**Intermediate Algebra** **Intermediate Algebra**

**by Gustafson and Frisk** **by Gustafson and Frisk**

### Chapter 1 Chapter 1

### A Review of Basic Algebra

### A Review of Basic Algebra

**Section 1.1: The Real Number System** **Section 1.1: The Real Number System**

**SETS: **

**SETS: ** collections of objects. collections of objects.

### Natural Numbers Natural Numbers

### Whole Numbers Whole Numbers

### Rational Numbers Rational Numbers

### Irrational Numbers Irrational Numbers

### Real Numbers Real Numbers

### Integers Integers

### Positive Numbers Positive Numbers

### Negative Numbers Negative Numbers

### Even Numbers Even Numbers

### Odd Numbers Odd Numbers

### Use { } {x | x > 5} {x | x > 5}

### is read “the set of all x such that is read “the set of all x such that

### x is greater than 5”

### x is greater than 5”

**Section 1.1: The Real Number System** **Section 1.1: The Real Number System**

### GRAPHS:

### GRAPHS: plot on the number line. plot on the number line.

### -1 0 1 2 3 4 -2

### -3

### Individual numbers are dots

### -1 0 1 2 3 4 -2

### -3

**Section 1.1: The Real Number System** **Section 1.1: The Real Number System**

### GRAPHS:

### GRAPHS: plot on the number line. plot on the number line.

### -1 0 1 2 3 4 -2

### -3

### Intervals including end points

### [

### [ ]

### -1 0 1 2 3 4 -2

### -3

**Section 1.1: The Real Number System** **Section 1.1: The Real Number System**

### GRAPHS:

### GRAPHS: plot on the number line. plot on the number line.

### -1 0 1 2 3 4 -2

### -3

### Intervals not including end points

### (

### ( )

**Section 1.2: Arithmetic & Properties of Real Numbers** **Section 1.2: Arithmetic & Properties of Real Numbers**

### OPERATIONS:

### OPERATIONS:

### Addition Addition

### Subtraction (the same as adding a Subtraction (the same as adding a number with the opposite sign)

### number with the opposite sign)

### Multiplication Multiplication

### Division (the same as multiplying by Division (the same as multiplying by the reciprocal)

### the reciprocal)

**Section 1.2: Arithmetic & Properties of Real Numbers** **Section 1.2: Arithmetic & Properties of Real Numbers**

### ADDITION:

### ADDITION:

### Addends that have opposite signs Addends that have opposite signs

### Subtract absolute values Subtract absolute values

### Keep the sign of the addend with the Keep the sign of the addend with the largest absolute value

### largest absolute value

### Addends that have the same signs Addends that have the same signs

### Add absolute values Add absolute values

### Keep the sign of the addends Keep the sign of the addends

**Section 1.2: Arithmetic & Properties of Real Numbers** **Section 1.2: Arithmetic & Properties of Real Numbers**

### MULTIPLICATION:

### MULTIPLICATION:

### Multiply absolute values Multiply absolute values

### If the factors have the same signs, If the factors have the same signs, the product is positive

### the product is positive

### If the factors have opposite signs, If the factors have opposite signs, the product is negative

### the product is negative

**Section 1.2: Arithmetic & Properties of Real Numbers** **Section 1.2: Arithmetic & Properties of Real Numbers**

### STATISTICS:

### STATISTICS: measures of central tendency measures of central tendency

### Mean Mean

### Median Median

### Mode Mode

**Section 1.2: Arithmetic & Properties of Real Numbers** **Section 1.2: Arithmetic & Properties of Real Numbers**

### Properties:

### Properties:

### Associative – addition, multiplication Associative – addition, multiplication

### Commutative – addition, multiplication Commutative – addition, multiplication

### Distributive – multiplication is Distributive – multiplication is distributed over addition

### distributed over addition a (b + c) = ab + ac

### a (b + c) = ab + ac

**Section 1.2: Arithmetic & Properties of Real Numbers** **Section 1.2: Arithmetic & Properties of Real Numbers**

### Identities:

### Identities:

### Addition – zero Addition – zero

### Multiplication – one Multiplication – one

### Inverses:

### Inverses:

### Addition – opposites Addition – opposites

### Multiplication – reciprocals Multiplication – reciprocals

**Section 1.3: Definition of Exponents** **Section 1.3: Definition of Exponents**

### EXPONENTS:

### EXPONENTS: repeated multiplication repeated multiplication In the expression: a

### In the expression: a

^{n}

^{n}

### a is the base and n is the exponent a is the base and n is the exponent

### Exponents are Exponents are NOT NOT factors factors

### Means to multiply “a” n times Means to multiply “a” n times

**Section 1.3: Definition of Exponents** **Section 1.3: Definition of Exponents**

### ORDER OF OPERATIONS:

### ORDER OF OPERATIONS:

### If an algebraic expression has more than one If an algebraic expression has more than one

### operation, the following order applies:

### operation, the following order applies:

1.1.

### Clear up any grouping. Clear up any grouping.

2.2.

### Evaluate exponents. Evaluate exponents.

3.3.

### Do multiplication and division from left to Do multiplication and division from left to right.

### right.

4.4.

### Do addition and subtraction from left to Do addition and subtraction from left to right.

### right.

**Section 1.5: Solving Equations** **Section 1.5: Solving Equations**

### Algebraic Expression vs. Equation Algebraic Expression vs. Equation

### Expressions: a combination of Expressions: a combination of numbers and operations

### numbers and operations

### Equation: a statement that two Equation: a statement that two expressions are equal

### expressions are equal

**Section 1.5: Solving Equations** **Section 1.5: Solving Equations**

### EXPRESSIONS:

### EXPRESSIONS:

### Terms Terms

### Like terms Like terms

### When multiplying, the terms do not When multiplying, the terms do not need to be alike

### need to be alike

### Can only add like terms! Can only add like terms!

**Section 1.5: Solving Equations** **Section 1.5: Solving Equations**

### TO SOLVE AN EQUATION IN ONE VARIABLE:

### TO SOLVE AN EQUATION IN ONE VARIABLE:

### If you see fractions, If you see fractions, multiply both sides by the LCD multiply both sides by the LCD . . This will eliminate the fractions.

### This will eliminate the fractions.

### Simplify Simplify the algebraic expressions on each side of the the algebraic expressions on each side of the equal sign (eliminate parentheses and combine like equal sign (eliminate parentheses and combine like

### terms).

### terms).

### Use the addition property of equality to Use the addition property of equality to isolate isolate the the variable terms from the constant terms

### variable terms from the constant terms on opposite on opposite sides of the equal sign.

### sides of the equal sign.

### Use the multiplication property to make the Use the multiplication property to make the coefficient of the variable equal to one.

### coefficient of the variable equal to one.

### Check your results by evaluating. Check your results by evaluating.

**Section 1.5: Solving Equations** **Section 1.5: Solving Equations**

### TYPES OF EQUATIONS:

### TYPES OF EQUATIONS:

### CONDITIONAL: if x equals this, then y CONDITIONAL: if x equals this, then y equals that.

### equals that.

### IDENTITY: always true no matter what IDENTITY: always true no matter what numbers you use.

### numbers you use.

### CONTRADICTION: never true no matter CONTRADICTION: never true no matter what numbers you use.

### what numbers you use.

### FORMULAS: conditional equations that FORMULAS: conditional equations that

### model a relationship between the variables.

### model a relationship between the variables.

**Section 1.6 & 1.7: Solving Problems, Applications** **Section 1.6 & 1.7: Solving Problems, Applications**

### TYPES OF PROBLEMS:

### TYPES OF PROBLEMS:

### Geometry Geometry

### Percent Percent

### Physics (forces) Physics (forces)

### Uniform motion Uniform motion

### Mixtures Mixtures

### Good ‘ole common sense analysis Good ‘ole common sense analysis

**Chapter 1: Basic Algebra Review** **Chapter 1: Basic Algebra Review**

### SUMMARY:

### SUMMARY:

### KNOW YOUR VOCABULARY! KNOW YOUR VOCABULARY! You can’t follow You can’t follow directions if you don’t know what the words directions if you don’t know what the words

### in the instructions mean.

### in the instructions mean.

### Memorize the processes and the properties. Memorize the processes and the properties.

### I will provide formulas for your reference. I will provide formulas for your reference.

### Ask questions if you are unsure. Ask questions if you are unsure.

### Always check your work to make sure that Always check your work to make sure that you answered the question, and that your you answered the question, and that your

### answer is reasonable.

### answer is reasonable.

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