MATH 251 Beginning Algebra

Transcription

1 Class Notes for MATH 251 Beginning Algebra Spring 2013 Prepared by Stephanie Sorenson

2 Table of Contents 1.1 Fractions (REVIEW) Exponents, Order of Operations, and Inequality Variables, Expressions, and Equations Real Numbers and the Number Line Adding and Subtracting Real Numbers Multiplying & Dividing Real Numbers Properties of Real Numbers Simplifying Expressions The Addition Property of Equality The Multiplication Property of Equality More on Solving Linear Equations An Introduction to Applications of Linear Equations Formulas and Additional Applications from Geometry Ratio, Proportion, and Percent Further Applications of Linear Equations Solving Linear Inequalities Linear Equations in Two Variables; The Rectangular Coordinate System Graphing Linear Equations in Two Variables The Slope of a Line Writing and Graphing Equations of Lines Graphing Linear Inequalities in Two Variables Introduction to Functions Solving Systems of Linear Equations by Graphing Solving Systems of Linear Equations by Substitution Solving Systems of Linear Equations by Elimination Applications of Linear Systems Solving Systems of Linear Inequalities The Product Rule and Power Rules for Exponents Integer Exponents and the Quotient Rule Scientific Notation Adding and Subtracting Polynomials Multiplying Polynomials Special Products Dividing Polynomials The Greatest Common Factor; Factoring by Grouping

3 6.2 Factoring Trinomials More on Factoring Trinomials Special Factoring Techniques Solving Quadratic Equations by Factoring Applications of Quadratic Equations The Fundamental Property of Rational Expressions Multiplying and Dividing Rational Expressions Least Common Denominators Adding and Subtracting Rational Expressions Complex Fractions Solving Equations with Rational Expressions Applications of Rational Expressions Variation Evaluating Roots Multiplying, Dividing, and Simplifying Radicals Adding and Subtracting Radicals Rationalizing the Denominator More Simplifying and Operations with Radicals Solving Equations with Radicals Using Rational Numbers as Exponents Solving Quadratic Equations by the Square Root Property Solving Quadratic Equations by Completing the Square Solving Quadratic Equations by the Quadratic Formula

4 Section Fractions (REVIEW) Example 1 Write the fraction in lowest terms. SOLUTION: Short-hand notation: We mentally cancel out the greatest common factor (GCF) 6. FASTER!! Example 2 Find the product, and write it in lowest terms. SOLUTION: Example 3 Find the product, and write it in lowest terms. SOLUTION: Short-hand notation: KICK the denominator. AKA: Cancel out the greatest common factor (GCF) 3. PUNCH the numerator. AKA: Multiply FASTER!!

5 2 Section 1.1 Example 4 Find the quotient, and write it in lowest terms. SOLUTION: Example 5 Find the sum, and write it in lowest terms. SOLUTION: The least common denominator (LCD) is 24.

6 Section Exponents, Order of Operations, and Inequality Exponent Notation Example 1 Find the value of the exponential expression. The base is ; The exponent is. (a) (b) (c) (d) Order of Operations P E M D A S Example 2 Find the value of each expression. (a) (b)

7 4 Section 1.2 Parentheses Exponents Multiplication Division Addition Subtraction (c) (d) Inequalities The alligator always chomps on the bigger number! is less than is greater than is not equal to Example 3 Determine whether each statement is true or false. If the statement is false, change the inequality sign to make a correct statement. (a) (b) (c)

8 Section More Inequalities is less than OR equal to is greater than OR equal to Example 4 Determine whether each statement is true or false. If the statement is false, change the inequality sign to make a correct statement. (a) (b) Example 5 Write each word statement in symbols. (a) Nine is equal to eleven minus two. (b) Fourteen is greater than twelve. (c) Two is greater than or equal to two. (d) Twelve is not equal to five.

9 6 Section 1.3 Consider the following question: 1.3 Variables, Expressions, and Equations There are 40 enrolled students. How many petitioners must there be if there are 45 students in class on the first day? We can represent this problem with an equation as follows: Enrolled students Petitioners Total students in class In algebra, instead of drawing a box with a question mark, we use a letter of the alphabet to represent the unknown number: A is a symbol, usually a letter such as unknown number. or, used to represent any An is a sequence of numbers, variables, operation symbols ( ), and/or grouping symbols that DOES NOT INCLUDE an equality (=) or inequality ( ) sign. The following is an algebraic expression: Example 1 Find the value of the algebraic expression when. Interpret your result in terms of what represents.

10 Section Use caution when writing expressions with exponents or finding their value. is different than Exponent ONLY applies to Exponent applies to the entire quantity in the parentheses Example 2 Find the value of each algebraic expression if. (a) (b) (c) Example 3 Find the value of each algebraic expression if and. (a) (b)

11 8 Section 1.3 Example 4 Write each word phrase as an algebraic expression, using as the variable. (a) The sum of 3 and a number (b) A number minus 8 (c) A number subtracted from 4 (d) Eight subtracted from a number (e) Two less than a number (f) The difference between a number and 2 (g) The difference between 5 and a number (h) 14 times a number (i) Twelve divided by a number (j) The quotient of a number and 8 (k) Nine multiplied by the sum of a number and 5 (l) The product of 3 and five less than a number ****************************************************************************** Let s take another look at the original question posed at the beginning of this section. There are 40 enrolled students. How many petitioners must there be if there are 45 students in class on the first day? We already found that we could represent the problem with the following equation: where represents the unknown number of petitioners. An is a statement that two algebraic expressions are equal. An equation ALWAYS INCLUDES an equality (=) sign. To an equation means to find the values of the variable that make the equation true. Such values of the variable are called the of the equation.

12 Section Example 5 Decide whether the given number is a solution of the equation. (a) (b) Example 6 Write the statement as an equation. Use as the variable. Then find all solutions from the set (a) The product of a number and 2 is 6. (b) One less than twice a number is 15. Example 7 Identify each as an expression or an equation. (a) (b) (c)

13 10 Section Real Numbers and the Number Line Natural Numbers: Whole Numbers: Integers: Rational Numbers: Includes: All integers, mixed fractions, terminating (ending) decimals, repeating decimals Irrational Numbers: Includes: All non-terminating, non-repeating decimals Real Numbers: The Real Number Line: 0

14 Section Ordering of Real Numbers For any two real numbers and, is less than if is to the of. Example 1 Determine whether the statement is true or false. (a) (b) (c) Additive Inverse The of a number is the number that is the same distance from 0 on the number line as, but on the opposite side of 0.

15 12 Section 1.4 Double Negative Rule For any real number, Absolute Value The absolute value, of a real number is: *Distance is NEVER negative. Therefore, the absolute value of a number is never negative. Example 2 Simplify. (a) (b) (c) Example 3 Decide whether each statement is true or false. (a) (b) (c) (d)

16 Section Adding and Subtracting Real Numbers To add two numbers with the same sign: Mentally ignore the signs and add. The answer has the same sign as the given numbers. eg. ; To add two numbers with different signs: Mentally ignore the signs and subtract the smaller number from the larger number. The answer has the same sign as the bigger number in the problem. eg. ; Subtraction is addition of the additive inverse: eg. eg. eg. Example 1 Perform each indicated operation Answers: 1) 2) 3) 4) 5) 6) 7) 8) 9) 10)

17 14 Section 1.5 Example 2 Write a numerical expression for each phrase and simplify. a) The sum of and 5 and b) added to the sum of and 11 c) The sum of and, increased by 14 d) more than the sum of and e) The difference between and f) less than g) The sum of 12 and, decreased by 14 h) 19 less than the difference between 9 and

18 Section Multiplying & Dividing Real Numbers Multiplication For all real numbers, Rules of Signs: Example 1 Find all integer factors of 50. Division Observe, and Any two numbers whose product is 1 are called reciprocals, or. The reciprocal of is. The reciprocal of is. What is the reciprocal of 0??. ****************************************************************************** Definition of Division: For any two real numbers and, ( ) In Words: divided by equals times the reciprocal of. ****************************************************************************** Note: Rules of Signs:

19 16 Section 1.6 Example 2 Perform the indicated operation(s). (a) (f) (b) (g) (c) (h) (d) (e) Example 3 Evaluate. Let (b) (c) (e) (f)

20 Section Example 4 Evaluate. Let (b) (c) (e) (f) Example 5 Evaluate each expression for,, and.

21 18 Section 1.6 Words or phrases that mean MULTIPLY: Product, times, double, twice, of, as much as Words or phrases that mean DIVIDE: Quotient, divided by, ratio of ******************************************************************** Example 6 Write a numerical expression for each phrase and simplify. (a) The product of 4 and, added to (b) Twice the product of and, subtracted from (c) Three subtracted from the product of 4 and (d) The product of and the difference between and (e) The quotient of and the sum of and (f) The sum of and, divided by the product of and (g) Three-fourths of the sum of and 12 (h) of the product of and 5

22 Section Example 7 Write each sentence as an equation, using integers between and, inclusive. as the variable. Then find the solution from the set of (a) The quotient of a number and 4 is. (b) 7 less than a number is 2.

23 20 Section Properties of Real Numbers For all real numbers, Commutative Properties (addition) (multiplication) Example 1 Use the commutative properties to complete the equality. (a) (b) For all real numbers, Associative Properties (addition) (multiplication) Example 2 Use the associative properties to complete the equality. (a) (b) For all real numbers Identity Properties and and (addition) (multiplication) Example 3 Use the identity properties to complete the equality. (a) (b)

24 Section For all real numbers Inverse Properties and and (addition) (multiplication) Example 4 Use the inverse properties to complete the equality. (a) (b) For all real numbers Distributive Property and and (addition) (subtraction) Example 5 Use the distributive property to complete the equality. (a) (b) ****************************************************************************** Example 6 Decide whether each statement is an example of the commutative, associative, identity, inverse or distributive property. (a) (b) (c) (d) (e)

25 22 Section 1.7 Example 7 Find each sum or product without using a calculator. (a) (b) (c) Example 8 Simplify each expression. (a) (b) (c)

26 Section Simplifying Expressions Example 1 Simplify each expression. (Use the properties of Section 1.7) (a) (b) A term is a number, a variable, or a product or quotient of numbers and variables raised to powers. The numerical coefficient of a term is the number in front of the variables. Identify the numerical coefficient of each term: Term Coefficient Terms with exactly the same variables that have the same exponents (but possibly different coefficients) are called like terms. Identify each group of terms as like or unlike. and and and and

27 24 Section 1.8 Using the distributive property ( in the reverse direction ) is called combining like terms. For example, It is important to be able to distinguish between terms (separated by a or ) and factors (multiplied): however, Example 2 Simplify each expression. (a) (b) (c) (d) (e) Example 3 Translate the phrase into a mathematical expression. Use as the variable. Combine like terms when possible. A number multiplied by 5, subtracted from the sum of 14 and eight times the number

28 Section The Addition Property of Equality Definition A linear equation in one variable can be written in the form where and are real numbers, and. Recall, a solution of an equation is a number that makes the equation true when it replaces the variable. An equation is solved by finding its solution set, the set of all solutions. Equations with exactly the same solution sets are equivalent equations. Example of equivalent equations: and Addition Property of Equality If and are real numbers, then the equations and are equivalent equations. A B *Since subtraction is addition of the additive inverse, we can also subtract the same number from each side of the equation without changing the solution. Example 1 Solve each equation. a) b) c) d)

29 26 Section 2.1 Example 2 Solve each equation. a) b) c) d) Example 3 Write an equation using the information given in the problem. Use equation. as the variable. Then solve the One added to three times a number is three less than four times the number. Find the number. 26

30 Section The Multiplication Property of Equality Multiplication Property of Equality If and ( ) are real numbers, then the equations and are equivalent equations. A B *Since division is multiplication by the reciprocal, we can also divide each side of an equation by the same nonzero number without changing the solution. Example 1 Solve each equation. a) b) c) d) e) f)

31 28 Section 2.2 Example 2 Solve each equation. a) b) c) d) Example 3 Write an equation using the information given. Use as the variable. Then solve the equation. When a number is multiplied by, the result is 10. Find the number.

32 Section Steps for solving a linear equation: 2.3 More on Solving Linear Equations Step 1: Simplify each side separately. Step 2: Collect all the variable term on one side. Step 3: Isolate the variable. You can check your answer by plugging your solution into the original equation. Example 1 Solve. (a) (b) (c) (d) ****************************************************************************** Where did the word Algebra come from? The word algebra comes from the work: Hisab al-jabr m al muquabalah written by Muhammed ibn Musa Al-Khowarizmi in the 800 s A.D. Translated, the title means: The Science of Transposition and Cancellation

33 30 Section 2.3 If fractions appear in an equation, CLEAR THE FRACTIONS by multiplying both sides of the equation (or every term!) by the least common denominator. If decimals appear in an equation, CLEAR THE DECIMALS by multiplying both sides of the equation (or every term!) by either 10, 100, 1000, etc Example 2 Solve.

34 Section There are 3 types of linear equations: 1) An equation with exactly one solution is called a. Solution Set: where is the solution 2) An equation for which every real number is a solution is called an. Solution Set: 3) An equation that has no solution is called a. Solution Set: Example 3 Solve each equation. Then state whether the equation is a conditional equation, an identity, or a contradiction. (a)

35 32 Section 2.3 (b) (c) Example 4 Perform each translation. a) Two numbers have a sum of 34. One of the numbers is. What expression represents the other number? b) The product of two numbers is. One of the numbers is. What expression represents the other number?

36 Section c) A football player gained yards on a punt return. On the next return, he gained 6 yd. What expression represents the number of yards he gained altogether? d) A hockey player scored 42 goals in one season. He scored n goals in one game. What expression represents the number of goals he scored in the rest of the games? e) Chandler is m years old. What expression represents his age 4 yr ago? 11 yr from now? f) Claire has y dimes. Express the value of the dimes in cents. g) A clerk has v dollars, all in $20 bills. What expression represents the number of $20 bills the clerk has? h) A concert ticket costs p dollars for an adult and q dollars for a child. Find an expression that represents the total cost for 4 adults and 6 children.

37 34 Section An Introduction to Applications of Linear Equations Solving an Applied Problem Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 Read the problem carefully until you understand what is given and what is to be found. Assign a variable to represent the unknown value, using diagrams or tables as needed. Write down what the variable represents. If necessary, express any other unknown values in terms of the variable. Write an equation using the variable expression(s). Solve the equation. State the answer. Does it seem reasonable? Check the answer in the words of the original problem. Example 1 If 5 is added to the product of 9 and a number, the result is 19 less than the number. Find the number. (Assign the variable. Write an equation and solve.) Example 2 In the 2006 Winter Olympics in Torino, Italy, Canada won 5 more medals than Norway. The two countries won a total of 43 medals. How many medals did each country win? (Assign the variable. Write an equation and solve.)

38 Example 3 The owner of Terry s Coffeehouse found that the number of orders for croissants was 1/6 the number of orders for muffins. If the total number for the two breakfast rolls was 56, how many orders were placed for croissants? (Assign the variable. Write an equation and solve.) Section Example 4 At a meeting of the local computer user group, each member brought two nonmembers. If a total of 27 people attended, how many were members and how many were nonmembers? (Assign the variable. Write an equation and solve.)

39 36 Section 2.4 Example 5 A piece of pipe is 50 in. long. It is cut into three pieces. The longest piece is 10 in. more than the middle-sized piece, and the shortest piece measures 5 in. less than the middle-sized piece. Find the lengths of the three pieces. (Assign the variable. Write an equation and solve.) Definitions Two integers that differ by 1 are called consecutive integers. Two integers that differ by 2 are called consecutive even integers or consecutive odd integers. Examples: Consecutive integers:,,,, Consecutive even integers:,, Consecutive odd integers:,,,

40 Section Consecutive Integer Word Problems If the lesser integer, then for any two consecutive integers, use three consecutive integers, use two consecutive even (or odd) integers, use three consecutive even (or odd) integers, use Example 6 Two back-to-back page numbers in this book have a sum of 569. What are the page numbers? (Assign the variable. Write an equation and solve.)

41 38 Section 2.4 Example 7 Find two consecutive even integers such that six times the lesser added to the greater gives a sum of 86. (Assign the variable. Write an equation and solve.)

42 2.5 Formulas and Additional Applications from Geometry Important Formulas from Geometry Figure Formulas Illustration Section *Square Perimeter: Area: *Rectangle Perimeter: Area: *Triangle Perimeter: Area: *Parallelogram Perimeter: Area: *You must know the formulas for these figures Example 1 Find the value of the remaining variable in the formula.

43 40 Section 2.5 Example 2 A farmer has 800 m of fencing material to enclose a rectangular field. The width of the field is 50 m less than the length. Find the dimensions of the field. Example 3 The longest side of a triangle is 1 in. longer than the medium side. The medium side is 5 in. longer than the shortest side. If the perimeter is 32 in., what are the lengths of the three sides?

44 Example 4 The area of a triangle is 120 m 2. The height is 24 m. Find the length of the base of the triangle. Section Geometry Review Type of Angles Measurement Figure Vertical Equal Measure Complementary Sum to Supplementary Sum to Example 5 Find the measure of each marked angle in the figure.

45 42 Section 2.5 Example 6 Solve each formula for the specified variable. for for for for

46 Section Ratio, Proportion, and Percent A ratio is a comparison of two quantities (with the same units) using a quotient: to Ratios are often used when comparison shopping at the grocery store! We set up the ratio of the price of the item to the number of units on the label to obtain the price per unit. The lowest price per unit is the better value. Example 1 The local supermarket charges the following prices for a jug of pancake syrup. Which size is the best buy? Size Price 12 oz $ oz $ oz $3.89 A proportion says that two ratios are equal is to as is to Beginning with the proportion, and multiplying each side by the common denominator yields: We see that the products and can also be found by multiplying diagonally. We call this cross multiplying.

47 44 Section 2.6 Example 2 Decide whether each proportion is true or false. Example 3 Solve the equation Example 4 If 12 gallons of gasoline costs $37.68, how much would 16.5 gallons of the same fuel cost?

48 Section Further Applications of Linear Equations Percents should be represented as decimals. For example, or Interest Problems: (Principal) x (% Interest Rate) = Annual Interest Example 1 Find the annual interest if $5000 is invested at 4%. Example 2 With income earned by selling a patent, an engineer invests some money at 5% and $3000 more than twice as much at 8%. The total annual income from the investments is $1710. Find the amount invested at 5%. Assign a variable: Principal % Interest Annual Interest Write an equation & solve:

49 46 Section 2.7 Mixture Problems: Solution Substance X (Amount of Solution) x (% concentration of substance X) = Amount of substance X Example 3 What is the amount of pure acid in 40 L of a 16% acid solution? Example 4 A certain metal is 40% copper. How many kilograms of this metal must be mixed with 80 kg of a metal that is 70% copper to get a metal that is 50% copper? Assign a variable: Amt of Metal (kg) % Copper Amt of Copper (kg) Write an equation & solve:

50 Section Money Denominations Problems: = (Number) x (Value of one item) = Total Value Example 5 A man has $2.55 in quarters and nickels. He has 9 more nickels than quarters. How many nickels and how many quarters does he have? Assign a variable: Number of coins Coin Value Total Value Write an equation & solve: Distance-Rate-Time Problems: 65 mph = (Rate) x (Time) = Distance Example 6 A new world record in the men s 100-m dash was set in 2005 by Asafa Powell of Jamaica, who ran it in 9.77 sec. What was his speed in meters per second?

51 48 Section 2.7 Example 7 Two airplanes leave Boston at 12:00 noon and fly in opposite directions. If one flies at 410 mph and the other 120 mph faster, how long will it take them to be 3290 mi apart? Assign a variable: Rate Time Distance Write an equation & solve: Example 8 Two buses left the downtown terminal, traveling in opposite directions. One had an average speed of 10 mph more than the other. Twelve minutes (1/5 hour) later, they were 12 mi apart. What were their speeds? Assign a variable: Rate Time Distance Write an equation & solve:

52 Section Solving Linear Inequalities Interval Notation Inequality Interval Notation Graph ) b ] b All real numbers ( a [ a ( a ( a [ a a ) b ] b ) b [ ] b Example 1 Write an inequality involving the variable ) that describes the set of numbers graphed ( ] Example 2 Write each inequality in interval notation, and graph the interval. a) b) c)

53 50 Section 2.8 A linear inequality in one variable can be written in the form where, and are real numbers, with. Consider the (true) inequality: Now add 4 to both sides: Is the inequality still true?? Again, consider the (true) inequality: Now subtract 4 from both sides: Is the inequality still true?? Addition Property of Inequality For any real numbers,, and, the inequalities Also, and and are equivalent. are equivalent. Example 1 Solve the inequality. Write the solution set in interval notation and graph it.

54 Section Consider the (true) inequality: Now multiply both sides by 4: Is the inequality still true?? Again, consider the (true) inequality: Now multiply both sides by : Is the inequality still true?? Multiplication Property of Inequality For any real numbers,, and, with, 1. If is positive, and are equivalent. 2. If is negative, and are equivalent. Since division is multiplication by the reciprocal, the property applies to division: 1. If is positive, and are equivalent. 2. If is negative, and are equivalent. In words: When multiplying or dividing by a negative #, remember to. Example 2 Solve the inequality. Write the solution set in interval notation and graph it. (a) (b)

55 52 Section 2.8 Example 3 Solve the inequality. Write the solution set in interval notation and graph it. (a) (b) When solving three-part inequalities, the goal is to isolate in the middle: Keep in mind that whatever you do to the middle expression, you must do to each of the three parts of the inequality! Also, sometimes through solving a three-part inequality, the signs reverse and you get the following: Do not leave your answer in this form!! You need to reverse the order so that all the inequalities point in the less than direction: Example 4 Solve the inequality. Write the solution set in interval notation and graph it. (a) (b)

56 Section Phrase A number is more than 4. A number is less than. A number is at least 6. A number is at most 8. Inequality Example 5 You score 98, 86 and 88 on the first three exams. If you want to average an A on all the exams (a score of at least 90), what score must you get on the 4th exam?

57 54 Section Linear Equations in Two Variables; The Rectangular Coordinate System A linear equation in two variables is an equation that can be written in the form: where and are real numbers and and are not both 0. Example 1 Write the following equations in the form. Then identify and. (a) (b) (c) A solution of a linear equation is a pair of numbers, and, that make the equation true. We write a solution of a linear equation as an ordered pair. A linear equation in two variables has infinitely many solutions. Example 2 List a few solutions of the linear equation. (Note that there are infinitely many solutions!)

58 Section Example 3 Decide whether the ordered pair is a solution of the equation. Example 4 Complete each ordered pair for the equation. (a) (b) In order to graph the solutions to linear equations in two variables, we need a number line for each value and. The horizontal number line used to represent the value of a solution is called the - axis. The vertical number line used to represent the value of a solution is called the - axis. When both axes are represented together, we call this the rectangular (or Cartesian) coordinate system. The numbers in the ordered pairs are called the coordinates of the point.

59 56 Section 3.1 Example 5 Complete the table of values for the equation. Then write the results as ordered pairs and plot the solutions Example 6 Complete the table of values for the equation. Then write the results as ordered pairs and plot the solutions.

60 Section Graphing Linear Equations in Two Variables Graph of a Linear Equation The graph of any linear equation in two variables,, is a straight line. Example 1 Graph. Example 2 Graph. The x-intercept is where a graph crosses the -axis. To find -intercepts,. The y-intercept is where a graph crosses the -axis. To find -intercepts,.

61 58 Section 3.2 Example 3 Find the intercepts for. Then sketch the graph. Example 4 Find the intercepts for. (What do you notice?) Then sketch the graph. The graph of a linear equation of the form passes through the origin, and so only has one intercept. Thus, to sketch the graph:

62 Section Example 5 Graph the line: Example 6 Graph the line: x y x y The graph of, where is a real number, is what kind of line? The graph of, where is a real number, is what kind of line?

63 60 Section The Slope of a Line Slope of a Line (Geometric Interpretation) vertical change in y = horizontal change in x Rise Run Right units Q(3,4) Up units P( 1,2) P(0, 1) Down units Q(4, 5) Right units Example 1 Find the slopes of the following lines. (2, 1) ( 4, 2) ( 3, 5) (3, 6)

64 Section Let s derive a formula for slope! ****************************************************************************** Slope Formula The slope of the line through the points and is Example 2 Find the slope of the line through the following two points: (a) and (b) and (c) and (d) and

65 62 Section 3.3 Positive Slope Line slants UPWARD from left to right Negative Slope Line slants DOWNWARD from left to right Slope = 0 Line is HORIZONTAL Slope is undefined Line is VERTICAL Example 3 Use the slope formula to find the slope of the line. (Hint: You will need two points on this line!) How to Find the Slope of a Line from its Equation Method #1 Method #2 (FASTEST method!) Step 1 Find any two points on the line. Step 1 Solve the equation for. Step 2 Use the slope formula. Step 2 The slope is the coefficient of. Example 4 Find the slopes of the lines: (a) (b)

66 Section Parallel lines have the same slope, but do not intersect (ie. have different -intercepts.) Slope Slope Example 5 Is the line that contains the points and parallel to the line that contains the points and? Perpendicular lines intersect at a right angle ( ). Their slopes are negative reciprocals. Slope Slope or Example 6 Is the line that contains the points and perpendicular to the line that contains the points and?

67 64 Section 3.3 Example 7 Decide whether the pair of lines is parallel, perpendicular, or neither.

68 Section Writing and Graphing Equations of Lines Recall from the previous section, if given the equation of a line, we may find the slope of the line as follows: 1) 2) Example 1 Given the following equation of a line, (a) Solve for. (b) What is the slope of the line? (c) Find the -intercept of the line. Slope-Intercept Form of the Equation of a Line

69 66 Section 3.4 Example 2 Identify the slope and -intercept of each line. (a) Slope: -int: (b) Slope: -int: (c) Slope: -int: (d) Slope: -int: Example 3 Find an equation of the line with slope and -intercept.

70 Example 4 Example 5 Graph the line with -intercept Graph the line passing through the and slope. point with slope. Section Example 6 Example 7 Graph the line passing through Graph the line passing through the the point with. point with undefined slope. Example 8 Graph the equation of the line by using the slope and -intercept.

71 68 Section 3.4 Example 9 Write an equation in slope-intercept form of the line passing through the point with slope. In this last example, we were given a point on the line, and the slope of the line. Since we didn t know the -intercept, we could not immediately write the equation of the line in slope-intercept form. We had to do a little bit of work to first find the -intercept. Wouldn t it be nice if there was a form of the equation of a line that didn t require that we know the -intercept? Let s derive another form of the equation of a line! Suppose you are given that the point is on a line with slope.

72 Section Point-Slope Form of the Equation of a Line Example 10 Write an equation for the line passing through the point with slope. Give the final answer in slope-intercept form. Example 11 Write an equation for the line passing through the point with undefined slope.

73 70 Section 3.4 Example 12 Write an equation for the line passing through the points and. Give the final answer in slope-intercept form. Example 13 Write an equation for the line passing through the points and. Example 14 Write an equation of the line passing through the point and perpendicular to.

74 Section Graphing Linear Inequalities in Two Variables How To Graph Linear Inequalities:,, 1) Graph the boundary line For and, use a dashed line to exclude the points on the boundary. For and, use a solid line to include the points on the boundary. 2) Choose a test point not on the line. If the point satisfies the original inequality, it is a solution! So shade the side of the boundary line containing the test point. However, if the point causes the inequality to be false, it is not a solution. So shade the side of the boundary line that doesn t contain the test point. Example 1 Graph the solution set Example 2 Graph the solution set

75 72 Section 3.5 Example 3 Graph the solution set Example 4 Graph the solution set Example 5 Graph the solution set

76 Section Introduction to Functions When the elements in one set are linked to elements in a second set, we call this a relation. Animal Dog Cat Duck Lion Rabbit Life Expectancy (years) Set of Inputs=Domain Set of Outputs=Range If is an element in the domain and is an element in the range, and if a relation exists between and, then we say that depends on, and we write x y. We can also represent this relation as a set of ordered pairs ( x, y), where represents the input and represents the output: {(Dog,11), (Cat,11), (Duck,10), (Lion,10), (Rabbit,7)} Any set of ordered pairs is a relation! Example 1 Set A Set B (a) Represent this relation as a set of ordered pairs ( x, y), where represents the input and represents the output. (b) Identify the domain and range of this relation.

77 74 Section 3.6 Example 2 Identify the domain and range of the relation: {(2,4), (2, -3), (1, 5)} Domain: Range: A function is a special relation. It is a set of ordered pairs in which each input corresponds to exactly one output. Example 3 Determine whether each relation is a function. (a) Animal Dog Cat Duck Lion Rabbit Life Expectancy (years) (b) Domain Range (c) {(-2,8), (-1,1), (0,0), (1,1), (2,8)} (d) {(5,2), (5,1), (3,4)}

78 Most useful functions have an infinite number of ordered pairs and are usually defined with equations that tell how to get the outputs given the inputs. Everyday Examples of Functions Section The cost in dollars charged by an express mail company is a function of the weight in pounds determined by the equation: 2. In Cedar Rapids, Iowa, the sales tax is 7% of the price of an item. The tax on a particular item is a function of the price :. 3. The distance a car moving at 45 mph travels is a function of the time One way to determine if a relation is a function is to look at the graph of the equation! y y x x x 2 y y x y x y x 2

79 76 Section 3.6 Vertical Line Test Intersects in one point Intersects in more than one point Passes the test Function Fails the test Not a Function If a vertical line intersects a graph in more than one point, then the graph is not the graph of a function. Example 4 Determine whether each relation is a function. (a) (b) (b) (c) (d)

80 Section Function Notation INPUT OUTPUT Say: of The letters,, and are commonly used to name functions. For example, since the equation y 4x 3 describes a function, we may use function notation: f ( x) 4x 3 If, then. The statement says that the value of is when is. The statement also indicates that the point lies on the graph of For functions, the notations and can be used interchangeably.

81 78 Section 3.6 Example 5 For each function, find the following. (a) (b)

82 Section Solving Systems of Linear Equations by Graphing Recall, an equation in two variables is linear provided it can be written in the form: Ax By C A system of linear equations is a grouping of two or more linear equations. A solution of a system of linear equations is an ordered pair that makes both equations true at the same time. A solution of an equation is said to satisfy the equation. Example 1 Decide whether the ordered pair ( 4, 1) is a solution of each system. 5x 6y 14 (a) 2x 5y 3 (b) x y 3 x y 3 A system of two equations containing two variables represents a pair of lines. The points of intersection are the solutions of the system. Hence, we can look at their graphs to solve the system! Their graphs can appear in one of 3 ways: Intersect at exactly one point Parallel They are the same line One solution: No solution: Infinite number of solutions: (Inconsistent System) (dependent equations)

83 80 Section 4.1 Example 2 Solve the system by graphing y x y x Example 3 Solve the system by graphing y x y x Example 4 Solve the system by graphing y x y x

84 4.2 Solving Systems of Linear Equations by Substitution Section Solving a system by graphing is very difficult, especially without graph paper! Thus, we prefer algebraic methods for solving systems of linear equations. There are two such algebraic methods: 1. Substitution 2. Elimination We look at the method of substitution in this section. Substitution Method Step 1 Solve one equation for either variable. Step 2 Substitute for that variable in the other equation. Step 3 Solve the equation from Step 2. Step 4 Substitute the result from Step 3 into the equation from Step 1 to find the value of the other variable. Step 5 Check the solution in both of the original equations. Example 1 Solve the system by the substitution method. 2x 7y 12 x 2y Example 2 Solve the system by the substitution method. x 1 4y 2x 5y 11

85 82 Section 4.2 Example 3 Solve the system by the substitution method. y 8x 4 16x 2y 8 Example 4 Solve the system by the substitution method. x 3y 7 4x 12y 28 Example 5 Solve the system by the substitution method. (Hint: Clear fractions) x y x 2y 2 2

86 4.3 Solving Systems of Linear Equations by Elimination Section Solving a system by graphing is very difficult, especially without graph paper! Thus, we prefer algebraic methods for solving systems of linear equations. There are two such algebraic methods: 3. Substitution 4. Elimination We look at the method of elimination in this section. Solving a Linear System by Elimination Step 1 Write both equations in standard form, Ax By C Step 2 Transform the equations as needed so that the coefficients of one pair of variable terms are opposites. Multiply one or both equations by appropriate numbers so that the sum of the coefficients of either the x- or y- terms is 0. Step 3 Add the new equations to eliminate a variable. Step 4 Solve the equation from Step 3 for the remaining variable. Step 5 Substitute the result from Step 4 into either of the original equations, and solve for the other variable. Step 6 Check the solution in both of the original equations. Then write the solution set. Example 1 Solve the system by the elimination method. 3x y 7 2x y 3

87 84 Section 4.3 Example 2 Solve the system by the elimination method. x 2 y 2x y 10 Example 3 Solve the system by the elimination method. 4x 5y 18 3x 2y 2

88 Section Example 4 Solve the system by the elimination method. 3y 8 4x 6x 9 2 y

89 86 Section 4.3 Example 5 Solve the system by the elimination method. 3x y 7 6x 2y 5 Example 6 Solve the system by the elimination method. 2x 5y 1 4x 10y 2

90 Section Applications of Linear Systems Solving an Applied Problem with Two Variables Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 Read the problem carefully until you understand what is given and what is to be found. Assign variables to represent the unknown values, using diagrams or tables as needed. Write down what each variable represents. Write two equations using both variables. Solve the system of two equations. State the answer. Does it seem reasonable? Check the answer in the words of the original problem. Example 1 In 2008, spending on sporting equipment and recreational transport totaled $51,879 million. Spending on recreational transport exceeded spending on sporting equipment by $2113 million. How much was spent on each? Assign variables: Write a system of two equations & solve:

91 88 Section 4.4 Example 2 For a production of Wicked at the Pantages Theatre in Los Angeles, main floor tickets cost $96 and mid-priced mezzanine tickets cost $58. If a group of 18 people attended the show and spent a total of $1234 for their tickets, how many of each kind of ticket did they buy? Assign variables: # of Tickets Price per ticket (in dollars) Total value Write a system of two equations & solve:

92 Section Example 3 How many liters of a 25% alcohol solution must be mixed with a 12% solution to get 13 L of a 15% solution? Assign variables: Liters of Solution % Concentration of alcohol Liters of pure alcohol Write a system of two equations & solve:

93 90 Section 4.4 Example 4 Two cars that were 450 mi apart traveled toward each other. They met after 5 hr. If one car traveled twice as fast as the other, what were their rates? Draw diagram/assign variables: Rate Time Distance Write a system of two equations & solve:

94 Section Water/Air Current Problems: Downstream (with current) Rate Upstream (against current) Rate Example 5 In two hours, Abby can row 4 mi against the current or 20 mi with the current. Find the speed of the current and Abby s speed in still water. Assign variables: Rate Time Distance Write a system of two equations & solve:

95 92 Section Solving Systems of Linear Inequalities To Solve a System of Linear Inequalities: Step 1: Graph the first inequality. a) Graph the boundary line (Dashed or Solid?) b) Test a point not on the line to shade the correct region. Step 2: Graph the second inequality. a) Graph the boundary line (Dashed or Solid?) b) Test a point not on the line to shade the correct region. Step 3: The solution is the INTERSECTION of the two regions. Example 1 Graph the solution set of the system: 3x y 6 x 2y 8 Step 1: Graph Step 2: Graph Step 3: The solution is the intersection of the two regions. Make this region stand out by shading it even darker!

96 Section Example 2 Graph the solution set of the system: x 2y 0 3x 4y 12 Example 3 Graph the solution set of the system: x 2 x 1 0

97 94 Section The Product Rule and Power Rules for Exponents Recall from Section 1.2: In the exponential expression, is the base and is the exponent or power. Let s evaluate it! Example 1 Name the base and the exponent. Then evaluate. Exponential Expression Base Exponent Evaluate ****************************************************************************** Now, evaluate the product: This suggests the following rule For any positive integers and, Product Rule for Exponents In words:

98 Section Product Rule for Exponents: Example 2 Use the product rule, if possible, to simplify each expression. Write each answer in exponential form. (a) (b) (c) (d) (e) (f) Note the difference between adding and multiplying: *********************************************************************************** Now, evaluate: This suggests the following rule For any positive integers and, Power Rule (a) for Exponents In words:

99 96 Section 5.1 Power Rule (a) for Exponents: Example 3 Use power rule (a) for exponents to simplify each expression. Write each answer in exponential form. (a) (b) Note the difference between the product rule and power rule (a): *********************************************************************************** Now, evaluate: This suggests the following rule For any positive integer, Power Rule (b) for Exponents In words:

100 Section Power Rule (b) for Exponents: Example 4 Use power rule (b) for exponents to simplify each expression. Write each answer in exponential form. (a) (b) (c) (d) Note that power rule (b) does not apply to a sum. *********************************************************************************** Now, evaluate: This suggests the following rule For any positive integer, Power Rule (c) for Exponents In words:

101 98 Section 5.1 Power Rule (c) for Exponents: Example 5 Use power rule (c) for exponents to simplify each expression. Write each answer in exponential form. Assume any variables in the denominator are non-zero. Summary of Rules for Exponents For positive integers and, 1) 3) 2) 4) Example 6 Simplify each expression by using a combination of the rules of exponents.

102 Section Example 7 Find an expression that represents the area of the figure. Example 8 Find an expression that represents the volume of the figure.

103 100 Section 5.2 Observe the following: 5.2 Integer Exponents and the Quotient Rule Do you see a pattern? Using this pattern we should be able to continue the list to include zero and negative exponents We just need to make sure the Rules for Exponents from Section 5.1 will apply to this definition of zero and negative exponents: If we define then: Using the Power Rule for Exponents: Do these agree?? If we define then: Using the Power Rule for Exponents: Do these agree??

104 Section Thus we define zero and negative exponents as follows: Zero Exponent For any nonzero real number, Negative Exponents For any nonzero real number and any integer, In Words: raised to is raised to. Example 1 Evaluate. Assume that all variables represent nonzero real numbers. (a) (b) (c) (d) (e) (f) (g) (h)

105 102 Section 5.2 Consider the following: Changing from Negative to Positive Exponents For any nonzero numbers and and any integers and, and as we already observed from the definition: Example 2 Simplify by writing with positive exponents. Assume that all variables represent nonzero real numbers. (a) (b) (c)

106 Section Be careful if the exponents occur in a sum or difference of terms. However, ****************************************************************************** Now, consider the quotient of two exponential expressions: Also, This suggests the following rule Quotient Rule for Exponents For any nonzero real number and any integers and, In words:

107 104 Section 5.2 Quotient Rule for Exponents: Example 3 Simplify by writing with positive exponents. Assume that all variables represent nonzero real numbers. (a) (b) (c) (d) (e) (f)

108 Section Summary of Definitions and Rules for Exponents For any integers and, Zero Exponent Power Rule (b) Negative Exponent Power Rule (c) Product Rule Negative-to-Positive Rules Quotient Rule Power Rule (a) Example 4 Simplify. Assume that variables represent nonzero real numbers. (a) (b)

109 106 Section 5.2 (c) (d)

110 Section Scientific Notation Use a scientific calculator to compute the following: 5,000,000 6,000,000 = Very large and very small numbers often occur in the sciences. For such numbers, we use scientific notation simply because it s easier to work with AND most calculators cannot display enough digits to give the answer in decimal form! The form for scientific notation is: where 1 a 10 and is an integer. Example 1 Circle the numbers written in scientific notation Observe the following: Do you notice a pattern?? How to write a number in decimal notation (without exponents): For numbers greater than 10, the exponent n is positive and equal to the number of places the decimal point in the number a moves to the right. scientific notation decimal notation (without exponents) For numbers less than 1, the exponent n is negative and equal to the number of places the decimal point in the number a moves to the left. scientific notation decimal notation (without exponents)

111 108 Section 5.3 Example 2 Write in decimal notation How to write a number in scientific notation: First write (where by placing the decimal after the first nonzero digit. For numbers greater than 10, the exponent n on the base 10 should be positive and equal to the number of places the decimal point in would need to move to the right to yield the number without exponents. decimal notation (without exponents) scientific notation For numbers less than 1, the exponent n on the base 10 should be negative and equal to the number of places the decimal point in would need to move to the left to yield the number without exponents. decimal notation (without exponents) scientific notation Example 3 Write each number in scientific notation , ,000,000

112 Example 4 Perform the indicated operations. Write each answer (a) in scientific notation, and (b) without exponents (decimal notation). Section

113 110 Section Adding and Subtracting Polynomials Name the coefficient of each term in the expression: 3 2 2x 5x x 4 Simplify by adding like terms: x 3xy 5x 10xy A polynomial in is a term or the sum of a finite number of terms of the form, for any real number and any whole number. Circle the polynomials: The degree of a term is the sum of the exponents on the variables: Examples: 7 2x degree: 5 degree: 2 4 3x y degree: The degree of a polynomial is the greatest degree of any nonzero term of the polynomial. Examples: x x degree: 4x 8 degree: x y 4xy 20 degree:

114 Section A polynomial with only one term is called a. Example: A polynomial with exactly two terms is called a. Example: A polynomial with exactly three terms is called a. Example: Example 1 For each polynomial, first simplify, if possible, and write it in descending powers of the variable. Then give the degree of the resulting polynomial and tell whether it is a monomial, binomial, trinomial, or none of these. (a) 6p 4p 8p 10p (b) 4 1 r r Example 2 Evaluate the polynomial at y y 8y 6

115 112 Section 5.4 To add two polynomials, add like terms. Example 3 Add. Horizontally: Vertically: Example 4 Subtract. Horizontally: Vertically:

116 Section Example 5 Find a polynomial that represents the perimeter of the rectangle. Example 6 Graph the equation by completing the table of values

117 114 Section Multiplying Polynomials To find the product of polynomials, we use the distributive property: Example 1 Find the product. (a) (b) Longer Method SHORT-CUT: To multiply two polynomials, multiply each term of the second polynomial by each term of the first polynomial and add the products. (c) Short-cut Method

118 Section Example 2 Multiply Vertically: 2 3x 4x 5 x 4 Example 3 Use the rectangle method to find the product. Example 4 Choose any method to multiply:

119 116 Section 5.5 First Outer ( )( ) 2 ax b cx d acx adx bcx bd Inner Last F. O. I. L. Example 5 Use the FOIL method to multiply. (a) (4x 3)( x 2) (b) 2 ( m 6) (c) ( 4 y x)(2y 3 x) (d) 3 3 x ( x 2)(2x 1)

120 Section Special Products Multiply: Square of a Binomial* *Note: It is your choice whether you want to be able to recognize and use these formulas, or just FOIL as usual. Example 1 Multiply. (a) (b) (c) (d)

121 118 Section 5.6 Multiply: Product of the Sum and Difference of Two Terms* *Note: It is your choice whether you want to be able to recognize and use this formula, or just FOIL as usual. Example 2 Multiply. (a) (b) (c) (d)

122 Section Example 3 Multiply. (a) (b)

123 120 Section Division by a Monomial 5.7 Dividing Polynomials Example 1 Divide by. Example 2 Divide Example 3 Divide

124 Section Division by a Non-Monomial Question: How should we divide Answer: Recall the long division process: Example 4 Divide

125 122 Section 5.7 Example 5 Divide Example 6 Divide

126 Section Example 7 Divide Example 8 Divide by.

127 124 Section The Greatest Common Factor; Factoring by Grouping To factor means to write a quantity as a product. Factoring is the opposite of multiplying. List all the positive factors of 12: List all the positive factor of 18: From the lists above, identify the greatest common factor (GCF) of 12 and 18: Example 1 Find the greatest common factor (GCF) for each list of numbers. (a) 20, 64 (b) 12, 18, 26, 32 (c) 12, 13, 14

128 Section Fact: The greatest common factor (GCF) will be the product of every common prime factor raised to the smallest exponent. Example 2 Find the greatest common factor of 72 and 240 by first factoring the numbers into prime factors. (A factor tree is very helpful!) Example 3 Find the greatest common factor for each list of terms. (a),, (b) (c)

129 126 Section 6.1 Example 4 Find the greatest common factor for each list of terms. (a) (b) The process of applying the distribute property (reverse direction) to write a sum as a product with the greatest common factor (GCF) as one of the factors is called factoring out the greatest common factor (GCF). Eg. Example 5 Factor out the greatest common factor (GCF). (a) (b) (c) (d) (e)

130 Section Example 6 Factor out the greatest common factor. (a) (b) When a polynomial has four terms, we can often factor by grouping. Example 7 Factor by grouping. (a) (b)

131 128 Section 6.1 (c) (d) (e)

132 Section Factoring Trinomials Observe: Here, we have factored the polynomial as a product of two binomials. Note the relationship between the coefficients of the original polynomial and those of the two binomials. Trinomials with a Leading Coefficient of 1 where and the product the sum If it is not possible to find such and then the polynomial cannot be factored, and we say the polynomial is. Example 1 Factor.

133 130 Section 6.2 Example 2 Factor. Example 3 Factor. Example 4 Factor.

134 Section Example 5 Factor. Example 6 Factor (Factor a trinomial with two variables) The 1 st step in EVERY factoring problem is to!!! Example 7 Factor

135 132 Section More on Factoring Trinomials Main Objective: Factor the general polynomial when the leading coefficient is not 1. Techniques: #1) By Grouping (a.k.a. AC Method) #2) By Using FOIL (a.k.a. Trial & Error Method) Technique #1: By Grouping Example 1 Factor by grouping method. Example 2 Factor by grouping method.

136 Section Example 3 Factor by grouping method. Example 4 Factor by grouping method. Technique #2: By Using FOIL Example 5 Factor by using FOIL (trial & error).

137 134 Section 6.3 Example 6 Factor by using FOIL (trial & error). Example 7 Factor by using FOIL (trial & error). Example 8 Factor by using FOIL (trial & error).

138 Section Example 9 Factor by using FOIL (trial & error). Example 10 Factor by using FOIL (trial & error). Example 11 Factor by using FOIL (trial & error).

139 136 Section Special Factoring Techniques Difference of Squares Example 1 Factor completely. (a) (b) (c) (d) (e) (f) (g)

140 Section Difference of Cubes Sum of Cubes Example 2 Factor completely. (a) (b) (c) (d) (e) (f)

141 138 Section 6.4 Perfect Square Trinomials Example 3 Factor completely. (a) (b) (c) (d) (e)

142 Section Solving Quadratic Equations by Factoring A quadratic equation is an equation that can be written in the form: Standard Form where and are real numbers and. How to recognize a quadratic equation: The highest power of is. Circle which of the following equations are quadratic? Zero-Factor Property If, then. Example 1 Solve the quadratic equation. (a) (b) (c)

143 140 Section 6.5 How to Solve a Quadratic Equation by. 1) 2) 3) Example 2 Solve the quadratic equation. (a) (b) (c)

144 Section Example 3 Solve the quadratic equation. (a) (b) (c)

145 142 Section 6.5 The zero-factor property may be used to solve equations that are not quadratic, but that are still factorable, as we will see in the next example Example 4 Solve. (a) (b) (c)

146 Section Applications of Quadratic Equations Important Formulas from Geometry Figure Formulas Illustration Square Area: Rectangle Area: Triangle Area: Parallelogram Area: Rectangular Box Volume:

147 144 Section 6.6 Example 1 The area of a triangle is 24. The base of the triangle measures more than the height. Find the measures of the base and the height. Example 2 A toolbox is 2 ft. high, and its width is 3 ft. less than its length. If its volume is 80 and width of the box., find the length

148 Section Example 3 The sides of one square have length 3 m more than the sides of a second square. If the area of the larger square is subtracted from 4 times the area of the smaller square, the result is 36. What are the lengths of the sides of each square?

149 146 Section 6.6 Recall from Section 2.4, Consecutive Integer Word Problems If the lesser integer, then for any two consecutive integers, use three consecutive integers, use two consecutive even (or odd) integers, use three consecutive even (or odd) integers, use Example 4 The product of the first and third of three consecutive odd integers is 16 more than the middle integer. Find the integers.

150 Section Pythagorean Theorem a c b Example 5 Two cars left an intersection at the same time. One traveled due north. The other traveled 14 mi farther, but to the east. How far apart were they at that time if the distance between them was 4 mi more than the distance traveled east? North Intersection East

151 148 Section The Fundamental Property of Rational Expressions Consider the following expressions: Each of these is a ratio of two. A rational expression is an expression of the form: where and are, with. Example 1 Find the value of the rational expression when (a) (b) (c)

152 Section Note: A rational expression is undefined for any values of the variable that cause the denominator to be zero. To find these values, we set the denominator equal to zero and solve. Example 2 Find any values of the variable for which each rational expression is undefined. Write answers with.

153 150 Section 7.1 A rational expression ( ) is in lowest terms if the numerator and denominator share no common factors (except 1). The Fundamental Property of Rational Expressions If ( ) is a rational expression and if represents any polynomial (, then To write a rational expression in lowest terms: 1) 2) Example 3 Write in lowest terms.

154 Section Example 4 Write in lowest terms. Example 5 Write four equivalent forms for the rational expression.

155 152 Section Multiplying and Dividing Rational Expressions Multiplying rational expressions Example 1 Multiply. Write your answer in lowest terms.

156 Section Dividing rational expressions Example 2 Divide. Write your answer in lowest terms.

157 154 Section Least Common Denominators Recall how to add or subtract two rational numbers with a common denominator: Recall how to add or subtract two rational numbers with different denominators: Before two rational expressions with different denominators can be added or subtracted, both rational expressions must be expressed in terms of a common denominator. This common denominator is called the LCD (Least Common Denominator). How to find the LCD of two (or more) rational expressions: 1) Factor each denominator into prime factors. 2) The LCD is the product of the LCM of the coefficients and each variable factor raised to the greatest power that occurs in any one factorization. Example 1 Find the LCD:

158 Section Example 2 Find the LCD: The next step after finding a common denominator is to rewrite each rational expression with the new denominator. Example 3 Rewrite each rational expression with the indicated denominator.

159 156 Section 7.3 Example 4 Rewrite each rational expression with the LCD as the denominator.

160 Section Adding and Subtracting Rational Expressions To add/subtract rational expressions with the same denominator, add/subtract the numerators and keep the same denominator. Recall how to write a rational expression in lowest terms: 1) Factor the numerator and denominator 2) Cancel common factors Example 1 Add/subtract. Write each answer in lowest terms.

161 158 Section 7.4 Add/subtract. Write each answer in lowest terms. To add/subtract rational expressions with different denominators: 1) Rewrite the problem with the denominators written in factored form. 2) Identify the LCD, and rewrite each rational expression with the LCD as the denominator. 3) Add/subtract the numerators and write this result over the LCD. 4) Write the answer in lowest terms (Factor and cancel common factors). Example 2 Add/subtract. Write each answer in lowest terms.