Elementary Algebra Exam 1 Material

Familiar Sets of Numbers Natural numbers Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers Includes zero and all natural numbers: 0, 1, 2, 3, … (Does not include negative numbers) Fractions Ratios of whole numbers where bottom number can not be zero:

Prime Numbers Natural Numbers, not including 1, whose only factors are themselves and 1 2, 3, 5, 7, 11, 13, 17, 19, 23, etc. What is the next biggest prime number? 29

Composite Numbers Natural Numbers, bigger than 1, that are not prime 4, 6, 8, 9, 10, 12, 14, 15, 16, etc. Composite numbers can always be “factored” as a product (multiplication) of prime numbers

Factoring Numbers To factor a number is to write it as a product of two or more other numbers, each of which is called a factor 12 = (3)(4) 3 & 4 are factors 12 = (6)(2) 6 & 2 are factors 12 = (12)(1) 12 and 1 are factors 12 = (2)(2)(3) 2, 2, and 3 are factors In the last case we say the 12 is “completely factored” because all the factors are prime numbers

Hints for Factoring Numbers To factor a number we can get two factors by writing any multiplication problem that comes to mind that is equal to the given number Any factor that is not prime can then be written as a product of two other factors This process continues until all factors are prime Completely factor 28 28 = (4)(7) 4 & 7 are factors, but 4 is not prime 28 = (2)(2)(7) 4 is written as (2)(2), both prime In the last case we say the 28 is “completely factored” because all the factors are prime numbers

Other Hints for Factoring Some people prefer to begin factoring by thinking of the smallest prime number that evenly divides the given number If the second factor is not prime, they again think of the smallest prime number that evenly divides it This process continues until all factors are prime Completely factor 120 120 = (2)(60) 60 is not prime, and is divisible by 2 120 = (2)(2)(30) 30 is not prime, and is divisible by 2 120 = (2)(2)(2)(15) 30 is not prime, and is divisible by 3 120 = (2)(2)(2)(3)(5) all factors are prime In the last case we say the 120 is “completely factored” because all the factors are prime numbers

Fundamental Principle of Fractions If the numerator and denominator of a fraction contain a common factor, that factor may be divided out to reduce the fraction to lowest terms: Reduce to lowest terms by factoring:

Summarizing the Process of Reducing Fractions Completely factor both numerator and denominator Apply the fundamental principle of fractions: divide out common factors that are found in both the numerator and the denominator

When to Reduce Fractions to Lowest Terms Unless there is a specific reason not to reduce, fractions should always be reduced to lowest terms A little later we will see that, when adding or subtracting fractions, it may be more important to have fractions with a common denominator than to have fractions in lowest terms

Multiplying Fractions Factor each numerator and denominator Divide out common factors Write answer Example:

Dividing Fractions Example: Invert the divisor and change problem to multiplication Example:

Adding Fractions Having a Common Denominator Add the numerators and keep the common denominator Example:

Adding Fractions Having a Different Denominators Write equivalent fractions having a “least common denominator” Add the numerators and keep the common denominator Reduce the answer to lowest terms

Finding the Least Common Denominator, LCD, of Fractions Completely factor each denominator Construct the LCD by writing down each factor the maximum number of times it is found in any denominator

Example of Finding the LCD Given two denominators, find the LCD: , Factor each denominator: Construct LCD by writing each factor the maximum number of times it’s found in any denominator:

Writing Equivalent Fractions Given a fraction, an equivalent fraction is found by multiplying the numerator and denominator by a common factor Given the following fraction, write an equivalent fraction having a denominator of 72: Multiply numerator and denominator by 4:

Adding Fractions Find a least common denominator, LCD, for the fractions Write each fraction as an equivalent fraction having the LCD Write the answer by adding numerators as indicated, and keeping the LCD If possible, reduce the answer to lowest terms

Example Find a least common denominator, LCD, for the rational expressions: Write each fraction as an equivalent fraction having the LCD: Write the answer by adding or subtracting numerators as indicated, and keeping the LCD: If possible, reduce the answer to lowest terms

Subtracting Fractions Find a least common denominator, LCD, for the fractions Write each fraction as an equivalent fraction having the LCD Write the answer by subtracting numerators as indicated, and keeping the LCD If possible, reduce the answer to lowest terms

Example Find a least common denominator, LCD, for the rational expressions: Write each fraction as an equivalent fraction having the LCD: Write the answer by adding or subtracting numerators as indicated, and keeping the LCD: If possible, reduce the answer to lowest terms

Improper Fractions & Mixed Numbers A fraction is called “improper” if the numerator is bigger than the denominator There is nothing wrong with leaving an improper fraction as an answer, but they can be changed to mixed numbers by doing the indicated division to get a whole number plus a fraction remainder Likewise, mixed numbers can be changed to improper fractions by multiplying denominator times whole number, plus the numerator, all over the denominator

Doing Math Involving Improper Fractions & Mixed Numbers Convert all numbers to improper fractions then proceed as previously discussed

Homework Problems Section: 1.1 Page: 11 Problems: Odd: 7 – 29, 33 – 51, 55 – 69 MyMathLab Homework 1.1 for practice MyMathLab Homework Quiz 1.1 is due for a grade on the date of our next class meeting

Exponential Expressions “3” is called the base “4” is called the exponent An exponent that is a natural number tells how many times to multiply the base by itself Example: What is the value of 34 ? (3)(3)(3)(3) = 81 An exponent applies only to the base (what it touches) Meanings of exponents that are not natural numbers will be discussed later

Order of Operations Many math problems involve more than one math operation Operations must be performed in the following order: Parentheses (and other grouping symbols) Exponents Multiplication and Division (left to right) Addition and Subtraction (left to right) It might help to memorize: Please Excuse My Dear Aunt Sally

Order of Operations Example: P E MD AS

Example of Order of Operations Evaluate the following expression:

Inequality Symbols An inequality symbol is used to compare numbers: Symbols include: greater than: greater than or equal to: less than: less than or equal to: not equal to: Examples: .

Expressions Involving Inequality Symbols Expressions involving inequality symbols may be either true or false Determine whether each of the following is true or false:

Translating to Expressions Involving Inequality Symbols English expressions may sometimes be translated to math expressions involving inequality symbols: Seven plus three is less than or equal to twelve Nine is greater than eleven minus four Three is not equal to eight minus six

Equivalent Expressions Involving Inequality Symbols A true expression involving a “greater than” symbol can be converted to an equivalent statement involving a “less then” symbol Reverse the expressions and reverse the direction of the inequality symbol 5 > 2 is equivalent to: 2 < 5 Likewise, a true expression involving a “less than symbol can be converted to an equivalent statement involving a “greater than” symbol by the same process 3 < 7 is equivalent to: 7 > 3

Homework Problems Section: 1.2 Page: 21 Problems: Odd: 5 – 19, 23 – 49, 53 – 79, 83 – 85 MyMathLab Homework 1.2 for practice MyMathLab Homework Quiz 1.2 is due for a grade on the date of our next class meeting

Terminology of Algebra Constant – A specific number Examples of constants: Variable – A letter or other symbol used to represent a number whose value varies or is unknown Examples of variables:

Terminology of Algebra Expression – constants and/or variables combined in a meaningful way with one or more math operation symbols for addition, subtraction, multiplication, division, exponents and roots Examples of expressions: Only the first of these expressions can be simplified, because we don’t know the numbers represented by the variables

Terminology of Algebra If we know the number value of each variable in an expression, we can “evaluate” the expression Given the value of each variable in an expression, “evaluate the expression” means: Replace each variable with empty parentheses Put the given number inside the pair of parentheses that has replaced the variable Do the math problem and simplify the answer

Example Evaluate the expression for : Consider the next similar, but slightly different, example

Example Evaluate the expression for : Notice the difference between this example and the previous one – it illustrates the importance of using a parenthesis in place of the variable

Example Evaluate the expression for :

Example Evaluate the expression for :

Translating English Phrases Into Algebraic Expressions Many English phrases can be translated into algebraic expressions: Use a variable to indicate an unspecified number Identify key words that imply: Add Subtract Multiply Divide

Phrases that Translate to Addition English Phrase A number plus 5 The sum of 3 and a number 4 more than a number A number increased by 8 Algebra Expression

Phrases that Translate to Subtraction English Phrase 4 less than a number A number subtracted from 7 6 subtracted from a number a number decreased by 9 2 minus a number Algebra Expression

Phrases that Translate to Multiplication English Phrase 7 times a number the product of 4 and a number double a number the square of a number Algebra Expression

Phrases that Translate to Division English Phrase the quotient of 2 and a number a number divided by 8 6 divided by a number Algebra Expression

Phrases Translating to Expressions Involving Multiple Math Operations English Phrase 4 less than 3 times a number the quotient of 5 and twice a number 6 times the difference between a number and 5 Algebra Expression

Phrases Translating to Expressions Involving Multiple Math Operations English Phrase the difference between 4 and 7 times a number the quotient of a number and 5, subtracted from the number the product of 3, and a number increased by 4 Algebra Expression

Equations Equation – a statement that two expressions are equal Equations always contain an equal sign, but an expression does not have an equal sign Like a statement in English, an equation may be true or false Examples: .

Equations Most equations contain one or more variables and the truthfulness of the equation depends on the numbers that replace the variables Example: What value of x makes this true? A number that can replace a variable to make an equation true is called a solution

Distinguishing Between Expressions & Equations Expressions contain constants, variables and math operations, but NO EQUAL SIGN Equations always CONTAIN AN EQUAL SIGN that indicates that two expressions have the same value

Solutions to Equations Earlier we said that any numbers that can replace variables in an equation to make a true statement are called solutions to the equation Soon we will learn procedures for finding solutions to an equation For now, if we have a set of possible solutions, we can find solutions by replacing the variables with possible solutions to see if doing so makes a true statement

Finding Solutions to Equations from a Given Set of Numbers From the following set of numbers, find a solution for the equation: Check x = 3 Check x = 4 Check x = 5

Writing Equations from Word Statements The same procedure is used as in translating English expressions to algebraic expressions, except that any statement of equality in the English statement is replaced by an equal sign Change the following English statement to an equation, then find a solution from the set of numbers Four more than twice a number is ten

Homework Problems Section: 1.3 Page: 29 Problems: Odd: 13 – 55, 59 – 81 MyMathLab Homework 1.3 for practice MyMathLab Homework Quiz 1.3 is due for a grade on the date of our next class meeting

Sets of Numbers Natural numbers Whole numbers Integers Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers Includes zero and all natural numbers: 0, 1, 2, 3, … (Does not include negative numbers) Integers Includes all whole numbers and their opposites (negatives): …, -3, -2, -1, 0, 1, 2, 3, …

Number Line Draw a line, choose a point on the line, and label it as 0 Choose some unit of length and place a series of points, spaced by that length, left and right of the 0 point Points to the right of zero are labeled in order 1, 2, 3, … Points to the left of zero starting at the point closest to zero and moving left are labeled in order, -1, -2, -3, … Notice that for any integer on the number line, there is another integer the same distance on the other side of zero that is the opposite of the first A number line is used for graphing integers and other numbers

Graphing Integers on a Number Line To graph an integer on a number line we place a dot at the point that corresponds to the given number and we label the point with the number The number label is called the “coordinate” of the point Graph -2:

Rational Numbers The next set of numbers to be considered will fill in some of the gaps between the integers on a number line Rational numbers Numbers that can be written as the ratio of two integers This includes all integers since they can be written as themselves over 1 This includes all fractions and their opposites (- ½ , ½, etc.) It also includes all decimals that either terminate ( .57 ) or have a a sequence of digits that form an infinitely repeating pattern at the end (.666…, written as .6, etc.)

Graphing Rational Numbers Positive rational numbers will correspond to a point right of zero and negative rational numbers will correspond to a point left of zero To find the location of the point, consider the mixed number equivalent of the given number If the number is positive: go to the right to the whole number divide the next interval into the number of divisions indicated by the denominator of the fraction continue to the right from the whole number to the division indicated by the numerator Place a dot at that point and label it with the coordinate If the number is negative: go to the left to the whole number continue to the left from the whole number to the division indicated by the numerator

Examples of Graphing Rational Numbers

Irrational Numbers It may seem that rational numbers would fill up all the gaps between integers on a number line, but they don’t The next set of numbers to be considered will fill in the rest of the gaps between the integers and rational numbers on a number line Irrational numbers Numbers that can not be written as the ratio of two integers This includes all decimals that do not terminate and do not have a sequence of digits that form an infinitely repeating pattern at the end Included in this set of numbers are any square roots of positive numbers that will not simplify to get rid of square root sign Examples:

Notes on Square Roots The square root of is written as and represents a number that multiplies by itself to give We know that the number that multiplies by itself to give is , so we write is a terminating decimal, so is a rational number We know of no number that multiplies by itself to give , but a calculator gives a decimal approximation that fills the screen without showing a repeating pattern at the end. is an irrational number Square roots may be rational, irrational, or neither

More Notes on Square Roots The square root of is written as , but it does not exist in the real number system (no real number multiplies by itself to give a negative is not rational or irrational. It’s not real, but is a type of number called an imaginary number, that will be studied in college algebra

Graphing Irrational Numbers Positive irrational numbers will correspond to a point right of zero and negative irrational numbers will correspond to a point left of zero To find the approximate location of the point, consider the decimal approximation If the number is positive: go to the right to the whole number divide the next interval into the number of divisions of accuracy desired (tenths, hundredths, etc.) continue to the right from the whole number to the division indicated by the digits right of the decimal point Place a dot at that point and label it with the coordinate If the number is negative: go to the left to the whole number continue to the left from the whole number to the division indicated by the digits right of the decimal point

Example of Graphing Irrational Numbers

Real Numbers The set of rational numbers and the set of irrational numbers have no numbers in common When the two sets of numbers are put together they make up a new set of numbers called “real numbers” Every real number is either rational or irrational There is a one-to-one correspondence between points on a number line and the set of real numbers There are some numbers that are not real numbers, an example is: . These type of numbers (complex numbers) will be discussed in college algebra.

Ordering Real Numbers Given two real numbers, represented by the variables a and b, one of the following order relationships is true: a = b a equals b if they graph at the same location a < b a is less than b, if a is left of b on a number line a > b a is greater than b, if a is right of b on a number line

Additive Inverses of Real Numbers Every real number has an additive inverse The additive inverse of a real number is the number located on a number line the same distance from zero, but in the opposite direction The additive inverse of a number is the same as its opposite The additive inverse of 5 is: The additive inverse of -3 is: Placing in negative sign in front of a number is a way of indicating the additive inverse of the number If we want to indicate the additive inverse of -7, we can place a negative sign in front of -7: - (-7) is the same as:

Absolute Value of Real Numbers Every real number has an absolute value The absolute value of a real number is its “distance” from zero Distance is never negative, so absolute value is never negative Absolute value of a number is indicated by placing vertical bars around the number The absolute value of 5 is shown by : and is equal to: The absolute value of -3 is shown by: and is equal to:

Homework Problems Section: 1.4 Page: 39 Problems: All: 9 – 20 Odd: 23 – 27, 35 – 63 MyMathLab Homework 1.4 for practice MyMathLab Homework Quiz 1.4 is due for a grade on the date of our next class meeting

Addition of Real Numbers Addition – like a game between two teams, “Positive” and “Negative,” the answer to the problem is the answer to the question, “Who won the game, and by how much?” Example: Reasoning: Negatives scored: Positives scored: _________ won by ____, so

Second Example of Addition Reasoning: Negatives scored: Positives scored: _________ won by ____, so:

Addition of Signed Fractions Addition rule is the same for all signed numbers, but you must first write each fraction as an equivalent fraction where all fractions have a common denominator Example: Reasoning: Negatives scored: Positives scored: _________ won by ________, so:

Addition of Signed Decimals Addition rule is the same for all signed numbers, but be sure to line up decimal points before adding or subtracting Example: Reasoning: Negatives scored: Positives scored: _________ won by ____, so:

Subtraction of Real Numbers Subtract means “add the opposite” All subtractions are changed to “add the opposite” and then the problem is done according to addition rules already discussed In identifying a subtraction problem remember that the same symbol, - , is used between numbers to mean “subtract” and in front of a number to mean “negative number” .

Problems Involving Both Addition and Subtraction Example: Identify subtraction: Add opposite: Reasoning: Negatives scored: Positives scored: _________ won by ____, so:

Homework Problems Section: 1.5 Page: 49 Problems: Odd: 7 – 97 MyMathLab Homework 1.5 for practice MyMathLab Homework Quiz 1.5 is due for a grade on the date of our next class meeting

Multiplying and Dividing Real Numbers Multiplication and Division of signed numbers follows the rule: Do problem as if both were positive Answer is positive if signs were the same Answer is negative if signs were opposite Examples: .

Multiplying Signed Fractions Basic rule has already been discussed Otherwise, remember to: Divide out factors common to top & bottom Multiply top factors to get top Multiply bottom factors to get bottom Example:

Dividing Signed Fractions Basic rule has already been discussed Otherwise, remember to: Invert the second fraction and change problem to multiplication Complete using rules for multiplication Example:

Division Involving Zero People are often confused when division involves zero – the rule must be memorized! Division by zero is always undefined Otherwise, division into zero is always zero Explanation comes from checking answer: .

Order of Operations Many math problems involve more than one math operation Operations must be performed in the following order: Parentheses (and other grouping symbols) Exponents Multiplication and Division (left to right) Addition and Subtraction (left to right) It might help to memorize: Please Excuse My Dear Aunt Sally

Homework Problems Section: 1.6 Page: 63 Problems: Odd: 11 – 73, 77 – 113 MyMathLab Homework 1.6 for practice MyMathLab Homework Quiz 1.6 is due for a grade on the date of our next class meeting

Averaging Real Numbers To average a set of real numbers we add all the numbers and then divide by the number of numbers in the set Find the average of the following set of numbers: .

Divisibility A real number is divisible by another if the division has no remainder On the following slides are tests for divisibility by all the numbers between 2 and 9, except for 7 (there is no test for divisibility by 7) Memorize these tests

Test for Divisibility by 2 A real number is divisible by 2 only if its last digit is even Which of the following numbers are divisible by 2? 31,976,104 257 1,348 35,750

Test for Divisibility by 3 A real number is divisible by 3 only if the sum of its digits is divisible by 3 Which of the following numbers are divisible by 3? 51,976,104 357 1,348 45,750

Test for Divisibility by 4 A real number is divisible by 4 only if the last two digits form a number that is divisible by 4 Which of the following numbers are divisible by 4? 51,976,104 357 1,348 45,750

Test for Divisibility by 5 A real number is divisible by 5 only if the last digit is 5 or 0 Which of the following numbers are divisible by 5? 51,976,104 357 1,348 45,750

Test for Divisibility by 6 A real number is divisible by 6 only if it passes both the test for divisibility by 2 and divisibility by 3 Which of the following numbers are divisible by 6? 51,976,104 357 1,348 45,750

Test for Divisibility by 8 A real number is divisible by 8 only if its last three digits form a number divisible by 8 Which of the following numbers are divisible by 8? 51,976,104 357 1,348 45,750

Test for Divisibility by 9 A real number is divisible by 9 only if the sum of its digits is divisible by 9 Which of the following numbers are divisible by 9? 51,976,104 357 1,348 45,750

Homework Problems Section: 1.6 Page: 63 Problems: All: 115 – 119, 121 – 127 MyMathLab Homework 1.6a for practice MyMathLab Homework Quiz 1.6a is due for a grade on the date of our next class meeting

Properties of Real Numbers Commutative Property – the order in which real numbers are added or multiplied does not effect the result: Associative Property – the way real numbers are grouped during addition or multiplication does not effect the result:

Properties of Real Numbers Commutative Property Examples: Associative Property Examples:

Properties of Real Numbers Identity Property for Addition – when zero is added to a number, the result is still the number: Identity Property for Multiplication – when one is multiplied by a number, the result is still the number:

Properties of Real Numbers Identity Property for Addition Example: Identity Property for Multiplication Examples:

Properties of Real Numbers Inverse Property for Addition – when the opposite (negative) of a number is added to the number, the result is zero: Inverse Property for Multiplication – when the reciprocal of a number is multiplied by the number, the result is one

Reciprocals of Real Numbers Zero has no reciprocal Reciprocals of other integers are formed by putting 1 over the number Reciprocals of fraction are formed by switching the numerator and denominator

Properties of Real Numbers Inverse Property for Addition Examples: Inverse Property for Multiplication Examples:

Properties of Real Numbers Distributive Property – multiplication can be distributed over addition or subtraction without changing the result

Illustration of Distributive Property

Illustration of Distributive Property Distributive Property works both directions: If two terms contain a common factor, that factor can be written outside parentheses with the remaining factors remaining as terms inside parentheses

Illustration of Distributive Property Use the Distributive Property “backwards” to write each of the following in a different way:

Homework Problems Section: 1.7 Page: 74 Problems: All: 1 – 30, 35 – 50, 55 – 80 MyMathLab Homework 1.7 for practice MyMathLab Homework Quiz 1.7 is due for a grade on the date of our next class meeting

Terminology of Algebra Constant – A specific number Examples of constants: Variable – A letter or other symbol used to represent a number whose value varies or is unknown Examples of variables:

Terminology of Algebra Expression – constants and/or variables combined with one or more math operation symbols for addition, subtraction, multiplication, division, exponents and roots in a meaningful way Examples of expressions: Only the first of these expressions can be simplified, because we don’t know the numbers represented by the variables

Terminology of Algebra Term – an expression that involves only a single constant, a single variable, or a product (multiplication) of a constant and variables Examples of terms: Note: When constants and variables are multiplied, or when two variables are multiplied, it is common to omit the multiplication symbol Previous example is commonly written:

Terminology of Algebra Every term has a “coefficient” Coefficient – the constant factor of a term (If no constant is seen, it is assumed to be 1) What is the coefficient of each of the following terms?

Like Terms Recall that a term is a _________ , a ________, or a _______ of a ________ and _________ Like Terms: terms are called “like terms” if they have exactly the same variables with exactly the same exponents, but may have different coefficients Example of Like Terms:

Determine Like Terms Given the term: Which of the following are like terms?

Adding Like Terms When “like terms” are added, the result is a like term and its coefficient is the sum of the coefficients of the other terms Example: The reason for this can be shown by the distributive property:

Subtracting Like Terms When like terms are subtracted, the result is a like term with coefficient equal to the difference of the coefficients of the other terms Example: Reasoning:

Simplifying Expressions by Combining Like Terms Any expression containing more than one term may contain like terms, if it does, all like terms can be combined into a single like term by adding or subtracting as indicated by the sign in front of each term Example: Simplify:

Review of Distributive Property Distributive Property – multiplication can be distributed over addition or subtraction Some people make the mistake of trying to distribute multiplication over multiplication Example: Associative Property justifies answer! !!

+ or – in Front of Parentheses When a + or – is found in front of a parentheses, we assume that it means “positive one” or “negative one” Examples:

Multiplying Terms Terms can be combined into a single term by addition or subtraction only if they are like terms Terms can always be multiplied to form a single term by using commutative and associative properties of multiplication Example:

Simplifying an Expression Get rid of parentheses by multiplying or distributing Combine like terms Example:

Homework Problems Section: 1.8 Page: 80 Problems: All: 5 – 30 Odd: 33 – 75 MyMathLab Homework 1.8 for practice MyMathLab Homework Quiz 1.8 is due for a grade on the date of our next class meeting