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

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”

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

[ [ ] 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

( ( ) 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

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)

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.

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”.

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

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.

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

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

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.

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.

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 2. 2 + 5 = 7; 5(2) = 10. Since 7 does not equal 10, we’ve proved x + 5 = 5x is false.

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)

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.

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

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.

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