Laurie s Notes Overview of Section 7. Introduction Students have developed a repertoire of equation-solving techniques throughout their study of algebra. In this lesson, additional techniques are presented. The ethod of cross products should be failiar to students fro solving proportions. What is different in this contet is the copleity of the proportion. When students solved proportions, they also used the strategy of ultiplying each side of the equation by the LCD. This strategy is also applied to rational equations. At the end of the lesson, students return to finding the inverse of a function. The inverse is then used in a contetual proble as yet another ethod of solving a rational equation. Dynaic Teaching Tools Dynaic Assessent & Progress Monitoring Tool Lesson Planning Tool Interactive Whiteboard Lesson Library Dynaic Classroo with Dynaic Investigations Teaching Strategy Students have solved equations graphically using a calculator. This approach can be used for solving rational equations, although students soeties have difficulty understanding what they are looking at. It is helpful when the two equations are displayed in different colors or thicknesses. Eaple can be solved using a graphing calculator. When entering the functions in the equation editor, students need to be sure that the correct order of operations is perfored. Using ore parentheses than ight be needed is safer than not using enough. In the standard viewing window, one solution appears. The graph also suggests that there is an additional solution, eaning the two graphs intersect around =. Students could spend tie zooing out looking for the second point of intersection. Looking at the table of values reveals that neither function is defined at =, and that = is not in the doain of one of the functions. 0 Y=/(X-) Y=X /(X -9)-X/(X+) Y= Y= Y= Y= Y7= 0 0 0 X Y Y - - - 0 - -. -. - - - ERROR ERROR. 0 - - ERROR X=- Pacing Suggestion Once students have copleted the eplorations, continue with the foral lesson. Section 7. T-90
What Your Students Will Learn Solve rational equations by cross ultiplying when each side of the equation is a single rational epression. Solve rational equations by using the least coon denoinator when the rational equation is not epressed as a proportion. Find inverses of rational functions. Laurie s Notes Eploration Motivate How any of you have wondered what you needed to score on a particular test in order to obtain a particular average? Answers will vary. Eplain that this type of proble can be answered by writing and solving a rational equation. Discuss Before students begin the eplorations, you ay wish to discuss ethods for solving systes of equations. One ethod is graphing. In the first eploration, students are shown the graphs of two functions, at least one of which is a rational function. Eploration Students have done atching activities like this before. That eperience, coupled with their knowledge of constant, quadratic, and rational functions, should ake this a fairly quick eploration. All of the solutions are integers. Matching the equations and graphs should not take long for students. What will take a bit longer is deciding where the two equations (syste) intersect. Students can find these graphs to be essy or confusing. They ust distinguish the rational functions fro the second function and fro the - and y-aes. The solutions are not the ordered pairs where the graphs intersect. The solutions to the equations are the -values of the points of intersection. Solving the equations graphically eans you are looking for the -values where the two function values are equal. How can you check the solution? Substitute the -values and see whether both sides of the equation are equal. Eploration Make Sense of Probles and Persevere in Solving The: Give tie for students to discuss alternate approaches for solving the equations in Eplorations (d) and (e). Can students describe an analytical approach, such as using the Cross Products Property and solving for? Reind students of work they did in earlier chapters when they solved equations. Discuss approaches suggested by students. Counicate Your Answer The equations in Question lend theselves to using cross products, or perhaps students will think of trying to rewrite each side of the equation with a like denoinator. Connecting to Net Step In the foral lesson, students will use an analytic approach to solve rational equations. Having a sense of how to solve the equations graphically should be helpful to students. T-9 Chapter 7
MAKING SENSE OF PROBLEMS 7. To be proficient in ath, you need to plan a solution pathway rather than siply juping into a solution attept.. b. Solving Rational Equations Essential Question How can you solve a rational equation? Solving Rational Equations Work with a partner. Match each equation with the graph of its related syste of equations. Eplain your reasoning. Then use the graph to solve the equation. a. = b. d. = e. = A. C. E. = = ( ) 0 = 0 = ( )( + ) = and = = = ( ) = = = c. = + B. D. F. Solving Rational Equations f. = Work with a partner. Look back at the equations in Eplorations (d) and (e). Suppose you want a ore accurate way to solve the equations than using a graphical approach. a. Show how you could use a nuerical approach by creating a table. For instance, you ight use a spreadsheet to solve the equations. b. Show how you could use an analytical approach. For instance, you ight use the ethod you used to solve proportions. Counicate Your Answer. How can you solve a rational equation?. Use the ethod in either Eploration or to solve each equation. a. + = + b. + = + Section 7. Solving Rational Equations 9 c. =. To solve a rational equation when each side of the equation is a single rational epression, use cross ultiplication. To solve any rational equation, ultiply each side of the equation by the least coon denoinator of the rational epression.. a. = 0 b. = 0, = c. = 0 Dynaic Teaching Tools Dynaic Assessent & Progress Monitoring Tool Lesson Planning Tool Interactive Whiteboard Lesson Library Dynaic Classroo with Dynaic Investigations. a. F; The graphed equations are y = and y = ; = b. D; The graphed equations are y = and y = ; = c. A; The graphed equations are y = and y = + ; = and = d. C; The graphed equations are y = and y = ; = and = e. B; The graphed equations are y = and y = ; = f. E; The graphed equations are y = and y = ; =. a. 0.7 0 0 undef. When = and =, the epressions are equal to each other. 0. 0. 0. 0 undef. 0. 0. undef. 0. When =, the epressions are equal to each other. Section 7. 9
English Language Learners Word Probles Eaple has any words that ay not be failiar to English learners. Read the proble aloud so that students can becoe failiar with the words. Then, reread the proble, one sentence at a tie, using siplified language. Etra Eaple Solve + = 0 +. The solution is. Etra Eaple White gold is an alloy coposed of 7% gold and % palladiu by weight. You have ounces of an alloy that is 0% gold and 0% palladiu. How uch pure gold should you i with this alloy to ake white gold? You should i 7. ounces of pure gold with the ounces of alloy. MONITORING PROGRESS. =. =. =, = 7. Lesson What You Will Learn Core Vocabulary cross ultiplying, p. 9 Previous proportion etraneous solution inverse of a function Check + =? 9 ( ) + =? 9 = Solve rational equations by cross ultiplying. Solve rational equations by using the least coon denoinator. Use inverses of functions. Solving by Cross Multiplying You can use cross ultiplying to solve a rational equation when each side of the equation is a single rational epression. Solve + = 9 +. Solving a Rational Equation by Cross Multiplying + = 9 + Write original equation. ( + ) = 9( + ) Cross ultiply. + = 9 + 9 + = 9 = Distributive Property Subtract 9 fro each side. Subtract fro each side. = Divide each side by. The solution is =. Check this in the original equation. Writing and Using a Rational Model An alloy is fored by iing two or ore etals. Sterling silver is an alloy coposed of 9.% silver and 7.% copper by weight. You have ounces of 00 grade silver, which is 0% silver and 0% copper by weight. How uch pure silver should you i with the 00 grade silver to ake sterling silver? percent of copper in iture = weight of copper in iture total weight of iture 7. 00 = (0.)() + is the aount of silver added. 7.( + ) = 00(0.)() Cross ultiply.. + 7. = 00 Siplify. 7. = 7. Subtract. fro each side. = Divide each side by 7.. 9 Chapter 7 Rational Functions You should i ounces of pure silver with the ounces of 00 grade silver. Monitoring Progress Help in English and Spanish at BigIdeasMath.co Solve the equation by cross ultiplying. Check your solution(s).. =. 7 + =. + = + Laurie s Notes Teacher Actions COMMON ERROR Although the Cross Products Property is failiar to students, they often forget the need to use the Distributive Property when binoials are involved. Make Sense of Probles and Persevere in Solving The: In Eaple, the second sentence gives a relationship about weight. Why do you need to add silver versus copper? The 00 grade silver is already 0% copper by weight, and sterling silver is only 7.% copper by weight. Help set up the equation. Let partners work to solve. 9 Chapter 7
Check + 7 =? 9 + =? 9 9 = 9 Solving by Using the Least Coon Denoinator When a rational equation is not epressed as a proportion, you can solve it by ultiplying each side of the equation by the least coon denoinator of the rational epressions. Solve each equation. a. + 7 = 9 a. + 7 = 9 ( + 7 Solving Rational Equations by Using the LCD b. = Write original equation. ) ( = 9 ) Multiply each side by the LCD,. 0 + 7 = Siplify. 7 = Subtract 0 fro each side. = Divide each side by 7. The solution is =. Check this in the original equation. b. = Write original equation. ( ) ( ) = ( ) ( ) = ( ) = + = 0 Multiply each side by the LCD, ( ). Siplify. ( )( ) = 0 Factor. Distributive Property Write in standard for. = or = Zero-Product Property The solutions are = and =. Check these in the original equation. Differentiated Instruction Organization If students struggle to follow their own work, provide gridded paper to help the organize the rational equations. You ay want to suggest that they write one nuber or sybol per bo and that they vertically align the equal signs. Etra Eaple Solve each equation. a. + = The solution is. b. = The solutions are and. MONITORING PROGRESS. = 0. =, =. =, = Check =? Substitute for. =? + =? Siplify. =? = = Monitoring Progress Help in English and Spanish at BigIdeasMath.co Solve the equation by using the LCD. Check your solution(s).. + = 7. + =. + + = + Section 7. Solving Rational Equations 9 Laurie s Notes Teacher Actions COMMON ERROR In part (a), when ultiplying the left side by the LCD, students do not always recognize the Distributive Property. Opposing Views: Ask the probing question, Could part (b) be solved using cross products? Eplain. Give students tie to consider their answer. Soe students will say yes, soe will say no, and soe will be uncertain. Ask a few students fro each viewpoint to share their thinking. The class should decide to write the left side as and then use cross products. Section 7. 9
Etra Eaple Solve 9 = + 9 + +. The apparent solution = is etraneous. So, the only solution is =. MONITORING PROGRESS 7. =. =, = When solving a rational equation, you ay obtain solutions that are etraneous. Be sure to check for etraneous solutions by checking your solutions in the original equation. Solve = 9 +. Solving an Equation with an Etraneous Solution Write each denoinator in factored for. The LCD is ( + )( ). = ( + )( ) + ( + )( ) = ( + )( ) ( + )( ) ( + )( ) + ( + ) = ( ) + = + 0 = + 0 = + 9 0 = ( )( + ) = 0 or + = 0 = or = Check ANOTHER WAY You can also graph each side of the equation and find the -value where the graphs intersect. 0 Check = : Check = : =? ( ) ( ) 9 =? 7 ( ) + =? = 9 =? ( ) ( ) 9 ( ) + =? 7 0 0 Division by zero is undefined. Intersection X=. Y=- 0 The apparent solution = is etraneous. So, the only solution is =. Monitoring Progress Solve the equation. Check your solution(s). Help in English and Spanish at BigIdeasMath.co 9 7. + + = 9 7. = 9 Chapter 7 Rational Functions Laurie s Notes Teacher Actions No-Hands Questioning and Popsicle Sticks: Students subtracted rational epressions in the last section. Now they can solve rational equations. Pose Eaple and give students tie to work independently and then with partners. Do not rush in. Use Popsicle Sticks to solicit responses. Why is it iportant to check solutions? to verify you have solved the equation correctly and to see whether etraneous solutions have been introduced Think-Pair-Share: Have students answer Questions 7 and, and then share and discuss as a class. 9 Chapter 7
Check REMEMBER 7 In part (b), the variables are eaningful. Switching the to find the inverse would create confusion. So, solve for without switching variables. f g 7 Using Inverses of Functions Finding the Inverse of a Rational Function Consider the function f() =. Deterine whether the inverse of f is a function. + Then find the inverse. Graph the function f. Notice that no horizontal line intersects the graph ore than once. So, the inverse of f is a function. Find the inverse. y = + = y + Set y equal to f(). Switch and y. ( y + ) = Cross ultiply. y + = y = Divide each side by. Subtract fro each side. So, the inverse of f is g() =. f() = + Solving a Real-Life Proble 0 + 000 In Section 7. Eaple, you wrote the function c =, which represents the average cost c (in dollars) of aking odels using a -D printer. Find how any odels ust be printed for the average cost per odel to fall to $90 by (a) solving an equation, and (b) using the inverse of the function. a. Substitute 90 for c and solve by b. Solve the equation for. cross ultiplying. 0 + 000 c= 0 + 000 90 = c = 0 + 000 90 = 0 + 000 0 = 000 c 0 = 000 = = 000 c 0 000 When c = 90, = 90 0 =. So, the average cost falls to $90 per odel after odels are printed. y Etra Eaple 7 Consider the function f() = +. Deterine whether the inverse of f is a function. Then find the inverse. No horizontal line intersects the graph of f ore than once, so the inverse of f is a function. The inverse of f is g() = 7. Etra Eaple In Section 7. Etra Eaple, you wrote n + 00 the function c =, which n represents the average cost (in dollars) of plotting n blueprints using a plotter. Find how any blueprints ust be plotted for the average cost to be $7 by each ethod. a. solving an equation n + 00 7 = ; after 0 blueprints n b. using the inverse of the function n = 00 ; after 0 blueprints c MONITORING PROGRESS 9. yes; g() = + 0. The average cost falls to $90 per odel after 0 odels are printed. Monitoring Progress Help in English and Spanish at BigIdeasMath.co 9. Consider the function f() =. Deterine whether the inverse of f is a function. Then find the inverse. 0 + 00 0. WHAT IF? How do the answers in Eaple change when c =? Section 7. Solving Rational Equations 9 Laurie s Notes Teacher Actions How can you tell whether a function has an inverse? Test whether a horizontal line intersects the function at two or ore points. How do you find the inverse? Write the equation, switch and y, and then solve for y. Have students graph the function and its inverse to see syetry about y =. Connection: Eaple was solved previously using a graphical approach. Two additional approaches are shown in this eaple. Closure Muddiest Point: Ask students to identify, aloud or on a paper to be collected, the uddiest point(s) about the lesson. What was difficult to understand? Section 7. 9
Assignent Guide and Hoework Check ASSIGNMENT Basic:,, 7 odd, 7 odd,, 7 odd,, Average:,, even, even,, Advanced:,, even, even, 9 HOMEWORK CHECK Basic:,,, 7, Average:,,,, Advanced: 0,, 0,,. when each side of the equation is a single rational epression; Saple answer: The equation is a proportion.. yes; When = is substituted into the original equation, the denoinators are zero.. =. =. =. = 7. =, = 7. =, = 9. =, = 0 0. =, =. serves. hits. 0. oz. about 0.07 L. ( + ). ( ) 7. ( + )( + ). ( + 9)( ) 9. = 0. =. = 7. =. =, =. =, =. no solution. no solution 7. =, =. = 0, = 7 9. = ± 9 0. = ± 79 7. Eercises Dynaic Solutions available at BigIdeasMath.co Vocabulary and Core Concept Check. WRITING When can you solve a rational equation by cross ultiplying? Eplain.. WRITING A student solves the equation = and obtains the solutions and. Are either of these etraneous solutions? Eplain. Monitoring Progress and Modeling with Matheatics In Eercises 0, solve the equation by cross ultiplying. Check your solution(s). (See Eaple.). = +. = 9 + 7. 9. + 7 = + = 9 Chapter 7 Rational Functions... 0. 9 = + = = + = 7. USING EQUATIONS So far in your volleyball practice, you have put into play 7 of the serves you have attepted. Solve the equation 90 00 = 7 + to find + the nuber of consecutive serves you need to put into play in order to raise your serve percentage to 90%.. USING EQUATIONS So far this baseball season, you have hits out of 0 ties at-bat. Solve the equation 0.0 = + to find the nuber of consecutive hits 0 + you need to raise your batting average to 0.0.. MODELING WITH MATHEMATICS Brass is an alloy coposed of % copper and % zinc by weight. You have ounces of copper. How any ounces of zinc do you need to ake brass? (See Eaple.). MODELING WITH MATHEMATICS You have 0. liter of an acid solution whose acid concentration is oles per liter. You want to dilute the solution with water so that its acid concentration is only oles per liter. Use the given odel to deterine how any liters of water you should add to the solution. Concentration of new solution = Concentration of Volue of original solution original solution Volue of original solution + Volue of water added USING STRUCTURE In Eercises, identify the LCD of the rational epressions in the equation.. + + = 7. + + + =. 7 = 9. + 9 + = 0 In Eercises 9 0, solve the equation by using the LCD. Check your solution(s). (See Eaples and.) 9.... 7. 9. 0. + = 0. + = + = + + = + = + + + = + + = + + = +... + = 0 + = + 9 0 + = +. + = + 9 Chapter 7
ERROR ANALYSIS In Eercises and, describe and correct the error in the first step of solving the equation... + = + = 7 + + + = 0 ( + ) 7 + + + = 0 ( + ). PROBLEM SOLVING You can paint a roo in hours. Working together, you and your friend can paint the roo in just hours. a. Let t be the tie (in hours) your friend would take to paint the roo when working alone. Copy and coplete the table. (Hint: (Work done) = (Work rate) (Tie)) Work rate Tie Work done You roo hours hours Friend hours b. Eplain what the su of the epressions represents in the last colun. Write and solve an equation to find how long your friend would take to paint the roo when working alone.. PROBLEM SOLVING You can clean a park in hours. Working together, you and your friend can clean the park in just. hours. a. Let t be the tie (in hours) your friend would take to clean the park when working alone. Copy and coplete the table. (Hint: (Work done) = (Work rate) (Tie)) Work rate Tie Work done You park. hours hours Friend. hours b. Eplain what the su of the epressions represents in the last colun. Write and solve an equation to find how long your friend would take to clean the park when working alone.. yes; y = 9. no; y = ±. no; y = ± + 7. a. about i/gal b. about i/gal. a. about 90. ft b. about 90. ft. OPEN-ENDED Give an eaple of a rational equation that you would solve using cross ultiplication and one that you would solve using the LCD. Eplain your reasoning.. OPEN-ENDED Describe a real-life situation that can be odeled by a rational equation. Justify your answer. In Eercises 7, deterine whether the inverse of f is a function. Then find the inverse. (See Eaple.) 7. f() = 7. f() = + 9. f() = 0. f() =. f() =. f() = 9 +. f() = +. f() = 7. PROBLEM SOLVING The cost of fueling your car for year can be calculated using this equation: Fuel cost for year = Miles driven Fuel-efficiency rate Price per gallon of fuel Last year you drove 9000 iles, paid $. per gallon of gasoline, and spent a total of $9 on gasoline. Find the fuel-efficiency rate of your car by (a) solving an equation, and (b) using the inverse of the function. (See Eaple.). PROBLEM SOLVING The recoended percent p (in decial for) of nitrogen (by volue) in the air that a diver breathes is given by p = 0.07, where d is the d + depth (in feet) of the diver. Find the depth when the air contains 7% recoended nitrogen by (a) solving an equation, and (b) using the inverse of the function. Section 7. Solving Rational Equations 97 Dynaic Teaching Tools Dynaic Assessent & Progress Monitoring Tool Interactive Whiteboard Lesson Library Dynaic Classroo with Dynaic Investigations. Both sides of the equation should be ultiplied by the sae epression; + =. Each ter of the equation should be ultiplied by the LCD; ( + )() (7 + ) + + ( + )() = 0 ( + )(). a. See Additional Answers. b. The su is the aount of tie it would take for you and your friend to paint the roo together; + t =, t =. h = h 0 in. a. See Additional Answers. b. The su is the aount of tie it would take for you and your friend to clean the park together; 0. +. =, t = h t. Saple answer: + + = +, Cross ultiplication can be used when each side of the equation is a single rational epression; Saple answer: + + + + = + ; Multiplying by the LCD can be used when there is ore than one rational epression on one side of the equation.. Saple answer: A rational equation can be used to deterine how long it would take you and your friend working together to clean a park. 7. yes; y = +. yes; y = 7 9. yes; y = + 0. yes; y = +. yes; y = + Section 7. 97
7. ±0.. ±0.77 9..7 0..99 +.. = ; The graphs intersect at =.. g() = +. g() = 7. y = b d c a. See Additional Answers. Mini-Assessent Solve the equation.. = 7 =. 9 + = =. = + + 0 + = ; The apparent solution = is etraneous.. Consider the function f() =. Deterine whether the inverse of f is a function. Then find the inverse. The inverse of f is a function. The inverse of f is g() = +. n + 000. The function c = n represents the average cost c (in dollars) of filling n orders using a copier. Find how any orders ust be filled for the average cost to be $ by each ethod. a. solving an equation n + 000 = ; n after 0 orders b. using the inverse of the function n = 000 ; after 0 orders c USING TOOLS In Eercises 7 0, use a graphing calculator to solve the equation f() = g(). 7. f() =, g() =. f() =, g() = 9. f() = +, g() = 0. f() = +, g() = +. MATHEMATICAL CONNECTIONS Golden rectangles are rectangles for which the ratio of the width w to the length is equal to the ratio of to + w. The ratio of the length to the width for these rectangles is called the golden ratio. Find the w value of the golden ratio using a rectangle with a width of unit.. HOW DO YOU SEE IT? Use the graph to identify the ( ) solution(s) of the rational equation = +. Eplain your reasoning. 9 Chapter 7 Rational Functions y ( ) y = y = + Maintaining Matheatical Proficiency USING STRUCTURE In Eercises and, find the inverse of the function. (Hint: Try rewriting the function by using either inspection or long division.). f() = + Is the doain discrete or continuous? Eplain. Graph the function using its doain. (Skills Review Handbook). f() = 7 +. ABSTRACT REASONING Find the inverse of rational functions of the for y = a + b. Verify your answer c + d is correct by using it to find the inverses in Eercises and.. THOUGHT PROVOKING Is it possible to write a rational equation that has the following nuber of solutions? Justify your answers. a. no solution b. eactly one solution c. eactly two solutions d. infinitely any solutions 7. CRITICAL THINKING Let a be a nonzero real nuber. Tell whether each stateent is always true, soeties true, or never true. Eplain your reasoning. a. For the equation a =, = a is an a etraneous solution. b. The equation a = has eactly a one solution. c. The equation a = + a + a a has no solution.. MAKING AN ARGUMENT Your friend clais that it is not possible for a rational equation of the for a = c, where b 0 and d 0, to have b d etraneous solutions. Is your friend correct? Eplain your reasoning. 9. The linear function y = 0. represents the aount of oney y (in dollars) of quarters in your pocket. You have a aiu of eight quarters in your pocket. 0. A store sells broccoli for $ per pound. The total cost t of the broccoli is a function of the nuber of pounds p you buy. Evaluate the function for the given value of. (Section.) Reviewing what you learned in previous grades and lessons. f() = + 7; =. g() = + 7 + ; =. h() = + + ; =. k() = + ; = If students need help... Resources by Chapter Practice A and Practice B Puzzle Tie Student Journal Practice Differentiating the Lesson Skills Review Handbook If students got it... Resources by Chapter Enrichent and Etension Cuulative Review Start the net Section 9 Chapter 7