About the Portfolio Activities. About the Chapter Project

Galileo is credited as the first person to notice that the motion of a pendulum depends only upon its length. About the Chapter Project Finding an average is something that most people can do almost instinctively. Currency echange, hourly wage, car mileage, and average speed are common daily topics of discussion. You can measure an average in various ways. Two common averages are the arithmetic mean and the harmonic mean. In the Chapter Project, Means to an End, you will use the data provided to determine the most appropriate average. After completing the Chapter Project, you will be able to do the following: Find the arithmetic mean and the harmonic mean of a set of data. Determine the relationship between the arithmetic and harmonic mean. Determine which of the averages arithmetic mean, harmonic mean, or weighted harmonic mean best represents a data set. About the Portfolio Activities Throughout the chapter, you will be given opportunities to complete Portfolio Activities that are designed to support your work on the Chapter Project. Eploring a historical representation of harmonic means is included in the Portfolio Activity on page 88. Eploring a geometric representation of harmonic means is included in the Portfolio Activity on page 97. Etending the definition of harmonic mean to n numbers is included in the Portfolio Activity on page. Using harmonic means to find average speeds is included in the Portfolio Activity on page 9. 79

Multiplying and Dividing Rational Epressions Why Multiplying and dividing rational epressions are sometimes used to solve real-world problems, such as analyzing the revenue and cost for a fund-raising activity. Objectives Multiply and divide rational epressions. Simplify rational epressions, including comple fractions. APPLICATION FUND-RAISING To analyze the revenue and costs from the sale of school-spirit ribbons, members of the Jamesville High School Home Economics Club used the revenue-to-cost ratio below. revenue from the sale of each ribbon cost of making each ribbon They represented the number of ribbons produced and sold by and the total production cost in dollars by 0.8. If the revenue for each ribbon was $, for how many ribbons was the revenue-to-cost ratio. or greater? Finding the answer to this question involves writing and simplifying a rational epression. You will answer this question in Eample 6. Simplifying Rational Epressions E X A M P L E TRY THIS To simplify a rational epression, divide the numerator and the denominator by a common factor. The epression is simplified when you can no longer divide the numerator and denominator by a common factor other than. Simplify 6 6 6 6 = ( 6) ( ) ( Factor the numerator and denominator. 6) ( 6) = ( 6) ( ) Divide out the common factor. ( 6) ( 6) = 6 Note that 6 and 6 are ecluded values of in the original epression. Simplify b 9 b 8b 7 98 CHAPTER 8

Multiplying Rational Epressions Multiplying rational epressions is similar to multiplying rational numbers. Rational Numbers Rational Epressions = 9 7 = 9 6 = / / / 7 = 0 7 / / / E X A M P L E Simplify. 7 = / / / / 7 7 = / / / / TRY THIS Simplify 8 a a 9 a To multiply one rational epression by another, multiply as with fractions. a b c = ac b,where b 0 and d 0 d d You can simplify the product by dividing out the common factors in the numerator and denominator before or after multiplying. E X A M P L E TRY THIS Simplify 6 6 = ( ( ) ( ) ) ( ) ( ) ( ) = (, or ) ( ) Simplify 6 Dividing Rational Epressions Dividing one rational epression by another is similar to dividing one rational number by another. Rational Numbers Rational Epressions 6 8 = 6 8 6 = 6 Multiply by the Multiply by the reciprocal of. reciprocal of. = 6 8 = / 6 / / =, or =, or LESSON 8. MULTIPLYING AND DIVIDING RATIONAL EXPRESSIONS 99

To divide one rational epression by another, multiply by the reciprocal of the divisor. a b c = a d b d c = a d,where b 0, c 0, and d 0 bc Simplify by dividing out common factors in the numerator and denominator. E X A M P L E Simplify ( ) ( ) = ( ) = ( ( ) ( ) ) ( ) ( ) ( ) = (, or ) ( ) Multiply by the reciprocal. Divide out common factors. TRY THIS PROBLEM SOLVING Simplify ( ) 9 8 You can use a graph to identify polynomials in a rational epression that cannot be factored with real numbers. For eample, to determine whether in the rational epression can be factored with real numbers, look for -intercepts in the graph of y =. The graph of y = has no -intercepts, so y = has no real zeros and cannot be factored with real numbers. Thus, the rational epression cannot be simplified further. Comple Fractions A comple fraction is a quotient that contains one or more fractions in the numerator, the denominator, or both. E X A M P L E a Simplify the comple fraction a. a a a = a a a Multiply by the reciprocal. a a a a = ( a ) ( a ) a Factor. ( a ) ( a ) a (a )(a )(a ) = Divide out common factors. (a )(a )(a ) = a a 00 CHAPTER 8

TRY THIS CRITICAL THINKING ( ) Simplify. ( ) y y Use mental math to simplify. y y E X A M P L E 6 APPLICATION ECONOMICS TECHNOLOGY GRAPHICS CALCULATOR Keystroke Guide, page Refer to the revenue-to-cost ratio given at the beginning of the lesson. For how many ribbons was the revenue-to-cost ratio. or greater? revenue from the sale of each ribbon = cost of making each ribbon 0.8 Simplify the comple fraction. = = 0.8 0.8 0.8 Enter y = 0.8 into a Since represents the graphics calculator, and number of ribbons, eamine a table of values. use an increment of. From the table, you can see that the revenueto-cost ratio, y,is greater than. when or more ribbons,, were produced and sold. Eploring Ecluded Values in Quotients TECHNOLOGY GRAPHICS CALCULATOR Keystroke Guide, page PROBLEM SOLVING CHECKPOINT You will need: a graphics calculator. Let f() =.When the comple fraction that defines f is simplified, it becomes Let g() =.Graph f and g on the same screen. What observations can you make about these graphs?. Make a table to evaluate f and g for -values of,,, 0,,, and. How do the entries in the table compare?. Repeat Steps and for the functions f() = and g() = 9 9 P. Let f() Q = and g() = P R,where P, Q, and R are polynomials. Eplain R Q how to find the ecluded values of f and why you should not try to find those values by eamining g. LESSON 8. MULTIPLYING AND DIVIDING RATIONAL EXPRESSIONS 0

Eercises Communicate. In what ways is the multiplication of two rational epressions similar to the multiplication of two rational numbers?. In what ways is the division of two rational epressions similar to the division of two rational numbers?. Eplain how to simplify a comple fraction such as. Compare the ecluded values of in the comple fraction and in its simplified form. Guided Skills Practice Simplify each rational epression. (EXAMPLES AND ).. 0 0 0 60 Simplify each rational epression. (EXAMPLES AND ) 6. 8 8 7. 9 8 6 8 6 8. Simplify the comple fraction 9 0. (EXAMPLES AND 6) 9 Homework Help Online Go To: go.hrw.com Keyword: MB Homework Help for Eercises 9 Practice and Apply Simplify each rational epression. 9. 8 0. 6 9 9.. 6 9 7. 0 9. 6 6. 9 8 7 6. 0 7. 8. 0 9 6 9. 9 0. 0 9 6 9. 6. 9 7. a a 7. ( y 6 b ay by 6. b ay by ) y.. 9 y y y y 8. rs r s s r 6 9 0 CHAPTER 8

Simplify each epression. 9 ( ) ( ) 9. 0. 9. 6 ( ) 8 7 ( ) 7 0 6 8... 7 9 8 7 0 7. 6. 7. ( ) ( y) y 6 y 7 ( y) 8. 9. 0. (y )(y ) y y y y y y y y y. 7 8. 7. 9 ( y)y ( y) ( y)y ( y) C H A L L E N G E C O N N E C T I O N A P P L I C A T I O N. Find the rational epression R whose numerator and denominator have degree and leading coefficients of such that 0 R = 8. GEOMETRY An open-top bo is to be made from a sheet of cardboard that is 0 inches by 6 inches. Squares with sides of inches are to be cut on cut one side and creased on another to form tabs. 6 in. When the sides are folded up, these tabs are glued crease to the adjacent sides to provide reinforcement. a. Show that (0 )(6 ) represents the 0 in. volume of the bo. b. Show that 0 represents the surface area of the bottom and sides of the inside of the bo. c. Write and simplify an epression for the ratio of the volume of the bo to the inside surface area of the bo. d. How does the ratio from part c change as increases? 6. ECONOMICS It costs Emilio and Maria Vianco $00 to operate their sandwich shop for one month. The average cost of preparing one sandwich is $.69. a. Using the menu shown, find the average revenue per sandwich. b. Let represent the number of sandwiches sold in one month. Write a function for the total cost, C, of operating the business for one month by using the average cost of preparing one sandwich. c. Write a function for the ratio, R,ofaverage revenue per sandwich to the average cost per sandwich. LESSON 8. MULTIPLYING AND DIVIDING RATIONAL EXPRESSIONS 0

A P P L I C A T I O N 7. PHYSICS The diagram below illustrates an ambulance traveling a definite distance in a specific period of time. The average acceleration, a,is defined as the change in velocity over the corresponding change in time. d t d t a. Simplify the epression at right that defines a. b. If the distance, d, is measured in feet and the time, t, is measured in seconds, in what units is acceleration measured? a = d d t t t t Look Back Write an equation in slope-intercept form for the line that contains the given point and is perpendicular to the given line. (LESSON.) 8. (8, ), y = 6 9. (, ), y = Graph each piecewise function. (LESSON.6) 0. f() = if <. g() = 9 if < if < 0 if 0 8 if > Factor each epression. (LESSON.). 8. 6. a a Simplify each epression. Write your answer in the standard form for a comple number. (LESSON.6) i i. 6. 7. i i 6 i i Write each product as a polynomial epression in standard form. (LESSON 7.) 8. ( 6 ) 9. ( )( 6 ) 60. ( )( 9) Factor each polynomial epression. (LESSON 7.) 6. 6. 7 6. 6 8 Look Beyond Simplify. 6. 8 8 6. 66. 67. 0 CHAPTER 8

Adding and Subtracting Rational Epressions Why Adding and subtracting rational epression can be used to solve real-world problems, such as calculating the average rate of speed for an entire trip. Objectives Objective Add and subtract rational epressions. APPLICATION TRAVEL A cab driver drove from the airport to a passenger s home at an average speed of miles per hour. He returned to the airport along the same highway at an average speed of miles per hour. What was the cab driver s average speed over the entire trip? The answer is not the average of and. To answer this question, you need to add two rational epressions. You will answer this question in Eample. Adding two rational epressions with the same denominator is similar to adding two rational numbers with the same denominator. Rational Numbers 7 7 = = 7 7 Common denominator Rational Epressions = = 8 Common denominator E X A M P L E Simplify. a. b. 9 a. = Note that is an ecluded ecluded value of value in the of original in the original epression. epression. b. 9 = 9 = ( ) ( ) = TRY THIS Simplify. a. b. 0 LESSON 8. ADDING AND SUBTRACTING RATIONAL EXPRESSIONS 0

To add two rational epressions with unlike denominators, you first need to find common denominators. The least common denominator (LCD) of two rational epressions is the least common multiple of the denominators. The least common multiple (LCM) of two polynomials is the polynomial of lowest degree that is divisible by each polynomial. Finding the LCM for two rational epressions is similar to finding the LCM for two rational numbers. Compare the procedures for rational numbers and for rational epressions shown below. Rational Numbers Rational Epressions 7 00 90 = 7 00( ) 90( 0 0 7 ) = 7 9 ( ) 9 ( ) = 0 = 900 9 Least common Least common = denominator denominator 9 00 Adding and Subtracting Rational Epressions To add or subtract two rational epressions, find a common denominator, rewrite each epression by using the common denominator, and then add or subtract. Simplify the resulting rational epression. E X A M P L E Simplify 8. 8 = 8 ( )( ) = ( ) 8 The LCD is ( )( ). ( )( ) ( = ( ) 8 ) ( ) = 8 ( ) ( ) Add the fractions. Write the numerator in standard form. = ( ( = ( ( = ) ( ) ) ( ) ) ( ) ) ( ) Factor the numerator. Divide out the common factors. Note that and are ecluded values of in the original epression. TRY THIS Simplify 0 CHECKPOINT Eplain how factoring a polynomial can help you to add two rational epressions. Illustrate your response by simplifying 6 9 06 CHAPTER 8

E X A M P L E Simplify 6. 6 = 6 ( ) ( ) The LCD is ( )( ). 6( ) ( ) = Subtract. ( )( ) = 0 ( )( ) = (, or )( ) 6 TECHNOLOGY GRAPHICS CALCULATOR Keystroke Guide, page 6 CHECK Graph y = 6 and y = 6 together on the same screen to see if the graphs are the same. You can also use a table of values to verify that the corresponding y-values are the same. The graphs appear to coincide. CHECKPOINT Identify the ecluded values of for the original epression and for the simplified epression in Eample. Are they the same or different? Eplain. TRY THIS Simplify 6 E X A M P L E Sometimes you need to rewrite comple fractions as rational epressions in order to add or subtract them, as shown in Eample. Simplify. a a = Add or subtract within the a a denominators. a a a a = a a a Multiply by the reciprocals. a = a a a a = a a a ( ) a a ( a a ) The LCD is (a )(a ). a = a a a a Multiply, and then add. (a ) (a ) = a a TRY THIS Simplify a a. a a a a LESSON 8. ADDING AND SUBTRACTING RATIONAL EXPRESSIONS 07

E X A M P L E APPLICATION TRAVEL Refer to the cab driver s round trip described at the beginning of the lesson. What is the cab driver s average speed for the entire trip? Let d represent the length of the trip one way, let t represent the cab driver s travel time to the passenger s home, and let t represent his travel time back to the airport. r = mph d = r t and d = r t Airport d Home t = d = d r t = d = d r r = mph average speed = to tal distance total time = d d t t = d d Substitute d for t and d for t. d d = The LCD for d and d d is 9. 9d d 9 = d 0 d 9 = d 9 Multiply by the reciprocal. 0d = 9. Thus, the cab driver s average speed was 9. miles per hour. The average speed was less than the average of and because he spent more time driving at miles per hour than at miles per hour. CRITICAL THINKING Suppose that the cab driver travels to the passenger s home at a miles per hour and returns along the same route at b miles per hour. Show that his average speed for the entire trip is not simply a b Eercises Communicate. Eplain how to find the least common denominator in order to add 7 6 6. Eplain how to use a graph to check your answer when you add two rational epressions.. Choose the two epressions below that are equivalent and eplain why they are equivalent. 7 a. b. 0 0 c. 7 d. 7 08 CHAPTER 8

Guided Skills Practice Simplify. (EXAMPLES AND ).. 6. A P P L I C A T I O N Simplify. (EXAMPLES AND ) 7. 8. t 9. TRAVEL Refer to the cab driver s round trip described at the beginning of the lesson. What is the average speed for the entire trip if he drives to the passenger s home at miles per hour and returns to the airport along the same route at 8 miles per hour? (EXAMPLE ) Homework Help Online Go To: go.hrw.com Keyword: MB Homework Help for Eercises, 7-8,,, Practice and Apply Simplify. 0. 6. 7 9. r 9 r. 7. 9. 6 9 6. 7. 6 8. 8 9. 7 0.... ( ).. 6. 7. 8. 9. 0 0.. y y. y y. a b a (ab) b Write each epression as a single rational epression in simplest form.. 0 7. 6 7 6. 7 7 7. 9 7 8. (a b) (a b) 9. (a b) (a b) 0. y y y r. y y r r s r s s r s C H A L L E N G E Find numbers A, B, C, and D such that the given rational epression equals the sum of the two simpler rational epressions, as indicated.. = A D. = B ( )( ) D. = A 0 C D. 6 7 = A C D LESSON 8. ADDING AND SUBTRACTING RATIONAL EXPRESSIONS 09

C O N N E C T I O N APPLICATIONS 6. GEOMETRY In the diagram at right, square A is unit on a side, square B is of a unit on a side, square C is of a unit on a side, and so on. a. Write a sum for the total area of squares A, B, C, and D, using only powers of. b. Rewrite the sum you wrote in part a as a single rational number. c. Suppose that two more squares, E and F,are added to the set of squares, continuing the pattern. Write a single rational number for the total area of squares A through F. d. Convert your answers from parts b and c to decimals rounded to the nearest ten-thousandth. What common fraction do the answers appear to be getting closer and closer to? 7. ELECTRICITY The effective resistance, R T,ofparallel resistors in an electric circuit equals the reciprocal of the sum of the reciprocals of the individual resistances. R T = R A R B R C A B C D Resistance in an electric circuit is measured in ohms. a. A circuit has three parallel resistors, R A, R B, and R C.Find R T to the nearest hundredth, given R A = ohms, R B = 8 ohms, and R C = ohms. b. Write R T as a rational epression with no fractions in the denominator. TRAVEL The diagram below shows the parts of a trip that Justine recently took. The distances between A and B, B and C, and C and D are all equal. The speed in each direction is shown in the diagram. Find each average speed listed below to the nearest tenth of a mile per hour. A B C D mph 0 mph mph 0 mph mph 0 mph 8. Justine s average speed for a trip from A to C and back to A 9. Justine s average speed for a trip from B to D and back to B 0. Justine s average speed for a trip from A to D and back to A 0 CHAPTER 8

Look Back State the property that is illustrated in each statement. All variables represent real numbers. (LESSON.). 8( ) = 0 6. ( ) = ( ) Find the discriminant, and determine the number of real solutions. Then solve. (LESSON.6). 0 =. = 0. = 0 A P P L I C A T I O N 6. PHYSICAL SCIENCE When sunlight strikes the surface of the ocean, the intensity of the light beneath the surface decreases eponentially with the depth of the water. If the intensity of the light is reduced by 7% for each meter of depth, what epression represents the intensity of light beneath the surface? (LESSON 6.) Write each epression as a single logarithm. Then evaluate. (LESSON 6.) 7. log log 8 8. log log 6 Use a graph and the Location Principle to find the real zeros of each function. (LESSON 7.) 9 d() = 6 60 f() = 6 f() = 6 6 g() = 8 8 Look Beyond 6 Find all real solutions of the rational equation. = ( ) ( ) Be sure to check for ecluded values. PORTFOLIO A C T I T Y I V The definition of a harmonic mean may be etended for,, or n numbers. For any three numbers a, b, and c,the harmonic mean is. a. Simplify this comple fraction. b c. Find the harmonic mean of the numbers,, and. For any four numbers a, b, c, and d, the harmonic mean is. a b c d. Simplify this comple fraction.. Find the harmonic mean of the numbers,, 6, and 8. WORKING ON THE CHAPTER PROJECT You should now be able to complete Activity of the Chapter Project. LESSON 8. ADDING AND SUBTRACTING RATIONAL EXPRESSIONS

Solving Rational Equations and Inequalities Objectives Solve a rational equation or inequality by using algebra, a table, or a graph. Solve problems by using a rational equation or inequality. Why There are many events in the real world that can be represented by a rational equation or inequality. For eample, you can write a rational equation to represent speed and distance information for a triathlon. APPLICATION SPORTS E X A M P L E Rachel finished a triathlon involving swimming, bicycling, and running in. hours. Rachel s bicycling speed was about 6 times her swimming speed, and her running speed was about miles per hour greater than her swimming speed. To find the speeds at which Rachel competed, you can solve a rational equation. A rational equation is an equation that contains at least one rational epression. Distance (mi) Speed (mph) Swimming d s = 0. s Bicycling d b = 6s Running d r = 6 s Find the speeds at which Rachel competed if she finished the triathlon in. hours.. Find the time for each part of the triathlon. Swimming time Bicycling time Running time d s = rt s d b = rt b d r = rt r 0. = st s = (6s)t b 6 = (s )t r 0. = ts = t 6 s 6 s b = s t r CHAPTER 8

TECHNOLOGY GRAPHICS CALCULATOR Keystroke Guide, page 6. Write a rational function to represent the total time, T,in hours for the triathlon in terms of the swimming speed, s,in miles per hour. T(s) = t s t b t r T(s) = 0. s 6 6s s T(s) = 0. s 6 s ) 6( ( s ) 6 ( s s s ) 6 s ( 6 s 6 ) The LCD is 6s(s ). s s s 6s T(s) = 6s(s ) T(s) = 6 s 0 6s(s ). Solve the rational equation. = 6 s 0 Reasonable 6s(s ) Graph y = 6 -values are > 0. 0 and y =., and find the 6( ) -coordinate of the point of intersection. Thus, Rachel swam at about.7 miles per hour, bicycled at about 6.7, or 6., miles per hour, and ran at about.7, or 7.7, miles per hour. CHECKPOINT Eamine the table of values at right for y = 6 0 Describe Rachel s triathlon times 6( ) if her swimming speed (-value) were less than.7 miles per hour and if her swimming speed were greater than.7 miles per hour. CRITICAL THINKING How can you solve. = 6 0 by using the 6( ) quadratic formula? E X A M P L E Solve = 6 TECHNOLOGY GRAPHICS CALCULATOR Keystroke Guide, page 6 Method Use algebra. = 6 6, ( ) = ( 6) = 6 6 = 0 ( )( ) = 0 = or = CHECK Let =. Let =. = = 6 6 6 =? 6 =? = True = True The solutions are and. Method Use a graph. Because it is not easy to see the intersection of y = and y = 6, use another graphing method. Write = 6 as = 6 0. Then graph y = 6, and find the zeros of the function. (, (, 0) (, 0) TRY THIS Solve = LESSON 8. SOLVING RATIONAL EQUATIONS AND INEQUALITIES

E X A M P L E Sometimes solving a rational equation introduces etraneous solutions.an etraneous solution is a solution to a resulting equation that is not a solution to the original equation. Therefore, it is important to check your answers, as shown in Eample. Solve = 8 9 Method Use algebra. Multiply each side of the equation by the LCD, ( )( ), or 9. = 8,where and 9 ( )( ) ( )( ) = 8 ( )( ) 9 ( ) ( ) = 8 6 = 8 8 = 0 ( 6) = 0 ( )( ) = 0 = or = CHECK Since = is an ecluded value of in the original equation, it is an etraneous solution. Check =. = 8 9 ( )? = ( ) 8 9 = 8 True Thus, the only solution is =. TECHNOLOGY GRAPHICS CALCULATOR Method Use a graph. Because it is not easy to see the intersection of y = and y = 8, 9 use another graphing method. Write = 8 as 9 8 = 0. Then graph 9 y = 8, and find any zeros of the function. 9 Keystroke Guide, page 6 The only zero of the function is at (, 0). The solution is. TRY THIS CRITICAL THINKING Solve = 6 Eplain why an etraneous solution is obtained in Eample above. CHAPTER 8

A rational inequality is an inequality that contains at least one rational epression. Solving Rational Inequalities TECHNOLOGY GRAPHICS CALCULATOR Keystroke Guide, page 6 CHECKPOINT You will need: a graphics calculator. Graph y =. Use the table feature and the graph to identify the values of for which y is 0, y is undefined, y is positive, and y is negative.. On the same screen, graph y =.. For what values of is y = y? y < y? y > y?. Eplain how to use a graph and a table of values to solve < and >. E X A M P L E Solve. Method Use algebra. To clear the inequality of fractions, multiply each side by. You must consider both cases: is positive or is negative., where > 0 or, where < 0 Change to. Change to. Change to. For this case, > because > 0. For this case, < because < 0. Therefore, the solution must satisfy Therefore, the solution must satisfy and > and < For this case,. Thus, the solution is or < For this case, < 0 TECHNOLOGY GRAPHICS CALCULATOR Keystroke Guide, page 7 Method Use a graph. Graph y = and y =, and find the values of for which the graph of y = is below the graph of y =. Thus, the solution is or < (, ) vertical asymptote at = TRY THIS Solve <. LESSON 8. SOLVING RATIONAL EQUATIONS AND INEQUALITIES

E X A M P L E Solve > ( ) In order to clear the inequality of fractions, you can multiply each side by the LCD, ( )( ). You must consider four possible cases: Case : is positive and is positive, or Case : is positive and is negative, or Case : is negative and is positive, or Case : is negative and is negative. The algebraic method of solution is beyond the scope of this tetbook, but with a graphics calculator, the solution is much easier to find. Rewrite > ( ) as ( > ) 0. Graph y = ( ), and find the values of for which y > 0. TECHNOLOGY GRAPHICS CALCULATOR Keystroke Guide, page 7 Any part of the graph that is above the -ais indicates solutions to the inequality. The graph shows that there are two intervals of for which y > 0. These two intervals can be found by using a table of values. One interval for which y > 0 is < <, as shown above. The other interval for which y > 0 is < < 6, as shown below. Thus, the solution is < < or < < 6. TRY THIS Solve < 6 CHAPTER 8

Eercises Communicate. Eplain what an etraneous solution is and how you can tell whether a solution to a rational equation is etraneous.. Eplain how to use a graph to check the solutions to a rational equation that are obtained by using algebra.. Eplain how to use the graphs of y = y and y =, shown at right, to solve < and >. 6 Guided Skills Practice A P P L I C A T I O N SPORTS Refer to Rachel s triathlon information given at the beginning of the lesson. At what swimming, bicycling, and running speeds must Rachel compete in order to finish the triathlon in hours? (EXAMPLE ) Solve each equation.. = (EXAMPLE ) 6. = Solve each inequality. 7. (EXAMPLE ) 8 < (EXAMPLE ) (EXAMPLE ) Practice and Apply Solve each equation. Check your solution. 9. = 8 0. y y = 6. n = 6. = m. z 8 =. z t = 8 t. y = y 6. n = n 8 7. = y y n n 8 8. = 9. = 0. =. 6 =. = c c. = 0. = 7 b. b b = 0 z 6. b b b 6 z z = z z 0 z 6 z 7z 6 7. = 8. = 9 LESSON 8. SOLVING RATIONAL EQUATIONS AND INEQUALITIES 7

Homework Help Online Go To: go.hrw.com Keyword: MB Homework Help for Eercises 9 7 Solve each inequality. Check your solution. 9. > 0. >. <. <. >. <. 6. 7 7. Use a graphics calculator to solve each rational inequality. Round answers to the nearest tenth. 8 > 9 0 t t t t a a a 6 > 9a a ( ) State whether each equation is always true, sometimes true, or never true. 8 7. = 6, ± 8. 6 =, 0 or ± 9. = 0. = 0. > 6 6. 8 C H A L L E N G E. Solve > 0 by using mental math. ( ). CULTURAL CONNECTION: ASIA A ninth-century Indian mathematician, Mahavira, posed the following problem: There are four pipes leading into a well. Individually, the four pipes can fill the well in,,, and of a day. How long would it take for the pipes to fill the well if they were all working simultaneously, and what fraction of the well would be filled by each pipe? C O N N E C T I O N APPLICATIONS Satellite in orbit over Earth.. GEOMETRY The length of a rectangle is more than its width. Find the length and the width of the rectangle if the ratio of the length to the width is at least. and no more than. 6. SPORTS Michael is training for a triathlon. He swims 0.6 miles, bicycles miles, and runs 8 miles. Michael bicycles about 9 times as fast as he swims, and he runs about 6 miles per hour faster than he swims. a. Write a rational function, in terms of swimming speed, for the total time it takes Michael to complete his workout. b. Find the speeds at which Michael must swim, run, and bike to complete his workout in. hours. 7. PHYSICS An object weighing w 0 kilograms on Earth is h kilometers above Earth. The function that represents the object s weight at that altitude is w(h) = w 0 ( 600 600 ).Find the approimate altitude of a satellite that h weighs 00 kilograms on Earth and 00 kilograms in space. 8 CHAPTER 8

Look Back Evaluate each epression. (LESSON.) 8. 8 9. 0 60. 9 6. 7 Find the inverse of each function. State whether the inverse is a function. (LESSON.) 6. {(, ), (, 8), (, ), (0, )} 6. {(, ),(, ), (, ), (0, )} 6. g() = 6. h() = Identify each transformation from the parent function f() = to g. (LESSON.7) 66. g() = 67. g() = ( ) 68. g() = ( ) 69. g() = 70. g() = () 7. g() = ( ) 6 Portfolio Etension Go To: go.hrw.com Keyword: MB QInequal Look Beyond 7 Graph the functions f() =, g() =, h() =, and k() = How are they alike? How are they different? (Hint: Use the fact that n = n.) PORTFOLIO A C T I T Y I V Refer to Eample on page 08. Notice that the cab driver s average speed is the total distance divided by the total time. The harmonic mean of the two speeds, miles per hour and miles per hour, gives the average speed for the entire trip.. Find the harmonic mean of and.. How does your answer to Step compare with the average speed found in Eample? Justin cycles for hours. He cycles along a level road for miles, and then he cycles up an incline for miles more. Justin immediately turns around and cycles back to his starting point along the same route. Justin cycles on level ground at a rate of miles per hour, uphill at a rate of miles per hour, and downhill at a rate of 8 miles per hour.. Eplain why Justin s average speed over the entire trip is the harmonic mean of,, 8, and.. Find Justin s average speed over the entire trip. WORKING ON THE CHAPTER PROJECT You should now be able to complete the Chapter Project. LESSON 8. SOLVING RATIONAL EQUATIONS AND INEQUALITIES 9

LESSON 8. Key Skills Identify all ecluded values, asymptotes, and holes in the graph of a rational function. y = ( ) = 6 ( )( ) The ecluded values are = and =. is a factor of the numerator and the denominator, so the graph has a hole when =. is a factor of only the denominator, so the vertical asymptote is =. The degree of the numerator is equal to the degree of the denominator, so the horizontal asymptote is y = =. hole when = 8 8 y 8 Eercises Identify all ecluded values, asymptotes, and holes in the graph of each rational function.. R() = 8 6. g() = 7. f() = 8. r(a) = a a a 9. s() = 9 0. M() = 0 9 7 y. h(y) = 6y 8y. r(t) = t t t t t LESSON 8. Key Skills Multiply, divide, and simplify rational epressions, including comple fractions. Simplify 6 6 = 6 = ( ) ( ) ( )( ) ( ) ( ) ( ) = Simplify comple fractions. = ( ) ( ) ( ) = ( ) ( ) ( ) ( ) =,or ( ) 8 0 Eercises Simplify each epression.. 6 0 6 0 8. 9 7 0 a 8. a a 0 a 0 a a 6. 9 6 z 7. z 8. z z 9. 0. ( ) a a (a ) a 6 CHAPTER 8 REVIEW 7

LESSON 8. Key Skills Add and subtract rational epressions. Simplify a a a a a a a = a a a ( a a ) a a ( a a ) a(a ) a(a ) = (a )(a ) a 9a = a a 0 Eercises Simplify each epression.. y y 6 y y.. b 9 9 7b 0 b 9y. y y y 8 y b. 6. LESSON 8. Key Skills Solve rational equations. Solve =. = = = 0 ( )( ) = 0 = or = i or = i The only real solution is =. Therefore, the only point of intersection for the graphs of y = and y = occurs when =. Solve rational inequalities. Solve., > 0 or, < 0 ( ) ( ) If > 0, then >. If < 0, then <. Thus, > and, Thus, < and, or simply >. or simply. The solution is > or. The graphs of y = and y = verify this solution. Eercises Solve each equation. 7. = 8. = 9. = 0. =. =. = Solve each inequality by using algebra.. <.. < 6. 7. < 8. < Solve each inequality by graphing. 9 0 < > < 8 CHAPTER 8

LESSON 8.6 Key Skills Find the inverse of a quadratic function. Find the inverse of y = 7 0. Interchange and y, and solve for y by applying the quadratic formula. = y 7y 0 (7) ± (7) y = ()(0 ) () y = 7 ± 9 Describe the transformations applied to the square-root parent function, f() = Describe the transformations applied to f() = to obtain y =. g() = = ( ) The parent function is stretched vertically by a factor of, compressed horizontally by a factor of,translated horizontally unit to the right, and translated vertically units up. Eercises Find the inverse of each quadratic function.. y = 6. y = 8 7. y = 6 8. y = 7 6 For each function, describe the transformations applied to f() =. 9. g() = 0. h() =. k() =. g() =. h() = 6 Evaluate each epression.. ( 7 ) 6. 8. r() = ( ) LESSON 8.7 Key Skills Simplify epressions involving radicals. (a Simplify 8 b ) a b. a b (a 8 b ) a b a b = 8 a b a b a b 96a = b 7 a b = a 0 b = a b 8 a b = a b a b Rationalize the denominators of epressions. Write with a rational denominator. = ( ) = Eercises Simplify each radical epression. Assume that the value of each variable is positive. 7. 6 y ( y) 8. (a b ) a b (s 9. 60. t 6 ) c d 7 t (6cd ) (6 y 7 ) y (m 9 n) 9m n 7 6. 6. mn Write each epression with a rational denominator and in simplest form. 6. 6. 7 6. 66. 67. 68. CHAPTER 8 REVIEW 9

LESSON 8.8 Key Skills Solve radical equations. Solve = = = = 0 ( )( ) = 0 = or = Check for etraneous solutions. = = ( )? = ) ( () =? () = True = False Solve radical inequalities. To solve, first solve 0. 0 Then solve the original inequality. ( ) Thus, and, or. The solution can be verified by graphing. Eercises Solve each radical equation by using algebra. If the inequality has no real solution, write no solution. Check your solution. 69. = 70. 7 8 = 6 7. = 7. 7 8 = 6 7. = 7. = 7. = 76. = 77. = 78. = 79. = 80. = Solve each radical inequality by using algebra. Check your solution. 8. 8. < 8. 8. > 8. 86. < 87. > 89. 6 < 0 88. < 90. <0 Solve each radical inequality by graphing. 9 9 Applications PHYSICS The weight of an object varies inversely as the square of the distance from the object to the center of Earth, whose radius is approimately 000 miles. 9. If an astronaut weighs 7 pounds on Earth, what will the astronaut weigh at a point 60 miles above Earth s surface? 9. If an astronaut weighs pounds at a point 80 miles above the Earth s surface, how much does the astronaut weigh on Earth? 0 CHAPTER 8

Chapter Test. y varies jointly as and z. If y = 6 when = 7 and z = 9, find y when = and z =. CONSTRUCTION The strength of a beam varies directly with the width of a beam and inversely as the cube of the depth. If a beam 0 mm wide by 0 mm deep will support 00 kg, how much will a beam 8 mm by mm support? Identify all ecluded values, asymptotes, and holes in the graph of each rational function.. f() 6. h() 8 6. g() 9 8 Simplify each epression. 6. 7. 9 6 8 6 7 9 0 6 8. 8 Simplify each epression. 9. 7 0.. GEOMETRY Find the area of the shaded region of the figure at right if the largest triangle has an area A =. Solve each equation or inequality... z z z z.. 7 For each function, describe the transformations applied to f() = 6. g() 7. h() Find the inverse of each quadratic function. 8. y 9. y Evaluate each epression. 0. ( 8 ). (9) ( 6 ) Simplify each epression. Assume that the value of each variable is positive.. 8 y 6 ( y) 8r 7 s 9. rs t. ( ) (7 8). ( )( ) Write each epression with a rational denominator and in simplest form. 6. 7. 8. 7 Solve each radical equation or inequality. If no solution, write no solution. 9. 7 0. 7. 7. CHAPTER 8 TEST

7. domain of all real numbers ecept = and = ; vertical asymptotes: = and = ; horizontal asymptote: y = 0; no holes y 8 9. domain of all real numbers; no vertical asymptotes; horizontal asymptote: y = 0; no holes y 8. domain of all real numbers ecept = and = 6; vertical asymptotes: = and = 6; no horizontal asymptote; no holes y 8. domain of all real numbers ecept = 0 and = ; vertical asymptote: = ; horizontal asymptote: y = 0; hole when = 0 y 8 8 8 8 8 8 8 8 8 8 8 8. f() = 7. f() = 9. f() = a. c > 9 ;ifthere are no vertical asymptotes, then c = 0 has no solutions and b ac 0, 9 c < 0. b. c = 9 ;ifthere is vertical asymptote, then c = 0 has solution and b ac = 0, 9 c = 0. c. c < 9 ;ifthere are vertical asymptotes, then c = 0 has solutions and b ac 0, 9 c 0. a. R() = b. all real numbers greater than 0; all real numbers greater than 0; all real numbers greater than 0 a. T() =. 0 b. C() = T( ) =. 0 7. < or > 8 9.. 6. 9 6. 9 6 7. ( ) 9. ( 7)( 7) 6. ( 6) LESSON 8. TRY THIS (p. 98) b 7 b TRY THIS (p. 99, E. ) 9 a TRY THIS (p. 99, E. ) 0 6 9 TRY THIS (p. 00) TRY THIS (p. 0) 6 Eercises.. 8 6. 7. 8. 8 9.. 9.. 7. 9... 8 8 6 a b ( ). 7. a y 9. b (. ).. 7. 9.. 9. y y a. V = w h; V = (0 )(6 ); V = (0 )(6 ) b. Total Area A = (0 )(6 ) (0 ) (6 ) = 0 0 0 = 0 c. R() = 8 80 80 dt d d. R() increases 7a. a = t t (t tt ) b. feet per second per second 9. y = 0. y 6 6. ( ) 8. i 7. i 9. 7 6. ( )( ) 6. ( 6 8) 6. 67. 6 Selected Answers SELECTED ANSWERS 07

Selected Answers LESSON 8. TRY THIS (p. 0) a. b. TRY THIS (p. 06) = ( ) ( ) TRY THIS (p. 07, E. ) = ( )( ) TRY THIS (p. 07, E. ) a = a a (a )(a )(a ) Eercises.. 8 6. = 8 ( ) ( ) 7. = t ( ) ( ) 8. 9. t.9 mph.. 0. 9 = ( ) 6 9 6 7. = 6 ( ) ( ) 7 9. = 7 ( )( ) 6. = ( ) ( ). = ( ). 7. 9. ( = ) ( ). b a 6.. b a 7. 6 = 6 ( ) ( ) ab 9. (a b) ( a b) = ab a a b b r r. ( r s)( s s r s) = r rs s r. B = and D = s. A = 6, C = 6, and D = 7 7a.. ohms R A R B R C b. R T = 9. 8.8 mph R B R C R A R C R A R B. Distributive Property. 7; 0; = ± i. ; ; = or = 7. log, or log 8 ; 9. = or = or = 6. = or = or = LESSON 8. TRY THIS (p. ) = or = TRY THIS (p. ) = TRY THIS (p. ) > or < 7 TRY THIS (p. 6) < 7 Eercises. Rachel must swim at about.6 miles per hour, bicycle at about 6(.6), or.6, miles per hour, and run at about.6, or 8.6, miles per hour.. = or = 6. no solution 7. < 0 8. < < 9. =. n = 0. z = 8. y = 7 7. = 9. = ± 0 0. = ± 7. no solution. b = 7. no solution 9. 0 < <. 0 < <. < < 9. < 0 7. < < 9. < 0 or... < 0 or 0 <... t < or t.. a < or a < 0 or a 7. always 9. never. sometimes. all real numbers ecept =.. w 0 and = w ; thus, 7.. 7. about 0 km 9. 6. 6.{(, ), (, ), (, ), (, 0)}; function 6. h () = ; function 67. a horizontal translation of units to the right 69. a vertical stretch by a factor of and then a vertical translation of units down 7. a horizontal translation units to the right, a reflection across the y-ais, a vertical stretch by a factor of, and a vertical translation 6 units down LESSON 8.6 TRY THIS (p. ) 8 or.6 TRY THIS (p. ) a. a vertical stretch by a factor of, a vertical translation units down, and a horizontal translation unit to the right b. a horizontal compression by a factor of, a vertical translation units up, and a horizontal translation unit to the left TRY THIS (p. ) Interchange the roles of and y. y = = y y y = ± y = y TRY THIS (p. ) 0, 6 ± y = 08 INFO BANK

.. or.6. never. never 7. sometimes 9. always 6a. a < 0 b. a 0 6a. 9,96 feet b. 6. miles c. 698 feet 6. y = 7 67. = or = 69. = 0.8 or =. 7. 6 7 7. 6 y 6 7. 77. 0 CHAPTER 8 REVIEW. y =. a = 0. ecluded values: = and = 6; 7 no holes; vertical asymptotes: = and = 6; horizontal asymptote: y = 0 7. ecluded values: = 7 and = ; no holes; vertical asymptotes: = 7 and = ; horizontal asymptote: y = 9. ecluded value: = ; no holes; vertical asymptote: = ; no horizontal asymptote. ecluded values: y = 0 and y = ; hole when y = 0; vertical asymptote: y = ; horizontal asymptote: h = 0 a.. 8a z 0 a 7. a 0 z 9. z y 9.. 0y 0. 7. = ± 9. = ± 9. = ±. > or < 0. < or > 7. < or > 9. or 0 <. < or < 0...7 < < or < < 0.. y = ± 9 7. y = 8± 9 9. The parent function is vertically compressed by a factor of. The parent function is horizontally translated units to the right and horizontally compressed by a factor of. The parent function is horizontally compressed by a factor of, vertically stretched by a factor, reflected across the -ais, and vertically translated 6 units down.. 7. y y 9. cd 7 6. y y 6. 6. 6 67. 6 69. no solution 6 7. = 0 7. = 0 or = 7. = 77. = 79. no solution 8. 0 8. 8. 87. > 89. no solution 9. 0 9. 69.9 pounds Chapter 9 LESSON 9. TRY THIS (p. 6) a. ellipse ( y = ± 6 ) b. parabola (y = ±6 ) TRY THIS (p. 6) 9,or about.9 TRY THIS (p. 6) (., 0.) TRY THIS (p. 66) center: (, ); C = 7π; A = 7π Eercises. hyperbola. circle 6. 8.76 7..6 miles 8. M (, ) 9. (, ); C = π7; A = 7π. circle. hyperbola. ellipse 7. parabola 9. circle. hyperbola. 7; M (, ). 6 8.06; M (, 9 ) 7. 7.; M (, )..0; M (. 9..8; M(9, ), ) M(.,.). 0.68; 7.; M ( 9, ) M (,7 ) M(., 7.9) 7. a 7. a ; M ( a, a ) 9. (,0 ); C = π; A = 6 9π. (6, ); C = π; A = π. (, ); C = π9; A = 9π. (, ) 7. (, ) 9. AB = ; BC = 0; AC = 8; not collinear. AB = 89; BC = 89; AC = 89; collinear a. AB = 7; BC = 7; CD = 7; DA = 7 b. yes a. AB: m = ; CD: m = ;the slopes of AD and BC are undefined. b. Yes, because AB and CD have the same slope, and AD and BC are both vertical. 7. scalene 9. y = 7 6. 6. = or = 6. = ± 9 67. = or = 6 69. 7. LESSON 9. TRY THIS (p. 7, E. ) y F(0, ) V(0, 0) y = TRY THIS (p. 7, E. ) = y 6 TRY THIS (p. 7, E. ) = (y ) 6 Selected Answers SELECTED ANSWERS 0