x | x > 2} and { x | 0 < x < 2} ( ) -1 3 1 4 2 -2 -3

6. Subtraction (the same as adding a number with the opposite sign)
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. Addends that have the same signs Add absolute values
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
No need to have rules for subtraction, just add the opposite.

8. Multiply absolute values
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
Division rules for the signs are the same. IMPORTANT: Division by zero is “undefined”.

9. Mean Median Mode STATISTICS: measures of central tendency
Section 1.2: Arithmetic & Properties of Real Numbers
STATISTICS: measures of central tendency
Mean
Median
Mode
meAn = Average
medIan = mIddle
mOde = mOst

10. Associative – addition, multiplication
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
Associative (grouping), Commutative (order)
Distributive (must have two operations, not just one). Can be multiplication or division, and addition or subtraction.

11. Multiplication – reciprocals
Section 1.2: Arithmetic & Properties of Real Numbers
Identities:
Addition – zero
Multiplication – one
Inverses:
Additive inverses: a and –a are inverses because when added, they equal the additive identity (zero).
Multiplicative Inverses: reciprocals are inverses because when multiplied, they equal the multiplicative identity (one).
Remember also, -(-a) = a. That is, the opposite of a negative number is a positive number with the same absolute value.
Addition – opposites
Multiplication – reciprocals

12. EXPONENTS: repeated multiplication
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
a5 = a  a  a  a  a, not a  5

13. Section 1.3: Definition of Exponents
ORDER OF OPERATIONS:
If an algebraic expression has more than one operation, the following order applies:
Clear up any grouping.
Evaluate exponents.
Do multiplication and division from left to right.
Do addition and subtraction from left to right.
Remember, addition and subtraction are basically the same thing, so can’t do one or the other first. You can always change subtraction to addition by adding the opposite.
Likewise for multiplication and division. You can change division to multiplying by the reciprocal.

14. Algebraic Expression vs. Equation
Section 1.5: Solving Equations
Algebraic Expression vs. Equation
Expressions: a combination of numbers and operations
Equation: a statement that two expressions are equal
2x + 5 is an algebraic expression as is a single number such as 9.
2x + 5 = 9 is an equation.

15. When multiplying, the terms do not need to be alike
Section 1.5: Solving Equations
EXPRESSIONS:
Terms
Like terms
When multiplying, the terms do not need to be alike
Can only add like terms!
A term is the combination of numbers and variables between “+” signs. 2x – 8 has two terms: 2x and -8.
The expression x + 5 contains two terms, x and 5. They are not alike because one is a variable, whereas 5 is a constant.
Therefore, x + 5 is simplified. No further work is required.
IMPORTANT: x + 5 does NOT equal 5x. 5x = x + x + x + x + x.
To prove x + 5 = 5x is false, pick a number (other than zero and one) and evaluate the statement.
For example, choose = 7; 5(2) = 10. Since 7 does not equal 10, we’ve proved x + 5 = 5x is false.

16. TO SOLVE AN EQUATION IN ONE VARIABLE:
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.
Example: 1/3x + 4 = 2(x – 1)

17. TYPES OF EQUATIONS: CONDITIONAL: if x equals this, then y equals that.
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.
The solution set for an identity is “all real numbers”, ℝ.
The solution set for a contradiction is the empty set, . That is, it has no solution.

18. TYPES OF PROBLEMS: Geometry Percent Physics (forces) Uniform motion
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.

20. This powerpoint was kindly donated to www.worldofteaching.com
Is home to well over a thousand powerpoints submitted by teachers. This a free site. Please visit and I hope it will help in your teaching 20