Rational Equations and Functions

Transcription

1 Rational Equations and Functions. Direct and Inverse Variation. Graphing Rational Functions. Simplifing Rational Epressions. Multipling and Dividing Rational Epressions. Dividing Polnomials. Adding and Subtracting Rational Epressions.7 Solving Rational Equations rk Descartes, in our homewo sed cha ng bei are ou m, ble pro b a cat-eating hena. Both of ou must sta on the graph of = /. The safe zone is the - ais. Descartes, I am keeping track of how man dogg treats m ow ner gives me each da. Can ou ever reach the safe zone? Eplain our reasoning. I am finding th at m happines s is directl prop ortional to the da of the wee k. MSCC_ALG_PE_00_co.indd 0 /7/ 8:: AM

2 What You Learned Before Your nam esake, Re ne Desca believed in rtes, rational th inking. Eample 0 Find +. 0 Eample 0 0 +=+ Find ind. = + 0 = 7 0 = = = Evaluate the epression.. + Eample. 8.. Solve =. = = Write the proportion. Use the Cross Products Propert. 8 = Multipl. 9. = Divide each side b. Solve the proportion.. = w 8. = 7. = MSCC_ALG_PE_00_co.indd /7/ 8::0 AM

3 . Direct and Inverse Variation How can ou recognize when two variables var directl? How can ou recognize when the var inversel? ACTIVITY: Recognizing Direct Variation Work with a partner. You hang different weights from the same spring. Equilibrium 0 kg 0. kg 0. kg 0. kg 0. kg 0. kg centimeters COMMON CORE Direct and Inverse Variation In this lesson, ou will identif direct and inverse variation. write and graph direct and inverse variation equations. Learning Standard A.REI.0 a. Describe the relationship between the weight and the distance d the spring stretches from equilibrium. Eplain wh the distance is said to var directl with the weight. b. Graph the relationship between and d. What are the characteristics of the graph? c. Write an equation that represents d as a function of. d. In phsics, the relationship between d and is described b Hooke s Law. How would ou describe Hooke s Law? d 0. kg kg Chapter Rational Equations and Functions

4 ACTIVITY: Recognizing Inverse Variation Math Practice Calculate Accuratel How can ou verif that our graph and equation represent the relationship between and? Work with a partner. The area of each rectangle is square inches. = in. = in. = in. = in. = in. = in. = in. = in. = in. = in. = 8 in. = 8 in. = in. = in. 0 0 a. Describe the relationship between and. Eplain wh is said to var inversel with. b. Graph the relationship between and. What are the characteristics of the graph? c. Write an equation that represents as a function of IN YOUR OWN WORDS How can ou recognize when two variables var directl? How can ou recognize when the var inversel?. Does the flapping rate of a bird s wings var directl or inversel el with the length of its wings? Eplain our reasoning. Use what ou learned ed about direct and inverse variation to complete Eercises and on page 7. Section. Direct and Inverse Variation

5 . Lesson Lesson Tutorials Ke Vocabular direct variation, p. inverse variation, p. Direct Variation Two quantities and show Inverse Variation Two quantities and show direct variation when = k, where k is a nonzero constant. inverse variation when =, k where k is a nonzero constant. Stud Tip The constant k is called the constant of proportionalit or the constant of variation. = k, k > 0 = k, k < 0 The ratio is constant. EXAMPLE k =, k > 0 k =, k < 0 The product is constant. Identifing Direct and Inverse Variation Tell whether and show direct variation, inverse variation, or neither. Eplain our reasoning. a. 0 0 The products are not constant. So, the table does not show inverse variation. Check each ratio : =, 0 =, =, 0 = The ratios are constant. So, and show direct variation. b. = = Divide each side b. k The equation is of the form =. So, and show inverse variation. Eercises Tell whether and show direct variation, inverse variation, or neither. Eplain our reasoning.. Chapter MSCC_ALG_PE_0.indd 8 Rational Equations and Functions. = + /7/ 8:8:9 AM

6 Reading EXAMPLE For direct variation equations, ou can sa varies directl with or is directl proportional to. For inverse variation equations, ou can sa varies inversel with or is inversel proportional to. Writing and Graphing a Direct Variation Equation The variable varies directl with. When =, =. Write and graph a direct variation equation that relates and. Find the value of k. = k Write the direct variation equation. = k() Substitute for and for. = k Divide each side b. So, an equation that relates and is =. EXAMPLE Writing and Graphing an Inverse Variation Equation The variable varies inversel with. When =, =. a. Write an inverse variation equation that relates and. Find the value of k. = k Write the inverse variation equation. = k Substitute for and for. 0 = k Multipl each side b. Stud Tip Notice that the equation = 0 is undefined when = 0. So, there is no point on the graph for = 0. So, an equation that relates and is = 0. b. Graph the inverse variation equation. Describe the domain and range. Make a table of values undef. Plot the ordered pairs. Draw a smooth curve through the points in each quadrant. Both the domain and range are all real numbers ecept Eercises 7. The variable varies directl with. When =, =. Write and graph a direct variation equation that relates and.. The variable varies inversel with. When =, =. Write and graph an inverse variation equation that relates and. Section. Direct and Inverse Variation

7 EXAMPLE Identifing Inverse Variation Which situation represents inverse variation? A B C D You bu several movie tickets for $7.0 each. You earn $0.0 for each pound of aluminum cans ou reccle. The cost of a $00 cabin rental is shared equall b a group of friends. You download several songs for $0.99 each. Make a table of values for each situation. A Number of tickets, Total cost, B Number of pounds, Total earned, C Number of people, Cost per person, D Number of songs, Total cost, The ratio is constant. The ratio is constant. The product is constant. The ratio is constant. The correct answer is C. EXAMPLE Real-Life Application You bike miles each morning. Your time t (in hours) to bike miles t 0 8 t r 8 0 r is given b t =, where r is our average speed (in miles per hour). r Graph the function. Make a conclusion from the graph. Because average speed cannot be negative, use onl nonnegative values of r. r 0 t undef. From the graph, ou can see that as our average speed increases, the time it takes ou to bike miles decreases. Eercises. The cost of a tai ride is shared equall b several friends. Does this situation represent direct variation or inverse variation? Eplain.. WHAT IF? In Eample, ou bike miles each morning. Write and graph a function that represents our time. Then make a conclusion from the graph. Chapter Rational Equations and Functions

8 . Eercises Help with Homework. VOCABULARY Eplain how direct variation equations and inverse variation equations are different.. WHICH ONE DOESN T BELONG? Which graph does not belong with the other three? Eplain our reasoning. g() Ź Ź f() Ź Ź Ź Ź Ź Ź Ź t() Ź Ź Ź Ź Ź m() Ź Ź Ź )= 9+(- )= +(- 9)= +(- = ) 9+(- Tell whether and show direct variation, inverse variation, or neither. Eplain our reasoning =. = The variable varies directl with. Write and graph a direct variation equation that relates and. 7. When =, =. 8. When =, =. 9. When = 0, =. The variable varies inversel with. Write and graph an inverse variation equation that relates and. 0. When =, =.. When =, = 9.. When =, =.. VOLUNTEERS You want to raise $00 for a charit. You volunteer h hours and raise r dollars each hour. The equation hr = 00 represents this situation. Does this represent direct variation, inverse variation, or neither? Eplain our reasoning. MSCC_ALG_PE_0.indd 7 Section. Direct and Inverse Variation 7 /7/ 8:8: AM

9 . ERROR ANALYSIS The variable varies inversel with. When = 8, =. Describe and correct the error in writing an inverse variation equation that relates and. Graph the equation. Describe the domain and range. = k = k(8) 8 = k So, = 8.. =. = 7. = 9 8. REASONING When varies directl with, does var directl with? If so, describe the relationship between the constants of proportionalit. Eplain our reasoning. The variable varies inversel with. Write an inverse variation equation that relates and. Then find the missing value of or. 9. When =, =. Find when =. 0. When =, =. Find when =.. When =, =. Find when =.. When = 0, =. Find when = 8. Determine whether the situation represents direct variation or inverse variation. Justif our answer.. You have enough mone to bu hats for $0 each or 0 hats for $ each.. Your cousin earns $0 for mowing lawns or $7 for mowing lawns.. The mone the swim team earns from a car wash is divided evenl among the members.. RUNNING You race in a 00-meter dash. Your average speed r (in meters per second) is given b r = 00, where t is the time t (in seconds) it takes ou to finish the race. Graph the function. Make a conclusion from the graph. 7. VACATION The amount v of vacation time (in hours) that an emploee earns varies directl with the amount t of time (in months) she works. An emploee who works months earns hours of vacation time. a. Write and graph a direct variation equation that relates v and t. b. How man hours of vacation time does the emploee earn after working months? 8 Chapter Rational Equations and Functions

10 8. REASONING Make a table using positive -values for the inverse variation equation v = and the direct variation equation d =. How does the rate of change of v differ from the rate of change of d? 9. THEATER A performing arts compan is hiring actors as etras for a theater performance. The amount t of performance time (in hours per person) varies inversel with the number p of etras hired. The director estimates that he will need 0 etras performing 0 hours each. a. Write an inverse variation equation that relates t and p. b. The director decides to hire etras. How much performance time will each etra receive? 0. STRUCTURE To balance the board in the diagram, the distance (in feet) of each animal from the center of the board must var inversel with its weight (in pounds). What is the distance of each animal from the fulcrum? d ft ft 7 lb fulcrum lb A function f is odd if f ( ) = f (). A function f is even if f ( ) = f (). Determine whether the function is odd, even, or neither.. f () =. f () =. f () =. f () =. f () =. f () = 7. REASONING Describe the smmetr shown in the graph of (a) an even function and (b) an odd function. Justif our answers. 8. Precision Are all direct variation and inverse variation equations odd functions? Eplain. Graph the function. Compare the graph to the graph of =. (Section 8. and Section 8.) 9. = 0. = +. =. MULTIPLE CHOICE What is the solution of the equation =? (Section 0.) A B C 9 D 8 Section. Direct and Inverse Variation 9

11 . Graphing Rational Functions rational function? What are the characteristics of the graph of a ACTIVITY: Graphing a Rational Function Work with a partner. As a fundraising project, our math club is publishing an optical illusion calendar. The cost of the art, tpesetting, and paper is $80. In addition to this one-time cost, the unit cost of printing each calendar is $.. a. Let A represent the average cost of each calendar. Write a rational function that gives the average cost of printing calendars. A = b. Make a table showing the average costs for several different production amounts. Then use the table to graph the average cost function. COMMON CORE Graphing Rational Functions In this lesson, ou will graph rational functions. identif asmptotes. compare graphs of rational functions. Learning Standards A.REI.0 F.BF.a A Chapter Rational Equations and Functions

12 Math Practice Justif Conclusions What information can ou use to justif our conclusion? ACTIVITY: Analzing the Graph of a Rational Function Work with a partner. Use the graph in Activit. a. What is the greatest average cost of a calendar? Eplain our reasoning. b. What is the least average cost of a calendar? Eplain our reasoning. What characteristic of the graph is associated with the least average cost? ACTIVITY: Analzing Profit and Revenue Work with a partner. Consider the calendar project in Activit. Suppose our club sells 00 calendars for $0 each. a. Find the revenue our club earns from the calendars. b. How much profit does our club earn? Eplain our reasoning.. IN YOUR OWN WORDS What are the characteristics of the graph of a rational function? Illustrate our answer with the graphs of the following rational functions. a. = + b. = + c. = + Use what ou learned about the graphs of rational functions to complete Eercises and on page. Section. Graphing Rational Functions

13 . Lesson Lesson Tutorials Ke Vocabular rational function, p. ecluded value, p. asmptote, p. The inverse variation equations in Section. are rational functions. Rational Function A rational function is a function of the form = polnomial, where the polnomial denominator does not equal 0. The most basic rational function is =. Because division b 0 is undefined, the value of the denominator of a rational function cannot be 0. So, the domain of a rational function ecludes values that make the denominator 0. These values are called ecluded values of the rational function. EXAMPLE Finding the Ecluded Value of a Rational Function Find the ecluded value of = +. Find the value of that makes the denominator 0. + = 0 Use the denominator to write an equation. = Subtract from each side. The ecluded value is =. EXAMPLE Graphing a Rational Function Graph =. Describe the domain and range. The ecluded value is =, so choose -values on either side of. Step : Make a table of values undef. Step : Plot the ordered pairs. Step : Draw a smooth curve through the points on each side of =. The domain is all real numbers ecept and the range is all real numbers ecept 0. Chapter Rational Equations and Functions

14 Find the ecluded value of the function. Eercises 7. =. = 8 + =. Graph the function. Describe the domain and range. 8. =. + = =. The ecluded value in Eample is =. Notice that the graph approaches the vertical line =, but never intersects it. The graph also approaches the horizontal line = 0, but never intersects it. These lines are called asmptotes. An asmptote is a line that a graph approaches, but never intersects. Asmptotes The graph of a rational function of a h the form = + k, where a 0, â à Ź â has a vertical asmptote = h and a horizontal asmptote = k. Ź 7 â Ź Ź EXAMPLE Identifing Asmptotes Identif the asmptotes of the graph of =. Then describe the domain and range. Rewrite the function to find the asmptotes. Check Ź 7 Horizontal Asmptote: = = + ( ) Vertical Asmptote: = The vertical asmptote is = and the horizontal asmptote is =. So, the domain of the function is all real numbers ecept and the range is all real numbers ecept. Ź9 Eercises 9 Identif the asmptotes of the graph of the function. Then describe the domain and range. 7. = + MSCC_ALG_PE_0.indd 8. + = Section = Graphing Rational Functions /7/ 8::7 AM

15 EXAMPLE Comparing Graphs of Rational Functions + Graph = +. Compare the graph to the graph of =. Stud Tip Step : Make a table of values. The vertical asmptote is =, so choose -values on either side of. Use the asmptotes to help ou draw the ends of the graph. â à à.. 0. undef.. Step : Use dashed lines to graph the asmptotes = and =. Then plot the ordered pairs. â Step : Draw a smooth curve through the points on each side of the vertical asmptote. + The graph of = + is a translation units up and units left of the graph of =. EXAMPLE Real-Life Application The French club is planning a trip to Québec Cit. The function Costs for Québec Cit trip = + 00 represents the cost (in dollars) per student when Le bus La nourriture L hôtel students and chaperones go on the trip. Use a graphing calculator to graph the function. How man students must go on the trip for the cost per student to be about $0? $800 $0 each $0 each Bon Voage! 80 Step : Use a graphing calculator to graph the function. Because the number of students cannot be negative, use onl nonnegative values of. Step : Use the trace feature to find where the value of is about About students must go on the trip for the cost per student to be about $0. = + Graph the function. Compare the graph to the graph of =. Eercises 8 0. =. =.. WHAT IF? In Eample, how man students must go on the trip for the cost per student to be about $80? Chapter MSCC_ALG_PE_0.indd Rational Equations and Functions /7/ 8::8 AM

16 . Eercises Help with Homework +. VOCABULARY Is = a rational function? Eplain.. VOCABULARY How is an ecluded value related to a vertical asmptote?. WRITING How can ou use asmptotes to help graph a rational function? )= 9+(- )= +(- 9)= +(- = ) 9+(- Describe the characteristics of the graph.. â 0. 0 à 0 0 â 0 Ź Ź0 Ź0 Ź Ź0 Ź Ź0 Find the ecluded value of the function = = 7. =. = 9 9. = + 0. = Graph the function. Describe the domain and range.. =. =. =. =. = = 8. HIKING You hike miles through a national forestt to a er hour) famous landmark. Your average speed (in miles per is represented b =, where is the total time (in hours) of the hike. a. Find the ecluded value of the function. b. Graph the function. Describe the domain and range. MSCC_ALG_PE_0.indd Section. Graphing Rational Functions Function n /7/ 8::0 AM

17 Identif the asmptotes of the graph of the function. Then describe the domain and range. 9. = 0. = + 8. = + 7. = +. =. = ERROR ANALYSIS Describe and correct the error in identifing the asmptotes of the graph of the function.. REASONING Describe the domain and range a of a rational function of the form = h + k. = + + The horizontal asmptote is =. The vertical asmptote is =. 7. OPEN-ENDED Write a rational function whose graph has the vertical asmptote = and the horizontal asmptote = 9. Graph the function. Compare the graph to the graph of =. 8. = + 9. = 0. = +. = =. = + 8. SOFTBALL A softball team bus a new $0 bat for a softball tournament. The cost of the bat is shared equall b the plaers on the team. Each plaer must also pa a $0 registration fee. The amount (in dollars) each plaer pas is represented b = 0 + 0, where p is the p number of plaers on the team. Graph the function. How man plaers must be on the team for the cost per plaer to be about $8? A. GEOMETRY The formula h = gives the b + b height h of a trapezoid, where A is the area and b and b are the base lengths. Suppose A = 0 and b = 8. a. Graph the function. Describe the domain and range. b. Use the graph to find b when h =.. ROAD TRIP The function t = 80 + models the total time t (in hours) it takes r to drive 80 miles at r miles per hour. The model allows for two half-hour breaks. Graph the function. What does our average speed need to be for the total travel time to be hours? Chapter Rational Equations and Functions

18 Write a function for the graph REPEATED REASONING Use a graphing calculator to graph the a function = + for several values of a. How does the value of a affect the graph? Consider a < 0, a >, and 0 < a < in our answer.. THUNDERSTORM The time t (in seconds) it takes for sound to travel kilometer can be represented 000 b t =, where T is the temperature in degrees 0.T + Celsius. Use a graphing calculator to graph the function for 0 T 00. During a thunderstorm, lightning strikes kilometer awa. You hear the thunder.9 seconds later. What is the temperature? Graph the function. Identif the asmptotes.. =. = +. = +. Modeling To qualif for a mortgage, the ratio r of our epected monthl housing epenses to our gross monthl income cannot be greater than 0.8. Suppose our gross monthl income is $00 and ou epect to pa $00 per month in housing epenses. You also epect to get a raise of m dollars this month. a. Write and graph an equation that gives r as a function of m. b. How much must the raise be in order for ou to qualif for a mortgage? Does the equation represent a linear or nonlinear function? Eplain. (Section.). = 7. + = 8. = 8 9. MULTIPLE CHOICE Which function models eponential deca? (Section.) A = ( ) B = () C = () D = ( ) Section. Graphing Rational Functions 7

19 Etension. Inverse of a Function Lesson Tutorials Ke Vocabular inverse relation, p. 8 inverse function, p. 9 Recall that a relation pairs inputs with outputs. An inverse relation switches the input and output values of the original relation. For eample, if a relation contains (a, b), then the inverse relation contains (b, a). EXAMPLE COMMON CORE Graphing Rational Functions In this etension, ou will find inverse functions. Learning Standards A.REI.0 F.BF.a Find the inverse of each relation. a. (, 7), (, ), (0, ), (, ), (, ) b. Finding Inverse Relations (7, ), (, ), (, 0), (, ), (, ) Input 0 Output Inverse relation: Input Output 0 Switch the coordinates of each ordered pair. Switch the inputs and outputs. Find the inverse of the relation.. (, 8), (, ), (0, 0), (, ), (0, 8). (, ), (, 0), (, ), (0, 8), (, ), (, ), (, 0). Input 0 Output 0. Input 0 0 Output 7 8. WRITING How do the domain and range of a relation compare to the domain and range of its inverse relation? Eplain.. CRITICAL THINKING Recall that ou can use the Vertical Line Test to determine whether a graph represents a function. What kind of similar test do ou think ou could use to determine whether a function has an inverse that is also a function? Eplain. 8 Chapter Rational Equations and Functions

20 Reading The in f ( ) is not an eponent. When a relation and its inverse are functions, the are called inverse functions. The inverse of a function f is written as f (). To find the inverse of a function represented b an equation, switch and and then solve for. Stud Tip EXAMPLE The domain is nonnegative in Eample b, so the range of the inverse must be nonnegative. This is wh ou take onl the positive square root of each side. Finding Inverse Functions Find the inverse of each function. Graph the inverse function. a. f () = = Replace f() with. = Switch and. + = + Add to each side. = Divide each side b. + = f () Replace with f (). b. f () =, where 0 = Replace f() with. = Switch and. = = f () Take the positive square root of each side. Replace with f () f() f() f () 8 0 f () 7 Find the inverse of the function. Graph the inverse function. 7. f () = 8. f () = + 9. f () =, where 0 0. f () =, where 0. f () =. f () =. REASONING Suppose f and f are inverse functions and f ( ) =. What is the value of f ()?. REASONING Draw the line = on the graph in each part of Eample. What do ou notice?. LOGIC Suppose f and g are inverse functions. What do ou know about f (g ()) and g( f ())? Eplain. Etension. Inverse of a Function 9

21 . Simplifing Rational Epressions What are the ecluded values of a rational epression? How can ou simplif a rational epression? ACTIVITY: Simplifing a Rational Epression Work with a partner. Sample: You can see that the rational epressions + and + are equivalent b graphing the related functions = + and = +. COMMON CORE Radical Epressions In this lesson, ou will simplif rational epressions. Learning Standard A.SSE. Both functions have the same graph. Match each rational epression with its equivalent rational epression. Use a graphing calculator to check our answers. a. + b. + c. + d e A. + B. + C. + D. E. + 0 Chapter Rational Equations and Functions

22 ACTIVITY: Finding Ecluded Values Work with a partner. Are the graphs of = + and = + eactl the same? Eplain our reasoning. ACTIVITY: Simplifing and Finding Ecluded Values Math Practice Eplain the Meaning What does it mean for a simplified epression to have an ecluded value? Work with a partner. Simplif each rational epression, if possible. Then compare the ecluded value(s) of the original epression with the ecluded value(s) of the simplified epression. a. d. + b e. c. f. +. IN YOUR OWN WORDS How can ou simplif a rational epression? What are the ecluded values of a rational epression? Include the following rational epressions in our answer. a. ( + ) b c. + 9 Use what ou learned about simplifing rational epressions to complete Eercises on page. Section. Simplifing Rational Epressions

23 . Lesson Lesson Tutorials A rational epression is an epression that can be written as a fraction whose numerator and denominator are polnomials. Values that make the denominator of the epression zero are ecluded values. Ke Vocabular rational epression, p. simplest form of a rational epression, p. Simplifing Rational Epressions A rational epression is in simplest form when the numerator and denominator have no common factors ecept. To simplif a rational epression, factor the numerator and denominator and divide out an common factors. Words Stud Tip You can see wh ou can divide out common factors b rewriting the epression. ac bc a b = Let a, b, and c be polnomials, where b, c 0. Algebra c = b = b c a ac bc a a c b c Eample ( + ) = ; ( + ) a b == EXAMPLE Simplifing Rational Epressions Simplif each rational epression, if possible. State the ecluded value(s). Divide out the common factor. Simplif. a. = = The ecluded value is = 0. n n+8 b. The epression is in simplest form. The ecluded value is n = 8. Stud Tip ( 7) ( 7) Divide out the common factors. ( 7) Simplif. c. = Make sure ou find ecluded values using the original epression. = The ecluded values are = 0 and = 7. Simplif the rational epression, if possible. State the ecluded value(s). Eercises 8 Chapter MSCC_ALG_PE_0.indd. Rational Equations and Functions. 8( + ). m+ m(m + ) /7/ 8:: AM

24 EXAMPLE Simplifing Rational Epressions Simplif each rational epression, if possible. State the ecluded value(s). z z ( z)( + z) z Difference of Two Squares Pattern (z )( + z) z Rewrite z as (z ). = (z )( + z) z Divide out the common factor. = z Simplif. a. = = The ecluded value is z =. c + c c c 0 (c + )(c ) (c + )(c ) Factor. Divide out the common factor. c c Simplif. = b. = The ecluded values are c = and c =. EXAMPLE Real-Life Application In general, as the surface area to volume ratio of a substance increases, it reacts faster with other substances. Write and simplif this ratio for a block of ice that has the shape shown. Surface area Volume ( ) + ( ) ()() = Write an epression. 0 Simplif. Divide out the common factors. Simplif. = = Simplif the rational epression, if possible. State the ecluded Eercises 0 value(s). b + 8 7b a a a. z z 8 z 7. What is the surface area to volume ratio of a cube-shaped substance with edge length? MSCC_ALG_PE_0.indd Section. Simplifing Rational Epressions /7/ 8::7 AM

25 . Eercises Help with Homework. VOCABULARY Is a rational epression? Eplain. +. REASONING Wh is it necessar to state ecluded values of a rational epression? 9+(-)= +(-)= +(-9)= 9+(-)= Simplif the rational epression, if possible. State the ecluded value(s) w w 7. t t(t + ).. n n ERROR ANALYSIS Describe and correct the error in stating the ecluded value(s). ( ) = The ecluded value is =. Simplif the rational epression. State the ecluded value(s). 0. b + 9 8b +. z z. a + a 9a + 8a. 0. n + n + n + 8n +. + ( + )( ). WRITING Is in simplest form? ( )( ) Eplain. ( ) ( ) 7. RECYCLING You hang reccling posters on bulletin boards at our school. Simplif the dimensions of the poster. Chapter Rational Equations and Functions

26 Write and simplif a rational epression for the ratio of the perimeter of the figure to its area OPEN-ENDED Write a rational epression whose ecluded values are and.. WRITING Is equivalent to? Justif our answer. + ( ) in. in. h in.. PROBLEM SOLVING The candles shown have the same volume. Write and simplif an epression for the height of the cone-shaped candle. in. Sandbo A Sandbo B. SANDBOX The area of Sandbo B is square feet greater than the area of Sandbo A. Write and simplif an epression for the width w of Sandbo B. ft ( ) ft ( ) ft w ft. Find two polnomials whose simplified ratio is + and whose sum is +. Eplain our reasoning. Graph the function. Is the domain discrete or continuous? (Section.). Input Boes, Output Number of Shoes, 7. Input Months, Output Height of Plant, (inches) MULTIPLE CHOICE Consider f () =. What is the value of so that f () = 8? (Section.) A B C D 7 Section. Simplifing Rational Epressions

27 Stud Help Graphic Organizer You can use an eample and non-eample chart to list eamples and non-eamples of a vocabular word or term. Here is an eample and non-eample chart for inverse variation equations. Inverse Variation Equations Eamples Non-Eamples = = = = = = = = + Make eample and non-eample charts to help ou stud these topics.. direct variation equations. rational functions. ecluded values. asmptotes. rational epressions. simplest form of a rational epression After ou complete this chapter, make eample and non-eample charts for the following topics. 7. multipling and dividing rational epressions 8. least common denominator of rational epressions 9. adding and subtracting rational epressions 0. rational equations What do ou think of m eample & non-eample chart for popular cat tos? Chapter Rational Equations and Functions

28 .. Quiz Progress Check Tell whether and show direct variation, inverse variation, or neither. Eplain our reasoning. (Section.) The variable varies directl with. When =, =. Write and graph a direct variation equation that relates and. (Section.). The variable varies inversel with. When =, = 7. Write and graph an inverse variation equation that relates and. (Section.) Find the ecluded value of the function. (Section.). = 7. = Identif the asmptotes of the graph of the function. Then describe the domain and range. (Section.) 8. = 9. = 0 0. = Find the inverse of the function. Graph the inverse function. (Section.). f () = +. f () = +, where 0 Simplif the rational epression, if possible. State the ecluded value(s). (Section.).. z. 8 z DIMENSIONS Simplif the dimensions of the computer monitor. (Section.) 7. FISHING BOAT The cost c per person to charter a fishing boat varies inversel with the number n of people fishing. The cost to charter a boat for an entire da is $00. (Section.) a. Write an inverse variation equation that relates c and n. b. How much does each person pa when 8 people fish? Sections.. Quiz 7

29 . Multipling and Dividing Rational Epressions epressions? How can ou multipl and divide rational ACTIVITY: Matching Quotients and Products Work with a partner. Match each quotient with a product and then with a simplified epression. Eplain our reasoning. Quotient of Two Product of Two Simplified Rational Epressions Rational Epressions Epression a. 0 A. 0. b. 0 B c. 0 C. 0. d. 0 D e. + + E. +. f. + F.. + g. G COMMON CORE Rational Epressions In this lesson, ou will multipl and divide rational epressions. Learning Standard A.SSE. h. i. j. + H. ( ) I. ( ) J ( ) k. K.. + l. L. ( ). ( ) 8 Chapter Rational Equations and Functions

30 Math Practice Use Definitions How do previousl established definitions and results help ou to solve this puzzle? ACTIVITY: Solving a Math Crossword Puzzle Work with a partner. Solve the crossword puzzle Across. Inverse of subtraction. or 7. Greek mathematician 0. Longest side of a right triangle. ( + ). = k. = k ft or Down. 0. C of this is πr. Dimension of. 0. Is the same as 7. About. 8. Graph approaches = h Two numbers whose product is. in., 0,, etc. 9. in ( + ). IN YOUR OWN WORDS How can ou multipl and divide rational epressions? Include the following in our answer. a. + + b. + Use what ou learned about multipling and dividing rational epressions to complete Eercises and 0 on page 7. Section. Multipling and Dividing Rational Epressions 9

31 . Lesson Lesson Tutorials You can use the same rules that ou used for multipling and dividing fractions to multipl and divide rational epressions. Multipling and Dividing Rational Epressions Let a, b, c, and d be polnomials. Multipling: Dividing: a b c d = ac, where b, d 0 bd a b c d = a b d c = ad, where b, c, d 0 bc EXAMPLE Multipling Rational Epressions Find each product. Remember Remember that epressions ma have ecluded values. In Eample a, the ecluded values are = and = 0. a. + = ( + ) = ( + ) 0 = + Multipl numerators and denominators. Divide out the common factors. Simplif. b. h h + h + h + h h (h + )(h + ) = h + h Factor h + h +. h(h + )(h + ) = h (h + ) Multipl numerators and denominators. h(h + )(h + ) = h (h + ) Divide out the common factors. = h + h Simplif. Eercises 9 Find the product c (c 8). z z 7z Chapter Rational Equations and Functions

32 EXAMPLE Dividing Rational Epressions 8 Which epression is equivalent to w w when w? w A 8 w B w 8 C 8w (w ) D 8w w 8w + 8 w w w = 8 w w w 8(w ) = w(w ) 8(w ) = w(w ) = 8 w Multipl b the reciprocal. Multipl numerators and denominators. Divide out the common factor. Simplif. The correct answer is A EXAMPLE Dividing Rational Epressions Find the quotient p p (p ). p + p p p + p = p p p + p (p )(p + ) = p + (p )(p + ) (p )(p + ) = (p + )(p )(p + ) (p )(p + ) = (p + )(p )(p + ) p = (p + )(p ) Write p as a fraction. Multipl b the reciprocal. Factor. Multipl numerators and denominators. Divide out the common factor. Simplif. Find the quotient. Eercises 0. t t t t. (g + ) g + g g. d + d (d + d ) Section. Multipling and Dividing Rational Epressions 7

33 . Eercises Help with Homework. WRITING Describe how to multipl rational epressions.. WRITING Describe how to divide rational epressions.. NUMBER SENSE Consider the epressions and +. For what value(s) is the product of the epressions undefined? For what value(s) is the quotient of the epressions undefined? 9+(-)= +(-)= +(-9)= 9+(-)= Find the product.. c c (c ). n + 8n n 7. (d d) d k 8k + k k k 9. r r + 8r r + r r Find the quotient. 0. h h + 8. t t h + 8 9t t p p (p ). g g g + g + (g 7). z 7 z (z z + ) ERROR ANALYSIS Describe and correct the error in finding the quotient.. 7. w w + w w + = w w w + v v w + v v = v v v v w = w v(v ) = (w + )(w + ) v (v ) = 8w (w + ) = v Find the total area of the red rectangle in terms of w. 8. w 9. w w w 7 Chapter Rational Equations and Functions

34 Find the product or quotient. 0. b b b b 7 b b 8 b b REASONING What are the ecluded values of + + +? d d d. CAMPSITE A campsite is in the shape of a rectangle. The green region represents campsites with shade. The ellow represents campsites without shade. Your campsite is randoml assigned. What is the probabilit that our campsite has shade?. TECHNOLOGY You can use a graphing calculator to check our answers when multipling or dividing rational epressions. For instance, graph = + and = 0 from Eample a in the same + viewing window. a. What do ou notice about the graphs? b. How can ou use the table feature to find the ecluded values?. CHARITY The revenue R (in thousands of dollars) and the average ticket price P (in dollars) for a charit event 0 can be modeled b R = 0.0 and P = 0.0 +, where is the number of ears since 000. (Note: revenue = tickets sold ticket price) a. Write an equation that models the number T of tickets sold as a function of. b. In what ear will this model become invalid? Eplain our reasoning.. Write 8 + in simplest form. + Graph the function. Describe the domain and range. (Section 8.) 7. = 8. = + 9. = MULTIPLE CHOICE What is the distance between (, ) and (, )? (Section 0.) A B C D Section. Multipling and Dividing Rational Epressions 7

35 . Dividing Polnomials another polnomial? How can ou divide one polnomial b ACTIVITY: Dividing Polnomials Work with a partner. Si different algebra tiles are shown below. Sample: Step : Arrange tiles to model ( + + ) ( + ) in a rectangular pattern. Step : Complete the pattern. Divisor Quotient Dividend Step : Use the completed pattern to write ( + + ) ( + ) = +. Dividend Divisor = Quotient COMMON CORE Rational Epressions In this lesson, ou will divide polnomials b monomials. divide polnomials b binomials. Learning Standard A.SSE. Complete the pattern and write the division problem. a. b. 7 Chapter Rational Equations and Functions

36 Math Practice Make Sense of Quantities What do the algebra tiles represent? How can ou use the tiles to divide one polnomial b another polnomial? ACTIVITY: Dividing Polnomials Work with a partner. Write two different polnomial division problems that can be associated with the given algebra tile pattern. Check our answers b multipling. a. b. c. d. ACTIVITY: Dividing Polnomials Work with a partner. Solve each polnomial division problem. a. ( 8 ) ( ) b. (8 ) ( ). IN YOUR OWN WORDS How can ou divide one polnomial b another polnomial? Include the following in our answer. a. ( + 0 7) ( + 7) b. ( ) ( ) Use what ou learned about dividing polnomials to complete Eercises and on page 78. Section. Dividing Polnomials 7

37 . Lesson Lesson Tutorials To divide a polnomial b a monomial, divide each term of the polnomial b the monomial. EXAMPLE Dividing a Polnomial b a Monomial Find ( + ). ( + ) = + = + = + = + Write as a fraction. Divide each term b. Divide out the common factors. Simplif. Eercises and 7 Find the quotient.. (z 8z) z. (n n + 8) n. ( + 9) You can use long division to divide a polnomial b a binomial. EXAMPLE Dividing a Polnomial b a Binomial: No Remainder Find (m + m + ) (m + ). Step : Divide the first term of the dividend b the first term of the divisor. Stud Tip There is no remainder in Eample, so ou could have factored the dividend and divided out a common factor. m + m + m + (m + )(m + ) = m + = m + Align like terms in the quotient and dividend. m Divide: m m = m. m + ) m + m + m + m Multipl: m(m + ). m + Subtract. Bring down the. Step : Divide the first term of m + b the first term of the divisor. m + Divide: m m =. m + ) m + m + m + m m + m + Multipl: (m + ). 0 Subtract. So, (m + m + ) (m + ) = m +. 7 Chapter Rational Equations and Functions

38 When ou use long division to divide polnomials and ou obtain a nonzero remainder, use the following rule. Remainder Divisor Dividend Divisor = Quotient + EXAMPLE Dividing a Polnomial b a Binomial: Remainder Find ( 7 + ) ( ). Write the dividend in standard form. 7 + ) Multipl: ( ). + + Subtract. Bring down the. Multipl: ( ). 0 Subtract. 0 So, ( 7 + ) ( ) =. Find the quotient.. (s s 8) (s 7) Eercises 8 EXAMPLE. ( + ) ( + ) Inserting a Missing Term Find (q 8) (q ). Stud Tip When dividing polnomials using long division, first write the polnomials in standard form and insert an missing terms. ) Include a q-term with a coefficient of 0. q + q + 0q 8 q q q Multipl: q(q ). q 8 q Subtract. Bring down the 8. Multipl: (q ). Subtract. q So, (q 8) (q ) = q + +. Find the quotient. Eercises 9 MSCC_ALG_PE_0.indd 77. (z + ) (z + 9) 7. (9 ) ( + ) Section. Dividing Polnomials 77 /7/ 8:: AM

39 . Eercises Help with Homework. WRITING How do ou divide a polnomial b a monomial? b a binomial?. REASONING How can ou check our answer when dividing polnomials?. NUMBER SENSE How do ou know whether a binomial is a factor of a polnomial? )= 9+(- )= +(- 9)= +(- = ) 9+(- Use algebra tiles to find the quotient.. ( + 8) ( + ). ( ) ( + ) Find the quotient.. (8c + c 7) 8c 7. (n n + ) n 8. (m m ) (m + ) 9. (z + 0z + ) (z + ) 0. ( + 8) ( ). (h + h ) ( + h). ( a + a ) (a + ). ( + 8k 9k) (k ). ( + + 7) ( + ). (g 7 + g ) (g ) ERROR ANALYSIS Describe and correct the error in finding the quotient ) + + ( + ) ( + ) = ) + ( ) ( ) = + 8. AMUSEMENT PARK The cost of a field trip to an amusement park is represented b + 00, where is the number of students going on the trip. The cost is shared equall b all the students ecept for three students whose parents are acting as chaperones. Find ( + 00) ( ) to find an epression for how much each student pas. 78 Chapter MSCC_ALG_PE_0.indd 78 Rational Equations and Functions /7/ 8:: AM

40 Find the quotient. 9. (d 9) (d + ) 0. (r + 0) (r + ). (8n + ) (n ). (0 9) ( ). ERROR ANALYSIS Describe and correct the error in finding the quotient.. REASONING Find k when ( ) is a factor of + k.. CRITICAL THINKING When dividing polnomials, how are the degrees of the dividend, divisor, and quotient related? 9 + ) ( ) ( + ) = TECHNOLOGY Rewrite the rational function = 8 in the a form = + k. Graph both functions in the same viewing h window of a graphing calculator. a. What do ou notice about the graphs? b. What are the asmptotes of the graph of = 8? 7. GEOMETRY The volume of the rectangular prism is m m. Write an epression for the width of the prism. m m 8. CHOOSE TOOLS Would ou use factoring or long division to simplif 8? Eplain our reasoning. 9. Repeated Reasoning Find each quotient in the table and identif the pattern. Then predict the quotient ( + + ) ( + ) without calculating. Verif our prediction. Quotient ( + ) ( + ) ( + ) ( + ) ( + + ) ( + ) Solve the equation b completing the square. (Section 9.) 0. = = 0. 8 = 0. MULTIPLE CHOICE What is the solution of = +? (Section.) A B C D Section. Dividing Polnomials 79

41 7.. Adding and Subtracting Factoring Polnomials Using the GCF Rational Epressions epressions? How can ou add and subtract rational ACTIVITY: Adding Rational Epressions Work with a partner. You and a friend have a summer job mowing lawns. Working alone it takes ou 0 hours to mow all of the lawns. Working alone it takes our friend 0 hours to mow all of the lawns. a. Write a rational epression that represents the portion of the lawns ou can mow in t hours. Portion ou mow in t hours = Time Rate b. Write a rational epression that represents the portion of the lawns our friend can mow in t hours. Portion our friend mows in t hours = Time Rate COMMON CORE Rational Epressions In this lesson, ou will add and subtract rational epressions. find least common denominators of two rational epressions. Learning Standard A.SSE. c. Add the two epressions to write a rational epression for the portion of the lawns that the two of ou working together can mow in t hours. Time + = Rate Time Rate d. Use the epression in part (c) to find the total time it takes both of ou working together to mow all of the lawns. Eplain our reasoning. 80 Chapter Rational Equations and Functions

42 ACTIVITY: Adding Rational Epressions Math Practice Appl Mathematics How do the units of measure in a problem help ou choose a formula? How does the formula help ou write an epression? Work with a partner. r. You are hang gliding. For the first 0,000 feet, ou travel feet per minute. You then enter a valle in which the wind is greater, and for the net 000 feet, ou travel feet per minute. a. Use the formula d = rt to write a rational epression that represents the time it takes ou to travel the first 0,000 feet. Time to travel first 0,000 feet = Distance Rate b. Use the formula d = rt to write a rational epression that represents the time it takes ou to travel the net 000 feet. Time to travel net 000 feet = Distance Rate c. Add the two epressions to write a rational epression that represents the total time it takes ou to travel,000 feet. + = d. Use the epression in part (c) to find the total time it takes ou to travel,000 feet when our rate during the first 0,000 feet is 000 feet per minute.. IN YOUR OWN WORDS How can ou add and subtract rational epressions? Include the following in our answer. a. d. + 0 b. e. + + c. f. 9 + Use what ou learned about adding and subtracting rational epressions to complete Eercises on page 8. Section. Adding and Subtracting Rational Epressions 8

43 . Lesson Lesson Tutorials Ke Vocabular least common denominator of rational epressions, p. 8 You can use the same rules that ou used for adding and subtracting fractions to add and subtract rational epressions. Adding and Subtracting Rational Epressions with Like Denominators Let a, b, and c be polnomials, where c 0. Adding: a c + b c = a + b a Subtracting: c c b c = a b c EXAMPLE Adding and Subtracting with Like Denominators Find the sum or difference. a. + 7 = + 7 = = = Add the numerators. Simplif. Divide out the common factor. Simplif. Common Error When subtracting rational epressions, remember to distribute the negative to each term of the numerator of the epression being subtracted. b = ( 8) + = = ( + ) = + = Simplif. Subtract the numerators. Use the Distributive Propert. Combine like terms. Factor. Divide out the common factor. Eercises Find the sum or difference.. 9z 8 9z. w + w + w w Chapter Rational Equations and Functions

44 To add or subtract rational epressions with unlike denominators, rewrite the epressions so the have like denominators. You can do this b finding the least common multiple of the denominators, called the least common denominator (LCD). EXAMPLE Finding the LCD of Two Rational Epressions Find the LCD of 0g and g. First write the prime factorization of each denominator. 0g = g g = g Use the greatest power of each factor that appears in either denominator to find the LCM of the denominators. LCM = g = 0g So, the LCD of 0g and g is 0g. Eercises 8 Find the LCD of the rational epressions.. 7g, g. 8 n, n n +. t t, 9 t 7. +, ( ) Stud Tip EXAMPLE To rewrite each epression using the LCD, multipl the numerator and denominator of each epression b the factor that makes its denominator the LCD. Adding with Unlike Denominators Find the sum 8 +. Because the epressions have unlike denominators, find the LCD. 8 = = The LCD is =. 8 + = () 8() + ( )() () = = = 7 8 Rewrite using the LCD,. Simplif. Add the numerators. Simplif. Section. Adding and Subtracting Rational Epressions 8

45 EXAMPLE Subtracting with Unlike Denominators Find the difference ( )( ) Factor ( )( ) ( ) ( )( ) Rewrite using the LCD, ( )( ). = = ( + ) ( ) ( )( ) Subtract the numerators. + 7 ( )( ) Simplif. = = EXAMPLE Real-Life Application You row our kaak miles downstream from our campsite to a dam, and then ou row back to our campsite. You row miles per hour during the entire trip, and the river current is mile per hour. Write an epression for the total time of the trip. d r ( ) mi/h ( ) mi/h Solving the formula d = rt for time t gives t =. Use this to write an epression for the total time of the trip. Time downstream Time upstream Distance Speed + downstream downstream + ( ) ( + )( ) Distance upstream ( + ) ( )( + ) Speed upstream Write an epression. Rewrite using the LCD. + = + ( ) + ( + ) ( + )( ) Add the numerators. 0 ( + )( ) Simplif. = = Find the sum or difference. Eercises k k+ k k WHAT IF? In Eample, the river current is miles per hour. Write an epression for the total time of the trip. 8 Chapter MSCC_ALG_PE_0.indd 8 Rational Equations and Functions /7/ 8:: AM

46 . Eercises Help with Homework. WRITING Eplain how finding the least common denominator of two rational epressions is similar to finding the least common denominator of two numeric fractions.. REASONING Describe how to rewrite the epressions + and so that the have the same denominator. 9+(-)= +(-)= +(-9)= 9+(-)= Find the sum or difference.. s 9 + s 9. r 8 r. w + w z (z ) + z (z ) 9. t t t + t 0. n + n n + n n n. p p p + p p +. ERROR ANALYSIS Describe and correct the error in adding the rational epressions. + = Find the LCD of the rational epressions.. 9, 7., 8. g, g 7. h h +, h + h. m m +, 9 m s 7 8. s s 8, s 9. CEREAL The height of a cereal bo is given S b ( + w) w, where S is the surface ( + w) area, is the length, and w is the width. Find the difference. Section. Adding and Subtracting Rational Epressions 8

47 Find the sum or difference c c +. m m m p + p 7p + p. ERROR ANALYSIS Describe and correct the error in adding the rational epressions. + + = ( ) + ( + ) ( )( + ) = + + ( )( + ) = + + ( )( + ) 7. REASONING Can ou find a common denominator of two rational epressions b finding the product of the denominators? Is this product alwas going to be the least common denominator? Justif our answers. 8. RUNNING You run miles up a hill and miles down the hill. You run % faster going down the hill than going up the hill. Let r be our speed (in miles per hour) while running up the hill. Write an epression that represents the amount of time ou spend running on the hill. 9. OPEN-ENDED Write two rational epressions with unlike denominators. a. Find the least common denominator of the two epressions. b. Add the two epressions. Write an epression for the perimeter of the figure. 0. p p. p p 0 p p 0. p p p p p p 0 p p 8 Chapter Rational Equations and Functions

48 Simplif the epression.. c + c + c + c c + c c d 8. d + d 0 + d d + d HOMEWORK You have 0 more math eercises for homework than biolog eercises. You finish eercises before dinner and 8 eercises after dinner. Write an epression that represents the portion of eercises that are complete. 8. WAKEBOARDING You are wakeboarding on a river. You travel miles downstream to a marina for supplies, and then ou travel miles upstream to a dock. The boat travels miles per hour during the entire trip, and the river current is miles per hour. a. Write an epression that represents the total time of the trip. b. How long will the trip take when the speed of the boat is 8 miles per hour? 9. Logic Let a, b, c, and d be polnomials. Find two rational epressions a b and c d so that a b c d = + ( + )( + ). Graph the sstem of linear inequalities. (Section.) 0. > +.. < + < 7 +. MULTIPLE CHOICE The graph of which function is shown at the right? (Section 0.) A = B = C = D = 7 7 Section. Adding and Subtracting Rational Epressions 87

49 .7 Solving Rational Equations How can ou solve a rational equation? ACTIVITY: Solving Rational Equations Work with a partner. A hocke goalie faces 799 shots and saves 707 of them. a. What is his save percentage? Save Percentage = Shots saved Shots faced National Hocke League goalies tpicall have a save percentage above.900. b. Suppose the goalie has additional consecutive saves. Write an epression for his new save percentage. Save Percentage = 707 plus additional saves 799 plus additional shots faced COMMON CORE Rational Functions In this lesson, ou will solve rational equations using cross products. solve rational equations using least common denominators. solve real-life problems. Appling Standard A.CED. c. Complete the table showing the goalie s save percentage as increases. Additional Saves, Save Percentage d. The goalie wants to end the season with a save percentage of.900. How man additional consecutive saves must he have to achieve this? Justif our answer b solving an equation. 88 Chapter Rational Equations and Functions

50 ACTIVITY: Solving Rational Equations Math Practice Find General Methods What method did ou use to complete the table? How can ou use this information to write and solve an equation in part (d)? Work with a partner. A baseball plaer has been at bat 7 times and has 8 hits. a. What is his batting average? Batting Average = b. Suppose the plaer has additional consecutive hits. Write an epression for his new batting average. Batting Average = The league batting average in Major League Baseball is usuall between.0 and.70. c. Complete the table showing the plaer s batting average as increases. Additional Hits, 0 7 Batting Average d. The plaer wants to end the season with a batting average of.0. How man additional consecutive hits must he have to achieve this? Justif our answer b solving an equation.. IN YOUR OWN WORDS How can ou solve a rational equation? Include the following in our answer. a. = b. + = c. + = Use what ou learned about solving rational equations to complete Eercise on page 9. Section.7 Solving Rational Equations 89

51 .7 Lesson Lesson Tutorials Ke Vocabular rational equation, p. 90 A rational equation is an equation that contains rational epressions. One wa to solve rational equations is to use the Cross Products Propert. You can use this method when each side of a rational equation consists of one rational epression. EXAMPLE Solving Rational Equations Using Cross Products Solve each equation. Check + = + =? 8 = 8 a. b. + = = 7 + = Write the equation. ( ) = ( + ) Cross Products Propert 0 = + = + = = 7 Distributive Propert Add 0 to each side. Subtract from each side. Write the equation. (7) = ( ) Cross Products Propert = Simplif. 0 = Subtract from each side. 0 = ( 7)( + ) Factor. 7 = 0 or + = 0 Zero-Product Propert = 7 or = Solve for. Check 7 =? 7 7 Substitute for. =? 7 7 = 7 Simplif. = Eercises 0 Solve the equation. Check our solution(s).. =. 7 z + = z z +. = Chapter Rational Equations and Functions

52 When there is more than one rational epression on one or both sides of a rational equation, multipl each side b the LCD and then solve. EXAMPLE Solving a Rational Equation Using the LCD z Solve z = z. (z ) ( z z z (z ) (z ) z z z + = ) = (z ) = (z ) z z Multipl each side b the LCD, (z ). Multipl. Then divide out common factors. Simplif. z = Solve for z. Because each side of the equation is undefined when z =, it is an etraneous solution. The equation has no solution. EXAMPLE Real-Life Application Your starter deck for a collectible card game has 0 cards. The deck contains 7 creature cards. You add creature cards to the deck until it contains 0% creature cards. How man do ou add? Write an equation for the ratio of creature cards to total cards after adding creature cards. Fling Whenever Pro-Dragon deals combat damage to a plaer, ou ma have it deal that much damage to target creature that plaer controls. Creature cards Total cards = 0. 0.( + 0) = + 7 Cross Products Propert 0. + = + 7 Distributive Propert 8 = 0. Simplif. Desired percent of creature cards = Divide each side b 0.. You add creature cards to the deck. Eercises 8 Solve the equation. Check our solution(s).. p = 7. p n + n + = n +. a + = 9 a 7. WHAT IF? In Eample, ou add creature cards until the deck contains 0% creature cards. How man do ou add? Section.7 Solving Rational Equations 9

53 .7 Eercises Help with Homework. VOCABULARY Describe two methods for solving rational equations.. OPEN-ENDED Write a rational equation that can be solved b multipling each side b ( + ).. WRITING Wh should ou check the solutions of a rational equation? 9+(-)= +(-)= +(-9)= 9+(-)=. A basketball plaer attempts free throws and makes 0 of them. a. What is her free throw percentage? b. Suppose the plaer makes additional consecutive free throws. Write an epression for her new free throw percentage. c. The plaer wants to end the season with a free throw percentage of.800. How man additional consecutive free throws must she make to achieve this? Solve the equation. Check our solution(s).. b =. b + = + 8. z z = 9. k 8 z + 9 k + = k 7. m = m 0. w w + = w w + = + ( + ) = ( + ) + = + = 0 ( )( + ) = 0 = or = So, the solutions are = and =.. ERROR ANALYSIS Describe and correct the error in solving the equation.. WATER RESCUE The table shows information about a water rescue team. a. Solve the rational equation = 7 + to find the upstream speed of the rescue team. b. What is the downstream speed of the rescue team? Water Rescue Direction Distance Rate Time Upstream miles mi/h t hours Downstream 7 miles ( + ) mi/h t hours 9 Chapter Rational Equations and Functions

54 Solve the equation. Check our solution(s).. c = c. 0 d(d ) + d = d 7. a + + = 8 a. + = +. n n + = n = REASONING Eplain how ou can use the Cross Products Propert to solve + = PAINT A department store paint mier contains pints of equal amounts of ellow and red paint. The shade of red that ou want requires a paint miture that is 7% red and % ellow. How man pints of red paint need to be added to the paint mier?. RAPPELLING A rappelling club charters a bus for a trip to the mountains for $0. To lower the bus fare per person, the club invites some hikers on the trip. After 7 hikers join the trip, the bus fare per person decreases b $7. How man members of the rappelling club are going on the trip? To solve work problems, find the portion of the job each person completes in unit of time. The sum of these portions is the portion of the job completed in unit of time.. You can mop a floor in 8 minutes. Your friend can mop the same floor in minutes. Working together, how much time does it take to mop the floor?. You can mow a lawn in hours. Your friend can mow the same lawn in hours. Working together, how much time does it take to mow the lawn?. A roofing contractor can shingle a roof in half the time it takes his assistant. Working together, the can shingle the roof in 8 hours. How much time does it take the roofing contractor to finish the job alone? Solve the equation. Check our solutions. (Section.). =. 7 = = 8. MULTIPLE CHOICE What is the solution of < + 8? (Section.) A 7 < B 7 > C and > 7 D < 7 or Section.7 Solving Rational Equations 9

55 ..7 Quiz Progress Check Find the product or quotient. (Section.). c + c c. k k + k +. ab b a + a a + a ab b. m m m + m + m Find the quotient. (Section.). (j + j + 8j ) j. (m m + 9) (m 7) 7. (d d + 8) (d ) 8. (n + 7) (n ) Find the sum or difference. (Section.) 9. v + 0 v +. t 8 t + t + 7 t 0. r r + r r. p + 0 p + p 0 p Solve the equation. Check our solution. (Section.7). s = s h. + h + = 7h h +. = w +. g + g(g + ) = g + 7. PIGPEN You are installing a fence around a pigpen. Write an epression that represents the amount of fencing ou need. (Section.) f f f 8. RAKING You can rake our front ard in 0 minutes. Your friend can rake the same ard in 0 minutes. Working together, how much time does it take to rake the ard? (Section.7) 9 Chapter Rational Equations and Functions

56 Chapter Review Review Ke Vocabular direct variation, p. inverse variation, p. rational function, p. ecluded value, p. asmptote, p. inverse relation, p. 8 inverse function, p. 9 rational epression, p. Vocabular Help simplest form of a rational epression, p. least common denominator of rational epressions, p. 8 rational equation, p. 90 Review Eamples and Eercises. Direct and Inverse Variation (pp. 9) The variable varies inversel with. When =, =. a. Write an inverse variation equation that relates and. Find the value of k. = k = k Write the inverse variation equation. Substitute for and for. = k Multipl each side b. So, an equation that relates and is =. b. Graph the inverse variation equation. Describe the domain and range. Make a table of values. 0 undef. Plot the ordered pairs. Draw a smooth curve through the points in each quadrant. Both the domain and range are all real numbers ecept The variable varies directl with. When =, =. Write and graph a direct variation equation that relates and.. The variable varies inversel with. When =, = 8. Write and graph an inverse variation equation that relates and. Chapter Review 9

57 . Graphing Rational Functions (pp. 0 9) Graph =. Compare the graph to the graph of =. Step : Make a table of values. The vertical asmptote is =, so choose -values on either side of undef Step : Use dashed lines to graph the asmptotes = and =. Then plot the ordered pairs. Step : Draw a smooth curve through the points on each side of the vertical asmptote. The graph of = is a translation unit down and units right of the graph of =. Find the inverse of f () =. Graph the inverse function. = = Replace f () with. Switch and. = Multipl each side b. = Divide each side b. f () = Replace with f (). Graph the function. Compare the graph to the graph of =.. =. = +. = 7 + Find the inverse of the function. Graph the inverse function.. f () = + 7. f () = 8. f () = Chapter Rational Equations and Functions

58 . Simplifing Rational Epressions (pp. 0 ) Simplif v 9, if possible. State the ecluded value(s). v v v 9 v v = (v )(v + ) v(v ) (v )(v + ) = v(v ) = v + v Factor. Divide out the common factor. Simplif. The ecluded values are v = 0 and v =. Simplif the rational epression, if possible. State the ecluded value(s). 9. 8z z 0. n +. b + 9b + 8 n b + b 0. Multipling and Dividing Rational Epressions (pp. 8 7) Find the product or quotient. 7 a. + + = 7( + ) ( + ) 7( + ) = ( + ) = 7 Multipl numerators and denominators. Divide out the common factors. Simplif. b. t 0 t = t 0 t = t 0 (t ) (t ) = 0(t ) (t ) = 0(t ) = Multipl b the reciprocal. Rewrite t as (t ). Multipl numerators and denominators. Divide out the common factors. Simplif. Chapter Review 97

59 Find the product or quotient.. 0r 9 r. k + k + k k. h + 8h (h + 7h 8) h. Dividing Polnomials (pp. 7 79) Find ( ). ( ) = = = = + + Write as a fraction. Divide each term b. Divide out the common factors. Simplif. Find (z z ) (z + ). Step : Divide the first term of the dividend b the first term of the divisor. Align like terms in the quotient and dividend. z Divide: z z = z. z + z z z + z Multipl: z(z + ). z Subtract. Bring down the. Step : Divide the first term of z b the first term of the divisor. z Divide: z z =. z + z z z + z z z Multipl: (z + ). 0 Subtract. So, (z 0 z ) (z + ) = z + z +. Find the quotient.. (8n + n) n. (b ) (b ) 7. ( + + ) ( + ) 8. (c ) (c + ) 98 Chapter Rational Equations and Functions

60 . Adding and Subtracting Rational Epressions (pp ) Find the difference = ( + )() 7 () (7) (7) + = 8 + = Rewrite using the LCD,. Simplif. Subtract the numerators. Find the sum or difference. 9. h h 0 h h Solving Rational Equations (pp. 88 9) You own a farm in a computer game. Twent of the 0 animals on our farm are cows. You bu cows and increase the ratio of cows to total animals to :. How man cows do ou bu? Write an equation for the ratio of cows to total animals after buing cows. Cows Total animals = New ratio of cows to total animals (0 + ) = 0 + Cross Products Propert 00 + = 0 + Distributive Propert = 0 Simplif. = Divide each side b. You bu cows. Solve the equation. Check our solution(s).. + = = +. t = t. TENNIS A tennis plaer lands out of 0 first serves in bounds for a success rate of.%. How man more consecutive first serves must she land in bounds to increase her success rate to 70%? Chapter Review 99

61 Chapter Test Test Practice. The variable varies directl with. When =, = 8. Write and graph a direct variation equation that relates and.. The variable varies inversel with. When =, =. Write and graph an inverse variation equation that relates and. Graph the function. Compare the graph to the graph of =.. =. = +. = + Find the inverse of the function. Graph the inverse function.. f () = 7 7. f () = 7 8. f () = + Simplif k + k + k + k 0. 8r r r. + ( 0). p + p. 8z + 7 z + z + z z + 9. Find (d + 8d ) d.. Find (b b + 0) (b + ). Solve the equation. Check our solution(s). 7. = 8. a + 7 a + = a + 9. n n = n 0. BALANCE To balance the board in the diagram, the distance (in feet) of each object from the center of the board must var inversel with its weight (in pounds). What is the distance of the suitcase from the fulcrum? fulcrum 0 lb 80 lb 0 ft d ft Wind: 0 mph Denver 000 miles Speed of airplane in still air: r miles per hour Indianapolis. AIRPLANE An airplane makes a round trip between two cities. The airplane flies with the wind when heading east and against the wind when heading west. Write an epression for the total time of the trip.. DELIVERY TRUCK Working alone, it takes ou 0 minutes, our friend 0 minutes, and our supervisor minutes to unload a deliver truck. Working together, how much time does it take all three of ou to unload the truck? 00 Chapter Rational Equations and Functions

62 Standards Assessment. Which function is shown b the graph? (A.REI.0) Test-Taking Strateg Work Backwards A. = + B. = + + C. = + + D. = Work backwards b tring,,, and. You will see that gives ou a whole treat.. What is the value of c in the triangle shown? (8.G.7) 0 m c m F. m H. 9 m G. m I. 0 m. What is the -coordinate of the focus of the graph of = 8? (F.IF.). What is the simplest form of the rational epression? (A.SSE.) A. C. B. D. Standards Assessment 0