2014 Math 100 Developmental Math I Fall 2014 R. Getso South Texas College

Transcription

1 2014 Math 100 Developmental Math I Fall 2014 R. Getso South Texas College

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3 Course Contents Module I 1.7 Exponents and Order of Operations Introduction to Variables, Algebraic Expressions, and Equations Introduction to Integers Adding Integers Subtracting Integers Multiplying and Dividing Integers Order of Operations Simplifying Algebraic Equations Module II 4.1 Introduction to Fractions and Mixed Numbers Factors and Simplest Form Multiplying and Dividing Fractions Adding and Subtracting Like Fractions, Least Common Denominator, and Equivalent Fractions Adding and Subtracting UNLike Fractions Ratio and Proportion Percents, Decimals and Fractions Solving Percent Problems with Equations Solving Percent Problems with Proportions Applications of Percent Percent and Problem Solving: Sales Tax, Commission and Discount Module III 3.2 Solving Linear Equations: Review of Addition and Multiplication Properties Solving Linear Equations in One Variable Linear Equations in One Variable and Problem Solving Solving Equations containing Fractions Further Solving Linear Equations : Further Problem Solving Formulas and Problem Solving i

4 Module IV 10.1 Exponents Negative Exponents and Scientific Notation Introduction to Polynomials Adding and Subtracting Polynomials Multiplying Polynomials End of Course Topics 5.7Decimal Applications: Mean, Median and Mode Reading Pictographs, Bar Graphs, Histograms and Line Graphs Reading and Drawing Circle Graphs Counting and Introduction to Probability Square Roots and Pythagorean Theorem Similar Triangles Perimeter Area ii

5 Math 100 Class Notes 1 Module I 1.7 Exponents and Order of Operations Exponents An exponent is a shorthand notation for repeated multiplication means 3 is a factor 5 times Using an exponent, this product can be written as base exponent is read as three to the fifth power. This is called exponential notation. The exponent, 5, indicates how many times the base, 3, is a factor. 4 is read as four to the first power. 4 4=4 is read as four to the second power or four squared. Usually, an exponent of 1 is not written, so when no exponent appears, assume that the exponent is 1. For example, 2=2 and 7=7. To evaluate an exponential expression, we write the expressions a product and then find the value of the product. 3 = = means =32 Order of Operations 1. Perform all operations within parentheses ( ), brackets [ ], or other grouping symbols such as fraction bars, starting with the innermost set. 2. Evaluate any expressions with exponents. 3. Multiply or divide in order from left to right. 4. Add or subtract in order from left to right.

6 2 R. Getso Examples: Evaluate each of the following ( ) [3 + (9 7) 2 ] Introduction to Variables, Algebraic Expressions, and Equations A combination of operations on letters (variables) and numbers is called an algebraic expression. Algebraic Expressions means 4 and means Replacing a variable in an expression by a number and then finding the value of the expression is called evaluating the expression for the variable. Examples: 1. Evaluate + for = 5 and = 2 2. Evaluate ; for = 2, = 4, = 1

7 Math 100 Class Notes 3 When an equation contains a variable, deciding which values of the variable make an equation a true statement is called solving an equation for the variable. A solution of an equation is a value for the variable that makes an equation a true statement. Determine whether a number is a solution: 3. Is -2 a solution of the equation = 3? 4. Is 6 a solution of the equation 5 1 = 30? 5. 6( + 2) = 24; {1,2,3}

8 4 R. Getso Addition (+) Subtraction ( ) Multiplication ( ) Division ( ) sum difference product quotient plus Minus times divide added to subtract multiply shared equally among more than less than multiply by per increased by decreased by of divided by total less double/triple divided into Examples: Translate less than a number 7. twice a number 8. a number decreased by 3 9. a number increased by four times a number subtracted from a number 12. three times the difference of a number and the quotient of 5 and a number

9 Math 100 Class Notes Introduction to Integers We compare integers just as we compare whole numbers. For any two numbers graphed on a number line, the number to the right is the greater number and the number to the left is the smaller number. < means is less than Example: < means is less than > means is greater than Example: > means 12 is greater than 3 Insert <, > or =

10 6 R. Getso Since the absolute value of a number is that number s distance from 0, the absolute value of a number is always 0 or positive. It is never negative. = = zero a positive number Examples: Insert <, > or = ( 3) ( 7) ( 4) ( 15) 5 is the opposite of 5 and 5 is the opposite of 5. The opposite of 4 is 4 and is written as (4) = 4. The opposite of 4 is 4 and is written as ( 4) =4. If is a number, then ( ) =. Remember that 0 is neither positive nor negative. Therefore, the opposite of 0 is 0.

11 Math 100 Class Notes 7 Examples: Simplify. 10. ( 27) ( 33) Evaluate. 14., = 7 15., = , = Adding Integers Step 1: Add their absolute values. Step 2: Use their common sign as the sign of the sum. Step 1: Find the larger absolute value minus the smaller absolute value. Step 2: Use the sign of the number with the larger absolute value as the sign of the sum.

12 8 R. Getso If is a number, then is its opposite. + ( ) = 0 The sum of a + = 0 number and its opposite is 0 Helpful Hint: Don t forget that addition is commutative and associative. In other words, numbers may be added in any order. Examples: Simplify Evaluate + for = 5 and = Subtracting Integers If and are numbers, then = +( ). To subtract two numbers, add the first number to the opposite (called additive inverse) of the second number. Examples: ( 6) 4. 8 ( 2) ( 5) ( 7) 6. Evaluate for = 6 and = 8

13 Math 100 Class Notes Multiplying and Dividing Integers Multiplying Integers The product of two numbers having the same signs is a positive number. Product of Same Signs (+)(+) =+ ( )( ) =+ The product of two numbers having different signs is a negative number. Product of Different Signs ( )(+) = (+)( ) = Dividing Integers The quotient of two numbers having the same signs is a positive number. Quotient of Same Signs ( ) =+ ( ) =+ ( ) ( ) The quotient of two numbers having different signs is a negative number. Quotient of Different Signs ( ) = ( ) = ( ) ( ) Examples: Simplify each of the following

14 10 R. Getso 2.5 Order of Operations Examples: Simplify ( 14) + ( 20) 3. 4(5 2) ( 8) ( 13) ( 23) ( 4) (2 9) + ( 12) 3

15 Math 100 Class Notes 11 Remember: When simplifying expressions with exponents, parentheses make an important difference. ( 5) 2 and 5 2 do not mean the same thing. ( 5) 2 means ( 5)( 5) = means the opposite of 5 5, or 25. Note: Only with parentheses around it is the 5 squared. 3.1 Simplifying Algebraic Equations A term that is only a number is called a constant term, or simply a constant. A term that contains a variable is called a variable term ( 4 ) + 2 Variable Term Constant Term Variable Terms Constant Term The number factor of a variable term is called the numerical coefficient. The coefficient of the term 3 is simply 3. A numerical coefficient of 1 is usually not written. That is, we usually write, not 1. Terms that are exactly the same, except that they may have different numerical coefficients are called like terms. Distributive Property Like Terms 3, 2 5, Unlike Terms 6, 2, 7, 7 3, 4 5, 5 2, 5 3, 7 If a, b, and c are numbers, then + = ( + ) and = ( ) A sum or difference of like terms can be simplified using the distributive property. Here is an example of combining like terms using the distributive property =(7+5) =12 An algebraic expression is simplified when all like terms have been combined. The commutative and associative properties of addition and multiplication help simplify expressions.

16 12 R. Getso Commutative Properties of Addition and Multiplication If a, b, and c are numbers, then + = + Commutative Property of Addition = Commutative Property of Multiplication The order of adding or multiplying two numbers can be changed without changing their sum or product. The grouping of numbers in addition or multiplication can be changed without changing their sum or product. Associative Properties of Addition and Multiplication If a, b, and c are numbers, then ( + ) + = + ( + ) Associative Property of Addition ( ) = ( ) Associative Property of Multiplication Examples of Commutative and Associative Properties of Addition and Multiplication = Commutative Property of Addition 6 9 = 9 6 Commutative Property of Multiplication (3 + 5) + 2 = 3 + (5 + 2) Associative Property of Addition (7 1) 8 = 7 (1 8) Associative Property of Multiplication We can also use the distributive property to multiply expressions. The distributive property says that multiplication distributes over addition and subtraction. 2(5+ )=2 5+2 =10 +2 or 2(5 ) =2 5 2 =10 2 To simplify expressions, use the distributive property first to multiply and then combine any like terms. Examples: Simplify each of the following algebraic expressions. 1. 3(5 + ) ( 3) + 17

17 Math 100 Class Notes ( 7) ( + 5) Find the perimeter of the triangle. 6. Find the area of the rectangle. 3z feet 9z feet 7z feet 3 meters (2x 5) meters Helpful Hints: Perimeter: distance around measured in units Area: surface enclosed measured in square units

18 14 R. Getso Module II 4.1 Introduction to Fractions and Mixed Numbers Whole numbers are used to count whole things. To refer to a part of a whole, fractions are used. Parts of a Fraction A fraction is a number of the form, where a and b are integers and b is not 0. The parts of a fraction are Numerator Denominator a b Fraction Bar Helpful Hint: Remember that the bar in a fraction means division. Since division by 0 is undefined, a fraction with a denominator of 0 is undefined. Visualizing Fractions One way to visualize fractions is to picture them as shaded parts of a whole figure.

19 Math 100 Class Notes 15 Types of Fractions A proper fraction is a fraction whose numerator is less than its denominator. Proper fractions have values that are less than ,, An improper fraction is a fraction whose numerator is greater than or equal to its denominator. Improper fractions have values that are greater than or equal to ,, A mixed number is a sum of a whole number and a proper fraction ,3, Fractions on Number Lines Another way to visualize fractions is to graph them on a number line. Fraction Properties of 1 If n is any integer other than 0, then = 1 If n is any integer, then 1 = Fraction Properties of 0 If n is any integer other than 0, then 0 = 0 If n is any integer, then = undefined Writing an Improper Fraction as a Mixed Number Step 1: Divide the denominator into the numerator. Step 2: The whole number part of the mixed number is the quotient. The fraction part of the mixed number is the remainder over the original denominator.

20 16 R. Getso Examples: Write each improper fraction as a mixed number Writing a Mixed Number as a Fraction or Whole Number Step 1: Multiply the denominator of the fraction by the whole number. Step 2: Add the numerator of the fraction to the product from Step 1. Step 3: Write the sum from Step 2 as the numerator of the improper fraction over the original denominator. Examples: Write each mixed number as an improper fraction Factors and Simplest Form Prime and Composite Numbers A prime number is a natural number greater than 1 whose only factors are 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29,... A composite number is a natural number greater than 1 that is not prime. Helpful Hint: The natural number 1 is neither prime nor composite. Prime Factorization A prime factorization of a number expresses the number as a product of its factors and the factors must be prime numbers. Helpful Hint: Remember a factor is any number that divides a number evenly (with a remainder of 0). Every whole number greater than 1 has exactly one prime factorization. 12 = and 3 are prime factors of 12 because they are prime numbers and they divide evenly into 12.

21 Math 100 Class Notes 17 Examples: Find the prime factorization of the numbers given Divisibility A whole number is divisible by: 2 if its last digit is 0, 2, 4, 6, or 8. 3 if the sum of its digits is divisible by 3. 5 if the ones digit is 0 or 5. 4 if its last two digits are divisible by 4. 6 if it s divisible by 2 and 3. 9 if the sum of its digits is divisible by 9. Equivalent Fractions Fractions that represent the same portion of a whole or the same point on the number line are called equivalent fractions. Fundamental Property of Fractions If a, b, and c are numbers, then = and also = as long as b and c are not 0. If the numerator and denominator are multiplied or divided by the same nonzero number, the result is an equivalent fraction.

22 18 R. Getso Simplest Form A fraction is in simplest form, or lowest terms, when the numerator and denominator have no common factors other than 1. Using the fundamental principle of fractions, divide the numerator and denominator by the common factor of = = 2 3 Using the prime factorization of the numerator and denominator, divide out common factors = = 2 3 = 2 3 Writing a Fraction Simplest Form To write a fraction in simplest form, write the prime factorization of the numerator and the denominator and then divide both by all common factors. The process of writing a fraction in simplest form is called simplifying the fraction. Examples: Write each fraction in simplest form Helpful Hint: When all factors of the numerator or denominator are divided out, don t forget that 1 still remains in that numerator or denominator.

23 Math 100 Class Notes Multiplying and Dividing Fractions Multiplying TWO Fractions If a, b, c, and d are numbers and b and d are not 0, then = In other words, to multiply two fractions, multiply the numerators and multiply the denominators. Examples: Multiply Helpful Hint: Recall that when the denominator of a fraction contains a variable, such as assume that the variable is not 0., we Expressions with Exponential Bases 2 3 = = = 8 27

24 20 R. Getso Examples: Evaluate Reciprocal of a Fraction Two numbers are reciprocals of each other if their product is 1. The reciprocal of the fraction is because Dividing TWO Fractions If b, c, and d are not 0, then = = =1 = = = In other words, to divide fractions, multiply the first fraction by the reciprocal of the second fraction. Helpful Hint: Every number has a reciprocal except 0. The number 0 has no reciprocal. Why? Examples: Divide

25 Math 100 Class Notes 21 Evaluating with Fractions Examples: Given the following replacement values, evaluate for (a) xy and (b) x y 10. = ; = 11. = ; = (a) (a) (b) (b) 4.4 Adding and Subtracting Like Fractions, Least Common Denominator, and Equivalent Fractions Fractions that have the same or common denominator are called like fractions. Fractions that have different denominators are called unlike fractions. Like Fractions 2 5 and and 3 7 Unlike Fractions 2 3 and and 5 12 Adding and Subtracting Fractions If a, b, and c, are numbers and b is not 0, then + = + and also = Helpful Hint: To add or subtract fractions with the same denominator, add or subtract their numerators and write the sum or difference over the common denominator.

26 22 R. Getso Examples: Add Do not forget to write the answer in simplest form. If it is not in simplest form, divide out all common factors larger than 1. To add or subtract fractions that have unlike, or different, denominators, we write the fractions as equivalent fractions with a common denominator. Least Common Denominator The least common denominator (LCD) of a list of fractions is the smallest positive number divisible by all the denominators in the list. (The least common denominator is also the least common multiple (LCM) of the denominators.) Method 1: Finding the LCD of a List of Fractions Using Multiples of the Largest Number Step 1: Write the multiples of the largest denominator (starting with the number itself) until a multiple common to all denominators in the list is found. Step 2: The multiple found in Step 1 is the LCD. Example: Find the LCD of the list of fractions using Method and Method 2: Finding the LCD of a List of Denominators Using Prime Factorization Step1: Write the prime factorization of each denominator. Step 2: For each different prime factor in Step 1, circle the greatest number of times that factor occurs in any one factorization. Step 3: The LCD is the product of the circled factors. Example: Find the LCD of the list of fractions using Method and

27 Math 100 Class Notes 23 Example: Find the LCD. 6. and 7. and 8. and 9.,, Equivalent Fractions To add or subtract unlike fractions, first write equivalent fractions with the LCD as the denominator. To write an equivalent fraction, = = where, and are nonzero numbers. Examples: 10. Write 12 5 as an equivalent fraction with a denominator of Write 8 7 as an equivalent fraction with a denominator of Write 5 3 as an equivalent fraction with a denominator of 35. 2x 13. Write as an equivalent fraction with a 3 denominator of 12.

28 24 R. Getso 4.5 Adding and Subtracting UNLike Fractions Adding and Subtracting Fractions Step 1: Find the least common denominator (LCD) of the fractions. Step 2: Write each fraction as an equivalent fraction whose denominator is the LCD. Step 3: Add or subtract the like fractions. Step 4: Write the sum or difference in simplest form. Examples: Add or subtract as indicated

29 Math 100 Class Notes 25 Writing Fractions in Order One important application of the least common denominator is to use the LCD to help order or compare fractions. Insert < or > to form a true sentence.? The LCD for these fractions is 35. Write each fraction as an equivalent fraction with a denominator of 35. = = = = Compare the numerators of the equivalent fractions. Since 21 >20, then >. Thus, >. Examples: Insert <, > or = Ratio and Proportion Writing Ratio as Fractions A ratio is the quotient of two quantities. For example, a percent can be thought of as a ratio, since it is the quotient of a number and % = or the ratio of 53 to 100. The ratio of a number a to a number b is their quotient. Ways of writing ratios are: a to b, a : b, and

30 26 R. Getso Examples: Write the ratio in lowest terms hours to 36 hours minutes to 3 hours to $4 Solving Proportions A proportion is a statement that two ratios are equal. If and are ratios, then = is a proportion. Example: 4. Is = a true proportion? Examples: Solve each proportion. 5. = 6. =

31 Math 100 Class Notes = 8. A 16-oz Cinnamon Mocha Iced Tea at a local coffee shop has 80 calories. How many calories are there in a 28-oz Cinnamon Mocha Iced Tea? Helpful Hint: When writing proportions to solve problems, write the proportions so that the numerators have the same unit measures and the denominators have the same unit measures. For example, 2 inches 5 miles = 7 inches miles 6.2 Percents, Decimals and Fractions Understanding Percent The word percent comes from the Latin phrase per centum, which means per 100. Percent means per one hundred. The % symbol is used to denote percent. 1% = =0.01 Writing a Decimal as a Percent Multiply by 1 in the form of 100% = 0.65(100%) = 65.% or 65% Writing a Percent as a Decimal Replace the percent symbol % with its decimal equivalent, 0.01; then multiply. 43% = 43(0.01) = % = 100(0.01) = 1.00 or 1

32 28 R. Getso Writing percent as a fraction Replace the percent symbol, %, with its fraction equivalent, simplify the fraction, if possible. 43% = 43 = ; then multiply. Don t forget to Writing a Fraction as a Percent 3 5 = % = %=300 5 %=60% Helpful Hint: 100% = 1 Recall that when we multiply a number by 1 or 100%, we are not changing the value of that number. Summary To write a percent as a decimal, replace the % symbol with its decimal equivalent, 0.01; then multiply. To write a percent as a fraction, replace the % symbol with its fraction equivalent, ; then multiply. To write a decimal or fraction as a percent, multiply by 100%.

33 Math 100 Class Notes 29 Examples: Complete the Table. Percent % Decimal Fraction 85% %

34 30 R. Getso 6.3 Solving Percent Problems with Equations Key Words of means multiplication ( ) is means equals (=) what (or some equivalent) means the unknown number Let x stand for the unknown number. Helpful Hint: Remember that an equation is simply a mathematical statement that contains an equal sign (=). Examples: Translate and Solve: is what percent of 40? 2. What number is 18% of 66? 3. 25% of 68 is what number? is 6% of what number?

35 Math 100 Class Notes Solving Percent Problems with Proportions amount percent = base 100 Key Words When we translate percent problems to proportions, the percent, p, can be identified by looking for the symbol % or the word percent. The base, b, usually follows the word of. The amount, a, is the part compared to the whole. Examples: Write as a proportion and Solve: is what percent of 40? 2. What number is 18% of 66? 3. 25% of 68 is what number? is 6% of what number?

36 32 R. Getso 6.5 Applications of Percent Examples: Solve. 1. The freshman class of 862 students is 32% of all students at State College. How many students go to State College? 2. A family paid $22,100 as a down payment for a home. If this represents 13% of the price of the home, find the price of the home. 3. The number of people attending the local play increased from 285 on Friday to 333 on Saturday. Find the percent increase in attendance. 4. There are 120 calories in a cup of whole milk and only 73 calories in a cup of skim milk. In switching to skim milk, find the percent decrease in number of calories.

37 Math 100 Class Notes Percent and Problem Solving: Sales Tax, Commission and Discount Sales Tax Percents are frequently used in the retail trade. For example, most states charge a tax on certain items when purchased. This tax is called a sales tax, and retail stores collect it for the state. Sales tax is almost always stated as a percent of the purchase price. A 5% sales tax rate on a purchase of a $10.00 item gives sales tax = 5% of $10 =0.05 $10.00 =$0.50 sales tax = tax rate purchase price total price = purchase price + sales tax Examples: Solve. 1. If the sales tax rate is 7.75%, find the amount of sales tax due on a purchase of $ Also find the total that the customer must pay. 2. What is the sales tax on a jacket priced at $425 if the sales tax is 7%?

38 34 R. Getso 3. The purchase price of a camcorder is $974. What is the total price if the sales tax rate is 8.5%? 4. A new plasma screen TV has a purchase price of $4460. If the sales tax on this purchase is $267.60, find the sales tax rate. Commission A wage is payment for performing work. Hourly wage, commissions, and salary are some of the ways wages can be paid. Many people who work in sales are paid a commission. An employee who is paid a commission is paid a percent of his or her total sales. commission = commission rate sales Examples: Solve. 5. Adam is a salesperson at a furniture store and earns a 4.5% commission on everything he sells. Last week, he sold furniture totaling $38,957. How much did Adam earn in commissions last week? 6. How much commission will Jack make on the sale of a $188,900 house if he receives 8.3% of the selling price? 7. A salesperson earned a commission of $ for selling $9790 worth of paper products. Find the commission rate.

39 Math 100 Class Notes 35 Discount Suppose that an item that normally sells for $40 is on sale for 25% off. This means that the original price of $40 is reduced, or discounted, by 25% of $40, or $10. The discount rate is 25%, the amount of discount is $10, and the sale price is $40 - $10, or $30. amount of discount = discount rate original price sale price = original price - amount of discount Examples: Solve. 8. This week at Brown s Department Store, purses are discounted 15%. Find the sale price of a purse that costs $ Complete the table. Original Price Discount Rate Amount of Discount Sale Price $390 15%

40 36 R. Getso Module III 3.2 Solving Linear Equations: Review of Addition and Multiplication Properties Examples: Solve each of the following. Don t forget to check your solution. 1. 2(2 3) = (5 3) = = 5(4 6) 4. = 1 ( 8)

41 Math 100 Class Notes 37 Translate each of the following. 5. The sum of 7 and a number 6. Negative four times a number, increased by The product of 6 and the sum of a number and Twice the sum of a number and 5 9. The quotient of 20 and a number, decreased by three

42 38 R. Getso 3.3 Solving Linear Equations in One Variable 3x 2 = 7 is called a linear equation or first degree equation in one variable. The exponent on each x is 1 and there is no variable below a fraction bar. It is an equation in one variable because it contains one variable, x. Make sure you understand which property to use to solve an equation. Addition: Multiplication: + 5 = 8 3 = 12 To undo addition of 5, we subtract 5 from both To undo multiplication of 3, we divide both sides. sides by =8 5 3 Using Addition Property of Equality. 3 = 12 3 =3 Use Multiplication Property of Equality. = 4 Steps for Solving a One Variable Linear Equation: Step 1: If parentheses are present, use the distributive property. Step 2: Combine any like terms on each side of the equation. Step 3: Use the addition property to rewrite the equation so that the variable terms are on one side of the equation and constant terms are on the other side. Step 4: Use the multiplication property of equality to divide both sides by the numerical coefficient of the variable to solve. Step 5: Check the solution in the original equation.

43 Math 100 Class Notes 39 Examples: Solve each of the following equations = = (5 + ) = 5( 4) = 0 Translate. 5. The sum of 42 and 16 is Negative 2 times the sum of 3 and 12 is What s wrong with the following solution? 2(3 5) = = = = = = 2

44 40 R. Getso 3.4 Linear Equations in One Variable and Problem Solving 1. UNDERSTAND the problem. During this step, become comfortable with the problem. Some ways of doing this are: Read and reread the problem. Construct a drawing. Propose a solution and check. Pay careful attention to how you check your proposed solution. This will help when writing an equation to model the problem. Choose a variable to represent the unknown. Use this variable to represent any other unknowns. 2. TRANSLATE the problem into an equation. 3. SOLVE the equation. 4. INTERPRET the results. Check the proposed solution in the stated problem and state your conclusion. Examples: 1. Five times a number subtracted from 40 is the same as three times the number. Find the number. 2. Based on the 2000 Census, Florida has 28 fewer electoral votes for president than California. If the total number of electoral votes for these two states is 82, find the number for each state. 1. UNDERSTAND Choose a variable to represent the unknown. Use this variable to represent any other unknowns. 2. TRANSLATE the problem into an equation. 3. SOLVE the equation. 4. INTERPRET the results. Check the proposed solution in the stated problem and state your conclusion.

45 Math 100 Class Notes The product of number and 3 is twice the difference of that number and 8. Find the number. 4. An Xbox 360 game system and several games are sold for $560. The cost of the Xbox 360 is 3 times as much as the cost of the games. Find the cost of the Xbox 360 and the cost of the games.

46 42 R. Getso 4.8 Solving Equations containing Fractions Addition Property Let a, b, and c represent numbers. If a = b, then + = + and =. In other words, the same number may be added to or subtracted from both sides of an equation without changing the solution of the equation. Multiplication Property Let a, b, and c represent numbers and let c 0. If a = b, then = and = In other words, both sides of an equation may be multiplied or divided by the same nonzero number without changing the solution of the equation. Method One: LCD Method Step 1: If fractions are present, multiply both sides of the equation by the LCD of the fractions. Step 2: If parentheses are present, use the distributive property. Step 3: Combine any like terms on each side of the equation. Step 4: Use the addition property of equality to rewrite the equation so that variable terms are on one side of the equation and constant terms are on the other side. Step 5: Divide both sides of the equation by the numerical coefficient of x to solve. Step 6: Check the answer in the original equation. Examples: Solve. 1. = 2. = + 3. = 4. 2 =

47 Math 100 Class Notes 43 Solving Equations containing Fractions applying Properties First Solve: = To get alone, we add to both sides of the equation. 1 3 = = = + Original Equation Add on both sides Multiply by = + Common denominator is 12 = Add fractions = Simplify Examples: Solve. 5. = 6. 2 = 4 7. = 2 8. = +

48 44 R. Getso 9.3 Further Solving Linear Equations Solving linear equations in one variable Step 1: Step 2: Step 3: Step 4: Step 5: Step 6: If an equation contains fractions, multiply both sides by the LCD to clear the equation of fractions. Use the distributive property to remove parentheses if they are present. Simplify each side of the equation by combining like terms. Get all variable terms on one side and all numbers on the other side by using the addition property of equality. Get the variable alone by using the multiplication property of equality. Check the solution by substituting it into the original equation. Example: Solve. 1. = ( 8) = (70) = 2( + 1) (3 + 2) =

49 Math 100 Class Notes = 3( + 1) 6. 2 = 9.4: Further Problem Solving 1. UNDERSTAND the problem. During this step, become comfortable with the problem. Some ways of doing this are: Read and reread the problem. Choose a variable to represent the unknown. Construct a drawing. Propose a solution and check it. Pay careful attention to how you check your proposed solution. This will help when writing an equation to model the problem. 2. TRANSLATE the problem into an equation. 3. SOLVE the equation. 4. INTERPRET the results. Check the proposed solution in the stated problem and state your conclusion. Angles in a Triangle The sum of the measures of the angles of any triangle is 180. A+B+C=180

50 46 R. Getso Examples: Solve. 1. The measure of the second angle of a triangle is twice the measure of the smallest angle. The measure of the third angle of the triangle is three times the measure of the smallest angle. Find the measures of the angles. 2. A 10-foot board is to be cut into two pieces so that the length of the longer piece is 4 times the length of the shorter piece. Find the length of each piece. 3. Some states have a single area code for the entire state. Two such states have area codes that are consecutive odd integers. If the sum of these integers is 1208, find the two area codes.

51 Math 100 Class Notes Formulas and Problem Solving A formula is an equation that states a known relationship among multiple quantities (has more than one variable in it). = Area of a rectangle = length width = simple Interest = Principal Rate Time = + + Perimeter of a triangle = side + side + side = distance = rate time = h Volume of a rectangular solid = length width height = Degrees in Fahrenheit = 5 9 Degrees in Celsius + 32 Examples: Solve. 1. A flower bed is in the shape of a triangle with one side twice the length of the shortest side, and the third side is 30 feet more than the length of the shortest side. Find the dimensions if the perimeter is 102 feet. 2. The average temperature for January in Algiers, Algeria is 59 Fahrenheit. Find the equivalent temperature in Celsius. Solving a Formula for a Variable It is often necessary to rewrite a formula so that it is solved for one of the variables. To solve a formula or an equation for a specified variable, we use the same steps as for solving a linear equation except that we treat the specified variable as the only variable in the equation.

52 48 R. Getso Solving a Formula for a Specified Variable Step 1: Multiply on both sides to clear the equation of fractions if they appear. Step 2: Use the distributive property to remove parentheses if they appear. Step 3: Simplify each side of the equation by combining like terms. Step 4: Get all terms containing the specified variable on one side and all other terms on the other side by using the addition property of equality. Step 5: Get the specified variable alone by using the multiplication property of equality. Examples: Solve for the indicated variable. 3. = 2 ; solve for 4. = + + ; solve for 5. = ; solve for = 9; solve for

53 Math 100 Class Notes 49 Module IV 10.1 Exponents Exponents that are natural numbers are shorthand notation for repeating factors. 3 = is the base and 4 is the exponent (also called power or degree) Note: By the order of operations, exponents are calculated before other operations. Examples: Evaluate ( 5) (2 4) Find 3 when = 5 7. Find 2 when = 1 Product Rule If m and n are positive integers and a is a real number, then = Examples: Evaluate (3 )( 4 ) In other words, to multiply two exponential expressions with the same base, we keep the base and add the exponents. We call this simplifying the exponential expression. Don t forget that if no exponent is written, it is assumed to be 1.

54 50 R. Getso Power Rule If m and n are positive integers and a is a real number, then ( ) = Examples: Evaluate. 12. (2 ) 13. ( ) Power of a Product Rule If n is a positive integer and a and b are real numbers, then ( ) = Examples: Evaluate. 14. (5 ) 15. ( 2 ) Power of a Quotient Rule If n is a positive integer and a and b are real numbers, then =, 0 Examples: Evaluate Quotient Rule If m and n are positive integers and a is a real number, then =, 0 Examples: Evaluate

55 Math 100 Class Notes 51 Zero Exponents =1, 0 Note: 0 is undefined. Examples: Evaluate ( ) Negative Exponents and Scientific Notation Negative Exponents Using the quotient rule from section 10.1, =, 0, we have = = Note: 0 But what does mean? = = 1 = 1 Note: 0 means 1 In order to extend the quotient rule to cases where the difference of the exponents would give us a negative number we define negative exponents as follows. If a is a real number other than 0, and n is an integer, then = or = Examples: Evaluate ( 3) 6. 7.

56 52 R. Getso Summary of Exponent Rules If m and n are integers and a and b are real numbers, then: = Product Rule for exponents ( ) = Power Rule for exponents ( ) = Power of a Product =, 0 Power of a Quotient =, 0 Quotient Rule for exponents = 1, 0 Zero exponents = 1, 0 or = 1, 0 Negative exponents Examples: Simplify Scientific Notation In many fields of science we encounter very large or very small numbers. Scientific notation is a convenient shorthand for expressing these types of numbers. A positive number is written in scientific notation if it is written as the product of a number a, where 1 a < 10, and an integer power r of 10: 10

57 Math 100 Class Notes 53 To Write a Number in Scientific Notation Step 1: Move the decimal point in the original number so that the new number has a value between 1 and 10. Step 2: Count the number of decimal places the decimal point is moved in Step 1. If the original number is 10 or greater, the count is positive. If the original number is less than 1, the count is negative. Step 3: Multiply the new number in Step 1 by 10 raised to an exponent equal to the count found in Step 2. Examples: Write the number using scientific notation ,000,000, In general, to write a scientific notation number in standard form, move the decimal point the same number of spaces as the exponent on 10. If the exponent is positive, move the decimal point to the right. If the exponent is negative, move the decimal point to the left. Examples: Write the number using standard form ( )( ) ( )(3 10 ) 20.

58 54 R. Getso 10.3 Introduction to Polynomials Terminology Term a number or a product of a number and variables raised to powers Coefficient numerical factor of a term Constant term which is only a number (no letters) Degree the exponent on the variable Degree of a Polynomial with One Variable the greatest degree of any term of the polynomial Degree of a Term with More Than One Variable the sum of the exponents on the variables Types of Polynomials Monomial is a polynomial with exactly one term. Binomial is a polynomial with exactly two terms. Trinomial is a polynomial with exactly three terms. Polynomial is a finite sum of terms of the form ax n, where a is a real number and n is a whole number. Which term is missing in the polynomial? Degree The degree of a term in one variable is the exponent on the variable. The degree of a constant is 0. The degree of a polynomial is the greatest degree of any term of the polynomial. The degree of 9x 3 4x is 3. The degree of a term with more than one variable is the sum of the exponents on the variables. The degree of the term 5a 4 b 3 c is 8 (remember that c can be written as c 1 ). Evaluating a polynomial for a particular value involves replacing the value for the variable(s) involved. Example: Find the value when x =

59 Math 100 Class Notes 55 Combining Like Terms Like terms are terms that contain exactly the same variables raised to exactly the same powers. Example: Combine like terms Adding and Subtracting Polynomials To Add Polynomials - Combine all the like terms. To Subtract Polynomials - Change the signs of the terms of the polynomial being subtracted, and then combine all the like terms.

60 56 R. Getso Example: Add or subtract as indicated. 1. (3 8) + ( ) 2. 4 ( 4) 3. ( + 1) ( 3 + 9) + ( ) 4. (3 8) ( ) Helpful Hint: Don t forget to change the sign of each term in the polynomial being subtracted Multiplying Polynomials If all of the polynomials are monomials, use the associative and commutative properties. If any of the polynomials are not monomials, use the distributive property before the associative and commutative properties. Then combine like terms. Multiply Two polynomials Multiply each term of the first polynomial by each term of the second polynomial, and then combine like terms.

61 Math 100 Class Notes 57 Example: Multiply. 1. (3 )( 2 ) 2. (4 )( ) DISTRIBUTION 3. (2 4)(7 + 5) FOIL 4. (3 7 )(7 + 2 ) 5. (3 + 4)

62 58 R. Getso End of Course Topics 5.7Decimal Applications: Mean, Median and Mode The mean, the median, and the mode are called measures of central tendency. They describe a set of data, or a set of numbers, by a single middle number. Finding the Mean The most common measure of central tendency is the mean (sometimes called the arithmetic mean or the average ). The mean (average) of a set of number items is the sum of the items divided by the number of items. Examples: Find the mean of the following lists of numbers , 5.1, 9.5, 6.8, , 49, 72, 91 Finding the Median You may have noticed that a very low number or a very high number can affect the mean of a list of numbers. Because of this, you may sometimes want to use another measure of central tendency, called the median. The median of an ordered set of numbers is the middle number. If the number of items is even, the median is the mean (average) of the two middle numbers. Helpful Hint: In order to compute the median, the numbers must first be placed in order. Examples: Find the median of the following lists of numbers , 5.1, 9.5, 6.8, , 49, 72, 91

63 Math 100 Class Notes 59 Finding the Mode The mode of a set of numbers is the number that occurs most often. (It is possible for a set of numbers to have more than one mode or to have no mode.) Helpful Hint: Don t forget that it is possible for a list of numbers to have no mode. For example, the list 2, 4, 5, 6, 8, 9 has no mode. There is no number or numbers that occur more often than the others. Examples: Find the mode of the following lists of numbers , 5.1, 9.5, 6.8, , 49, 72, Reading Pictographs, Bar Graphs, Histograms and Line Graphs Reading Pictographs A pictograph is a graph in which pictures or symbols are used. This type of graph contains a key that explains the meaning of the symbol used. An advantage of using a pictograph to display information is that comparisons can easily be made. A disadvantage of using a pictograph is that it is often hard to tell what fractional part of a symbol is shown. The pictograph shows the top eight most-spoken (primary) languages. Examples: 1. Approximate the number of people who speak primarily English. 2. Approximate the number of people who speak primarily Spanish.

64 60 R. Getso Reading and Constructing Bar Graphs Bar graphs can appear with vertical bars or horizontal bars. An advantage to using bar graphs is that a scale is usually included for greater accuracy. This bar graph shows the number of different endangered species. Examples: 1. Approximate the number endangered species that are crustaceans. 2. Which category has the most endangered species? 3. Draw a vertical bar graph using the information in the table below about electoral votes for selected states. Electoral State Votes Texas 34 California 55 Florida 27 Nebraska 5 Indiana 11 Georgia 15

65 Math 100 Class Notes 61 Reading and Constructing Histograms A histogram is a special bar graph. The width of each bar represents a range of numbers called a class interval. The height of each bar corresponds to how many times a number in the class interval occurred and is called the class frequency. The bars in a histogram lie side by side with no space between them. This histogram shows test scores of 36 students. Number of Students Examples: 4. How many students scored on the test? Use the histogram above Student Test Scores 5. How many students scored 70 or higher on the test? Use the histogram above. 6. Construct a histogram for the frequency distribution shown below.

66 62 R. Getso Reading Line Graphs Another common way to display information graphically is by using a line graph. An advantage of a line graph is that it can be used to visualize relationships between two quantities. A line graph can also be very useful in showing change over time. This is a line graph showing average daily temperatures. Examples: 7. During which month is the average daily temperature the highest? 8. During which months is the average daily temperature less than 30 F?

67 Math 100 Class Notes Reading and Drawing Circle Graphs A circle graph is often used to show percents in different categories, with the whole circle representing 100%. The following circle graph shows the percent of visitors to the United States in a recent year by various regions. Examples: 1. Use the circle graph shown above to determine the percent of visitors who came to the United States from Mexico and Canada. 2. Use the circle graph shown above to determine the percent of visitors who came to the United States from Europe, Asia and South America.

68 64 R. Getso 7.5 Counting and Introduction to Probability Using a Tree Diagram In our daily conversations, we often talk about the likelihood or the probability of a given result occurring for a chance happening. We call the chance happening an experiment. The possible results of an experiment are called outcomes. Flipping a coin is an experiment and the possible outcomes are heads (H) or tails (T) and are equally likely to happen. One way to picture the outcomes of an experiment is to draw a tree diagram. Each outcome is shown on a separate branch. For example, the outcomes of flipping a coin once are H T

69 Math 100 Class Notes 65 Examples: 1. Draw a tree diagram for tossing a coin three times. Then use the diagram to find the number of possible outcomes. Finding the Probability of an Event probability of an event = number of ways the event can occur number of possible outcomes To find the probability of an event, divide the number of ways that the event can occur by the number of possible outcomes. Note from the definition of probability that the probability of an event is always between 0 and 1, inclusive (i.e., including 0 and 1). A probability of 0 means an event won t occur, and a probability of 1 means that an event is certain to occur. 2. If a die is rolled one time, find the probability of rolling a 3 or a 4.

70 66 R. Getso 7.3 Square Roots and Pythagorean Theorem Square of a Number The square of a number is the number times itself. The square of 6 is 36 because 6 2 = 36. The square of 6 is also 36 because ( 6) 2 = ( 6) ( 6) = 36. Square Root of a Number A square root of 36 is 6 because 6 2 = 36. A square root of 36 is also 6 because ( 6) 2 = 36. We use the symbol number., called a radical sign, to indicate the positive square root of a nonnegative 16 =4 because 4 =16 and 4 is positive. 25 =5 because 5 =25 and 5 is positive. The square root, symbols,, of a positive number a is the positive number b whose square is a. In = if = 9 =3 because 3 =9 Also, 0 =0 Helpful hint: Remember that the radical sign is used to indicate the positive square root of a nonnegative number. Perfect Squares Numbers like,36, number or a fraction.,and 1 are called perfect squares because their square root is a whole Approximating a Square Root A square root such as 6 cannot be written as a whole number or a fraction since 6 is not a perfect square. It can be approximated by estimating, by using a table, or by using a calculator. Right Triangles A right triangle is a triangle in which one of the angles is a right angle or measures The hypotenuse of a right triangle is the side opposite the right angle. (degrees).

71 Math 100 Class Notes 67 The legs of a right triangle are the other two sides. Pythagorean Theorem If and are the lengths of the legs of a right triangle and is the length of the hypotenuse, + = (leg) +(other leg) =(hypotenuse) Examples: 1. Find the square root. 2. Determine what two whole numbers the square root is between without using a calculator or table. Then use a calculator or table to check Find the length of the missing side. 4. Find the height of the tree.

72 68 R. Getso 7.4 Similar Triangles Similar triangles are found in art, engineering, architecture, biology, and chemistry. Two triangles are similar when they have the same shape but not necessarily the same size. Examples: 1. Find the length of the side labeled n of the following pair of similar triangles n 8 2. Find. 3. Find. n

73 Math 100 Class Notes Perimeter The perimeter of a polygon is the distance around the polygon, which is the sum of the lengths of its sides. Circumference

74 70 R. Getso Examples: Find the perimeter of the figure A line-marking machine lays down lime powder to mark a diamond shape between each corner, how many feet of lime powder will be deposited on the path? 4. Wiley just bought a trampoline for his children to use. The trampoline has a diameter of 6 feet. If he wishes to put a netting to go around of the trampoline, how many feet of netting does he need? Give the exact circumference of the trampoline and then the approximation.

75 Math 100 Class Notes Area Helpful Hint: Area is always measured in square units. When finding the area of figures, check to make sure that all measurements are in the same units before calculations are made. Examples: Find the area of the geometric figure Give an exact area and then use as an approximation for to approximate the area. 3.