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MTH070 Elementary Algebra Chapter 1 Review of Real Numbers and Problem Solving Copyright © 2010 by Ron Wallace, all rights reserved.

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What is Algebra?

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A generalization of arithmetic. Mathematics Dictionary, 4 th Ed. by James/James Therefore, you need to know and understand arithmetic!

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What is Algebra? A generalization of arithmetic in which symbols, usually letters of the alphabet, represent numbers or members of a specified set of numbers and are related by operations that that obey specified laws. The American Heritage Dictionary, 2 nd College Edition

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What is Algebra? A branch of mathematics in which symbols, usually letters of the alphabet, represent numbers or members of a specified set and are used to represent quantities and to express general relationships that hold for all members of the set. www.dictionary.com

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Terminology & Notation Mathematical Vocabulary Precise Communication Notation Symbols instead of words Semantics & Syntax Multiplication & Division in Algebra? Yes, you are expected to be able to read, understand, and use correct mathematical terminology and notation.

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Terminology & Notation Example …

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Terminology Variable A symbol used to represent a value. Usually a letter (Latin or Greek) The value is … – unknown – to be determined – assigned different quantities

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Terminology Constant A specific value. – Integers – Fractions – Decimals – Special

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Terminology Algebraic Expression A legal combination of variables, constants, operators, and grouping symbols. AKA: Expression

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Terminology Equivalent Expressions Expressions that are equal for all values of their variables. Simplifying: Changing expressions into simpler equivalent expressions.

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Terminology Term A constant, variable, or product of a constant and one or more variables. When grouping symbols are removed, all expressions are a sum of terms. Coefficient – The constant part of a term Linear Term – A constant times a variable. x = 1x-x = -1x

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Terminology Factor The constants or variables that are multiplied together in a term. Common Factor – A constant or variable that is a factor of each term of an expression.

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Terminology Identities additive identity: 0 multiplicative identity: 1

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Terminology Inverses Opposite – The negative of the number. – aka: Additive inverse Reciprocal - One divided by the number. – aka: Multiplicative inverse

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Terminology Commutative Order of addition and multiplication may be reversed.

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Terminology Associative Sums and products may be grouped in any way. What about subtraction and division?

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Terminology Distributive (involves both operations) “3 is distributed over the sum x+5.” What about subtraction and division?

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Terminology Equation A mathematical statement that two expressions are equal. Three possibilities … – always true “identity” – always false “fallacy” – neither “conditional”

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Terminology Inequality A mathematical statement that one expression is... > < ≥ ≤ … a second expression.

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Terminology Solution A value for the variable that makes an equation or inequality a true statement. Solution Set: All such values.

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Terminology Absolute Value How far a number is from zero. AKA: Magnitude

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Terminology Real Numbers The set of all numbers that correspond to the points on a number line. – Can be thought of as directed distances from zero. 0 1xy

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Terminology Integers The set of positive and negative counting numbers and zero.

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Terminology Positive Integers The set of counting numbers. AKA: Natural Numbers

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Terminology Negative Integers The set of opposites of the positive integers.

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Terminology Non-Negative Integers The set of counting numbers and zero. AKA: Whole Numbers

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Terminology Rational Numbers Set of numbers that can be written as a quotient of two integers. Decimal forms will either be terminating decimals or repeating decimals.

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Terminology Irrational Numbers All Real numbers that are not Rational.

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Properties of Zero

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Addition Signed Number Arithmetic Think Position on # Line Start @ 1 st # –Negative to left of 0 –Positive to right of 0 Add 2 nd # –Negative move left –Positive move right Are you moving towards or away from zero? Think Money in Your Pocket Start w/ 1 st # –Negative in debt (owe) –Positive cash on hand Add 2 nd # –Negative spending –Positive receiving Are you … –increasing your debt? –increasing your savings? –reversing your state? “Rules” on Page 28.

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Subtraction Signed Number Arithmetic Think … –Add the opposite! “Rule” on Page 32. SEE THINK

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Multiplication Signed Number Arithmetic Multiplying by -1 “Rules” on Page 37. So … –Multiply numbers ignore signs –Even # of negatives Positive result –Odd # of negatives Negative result

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Division Signed Number Arithmetic Treat as a Fraction Therefore … reduce! “Rules” on Page 38. Division is … –multiplication by the reciprocal –sign of the result is just like multiplication.

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Division Signed Number Arithmetic Warning: Be careful w/ Cancelling “Rules” on Page 38.

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“Value of an Expression” The Question… Solution Replace x with its value and do the arithmetic! (be careful when x is negative)

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Order of Operations Grouping symbols { }, [ ], ( ), — may be nested Exponents Multiplication & Division left to right Addition & Subtraction left to right

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Simplify! Remove Grouping Symbols Distributive & Associative Laws

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Simplify! Remove Grouping Symbols Distributive & Associative Laws Group Like Terms Commutative & Associative Laws Combine Like Terms Distributive Law (i.e. add coefficients) Complete All Arithmetic

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Translate: Words Algebra Addition –added to; sum of; plus; more than; increased by Subtraction –subtracted from; difference of; minus; less than; decreased by Multiplication –multiplied by; product of; times; twice; of Division –divided by; quotient of; divided into; ratio of; per Table of Terms w/ Examples on Page 50. … and combinations of these (e.g. twice the sum of …)

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Problem Solving: The 4P’s Problem Solving? Analyzing a situation Organizing information Choosing a strategy Implementing the strategy The 4P’s Prepare –Read (not a novel, it’s technical) –What do you know? –What are you trying to find? Plan (i.e. strategy) –Guess & check? –Patterns? –Algebraic? Process (i.e. implementation) Ponder –Did you answer the question? –Does the solution make sense? –Units?

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