4.7 Solutions of Rational Equations

Transcription

1 Chapter 4. Rational Equations and Functions 4.7 s of Rational Equations Learning Objectives Solve rational equations using cross products. Solve rational equations using lowest common denominators. Solve real-world problems with rational equations. Introduction A rational equation is one that contains rational epressions. It can be an equation that contains rational coefficients or an equation that contains rational terms where the variable appears in the denominator. An eample of the first kind of equation is = 4. An eample of the second kind of equation is = 4 +. The first aim in solving a rational equation is to eliminate all denominators. In this way, we can change a rational equation to a polynomial equation which we can solve with the methods we have learned this far. Solve Rational Equations Using Cross Products A rational equation that contains two terms is easily solved by the method of cross products or cross multiplication. Consider the following equation. 5 = + 1 Our first goal is to eliminate the denominators of both rational epressions. In order to remove the five from the denominator of the first fraction, we multiply both sides of the equation by five: = Now, we remove the from the denominator of the second fraction by multiplying both sides of the equation by two. = 5( + 1) 1 1 The equation simplifies to = 5( + 1). Continuing we get: = = 5 Answer Notice that when we remove the denominators from the rational epressions we end up multiplying the numerator on one side of the equal sign with the denominator of the opposite fraction. 89

2 4.7. s of Rational Equations Once again, we obtain the simplified equation: = 5( + 1), whose solution is = 5. We check the answer by plugging the answer back into the original equation. Check On the left-hand side, if = 5, then we have 5 5 = 5 = 1 On the right hand side, we have + 1 = = = 1 Since the two epressions are equal, the answer checks out. Eample 1 Solve the equation = +. Use cross-multiplication to eliminate the denominators of both fractions. The equation simplifies to ( + )=( ) Simplify. Check. + 6 = 6 = 1 The answer checks out because the epressions are equal. 90 = 1 = 10 = = 1 + = 15 = 1 5

3 Chapter 4. Rational Equations and Functions Eample Solve the equation +4 = 5. Cross-multiply. The equation simplifies to = 5( + 4) Simplify. = Move all terms to one side of the equation. 5 0 = 0 Notice that this equation has a degree of two, that is, it is a quadratic equation. We can solve it using the quadratic formula. = 5± or 4.65 Answer It is important to check the answer in the original equation when the variable appears in any denominator of the equation because the answer might be an ecluded value of any of the rational epression. If the answer obtained makes any denominator equal to zero, that value is not a solution to the equation. Check: First we check.15 by substituting it in the original equations. On the left hand side we get the following. Now, check on the right hand side. + 4 = (.15) = Thus,.15 checks out. For 4.65 we repeat the procedure. 5 =

4 4.7. s of Rational Equations also checks out. + 4 = (4.65) = Solve Rational Equations Using the Lowest Common Denominators An alternate way of eliminating the denominators in a rational equation is to multiply all terms in the equation by the lowest common denominator. This method is suitable even when there are more than two terms in the equation. Eample Solve 5 = Find the lowest common denominator of all the terms in the equation. LCD = 105 Multiply each term by the LCD. Divide out common factors. 105 = The equation simplifies to = = 1 5 Move all terms to one side of the equation = 0 Solve using the quadratic formula. 0. or 0.75 Answer Check 9 = 9 ± 501 4

5 Chapter 4. Rational Equations and Functions We use the substitution 0.. Now we check the solution = ( 0.) = ( 0.) The answer checks out. 5 1 Eample 4 Solve = 10. Factor all denominators. 5 = (0.75) (0.75) The answer checks out Find the lowest common denominator ( + )( 5) Multiply all terms in the equation by the LCD. LCD =( + )( 5) ( + )( 5) Divide out the common terms. + ( + )( 5) 4 5 =( + )( 5) ( + )( 5) ( + ) 1 ( 5) ( + ) ( 5) 1 4 = ( + ) ( 5) ( + ) 1 ( 5) 1 The equation simplifies to Simplify. ( 5) 4( + )= = = 5 Answer 9

6 4.7. s of Rational Equations Check. The answer checks out. Eample 5 Solve = 1. Find the lowest common denominator = = ( 5) ( 5) Multiply all terms in the equation by the LCD. LCD =( + 1)( + 4) Cancel all common terms. ( + 1)( + 4) +( + 1)( + 4) =( + 1)( + 4) ( + 1) 1 ( + 4) +( + 1) ( + 4) =( + 1)( + 4) 4 1 The simplified equation is Eliminate parentheses. ( + 4)+( + 1)=( + 1)( + 4) Collect like terms = Check. = 4 = = ± Answer = = 1. The answer checks out. + 4 ( ) + 4 = ( ) = 1. The answer checks out

7 Chapter 4. Rational Equations and Functions Solve Real-World Problems Using Rational Equations Motion Problems A motion problem with no acceleration is described by the formula distance = speed time. These problems can involve the addition and subtraction of rational epressions. Eample 6 Last weekend Nadia went canoeing on the Snake River. The current of the river is three miles per hour. It took Nadia the same amount of time to travel 1 miles downstream as three miles upstream. Determine the speed at which Nadias canoe would travel in still water. 1. Define variables Let s = speed of the canoe in still water Then, s + = the speed of the canoe traveling downstream s = the speed of the canoe traveling upstream. Construct a table. We make a table that displays the information we have in a clear manner: TABLE 4.9: Direction Distance (miles) Rate Time Downstream 1 s + t Upstream s t. Write an equation. Since distance = rate time, we can say that: time = distance rate. The time to go downstream is The time to go upstream is t = 1 s + t = s Since the time it takes to go upstream and downstream are the same then: s = 1 s+ 4. Solve the equation Cross-multiply. (s + )=1(s ) Simplify. 95

8 4.7. s of Rational Equations s + 9 = 1s 6 Solve. s = 5 mi/hr Answer Nadia would travel 5 mi/hr or 5 mph in still water. 5. Check Downstream: t = 1 5+ = 1 8 = 1 1 hour; Upstream: t = 5 = = 1 1 hour. The answer checks out. Eample 8 Peter rides his bicycle. When he pedals uphill he averages a speed of eight miles per hour, when he pedals downhill he averages 14 miles per hour. If the total distance he travels is 40 miles and the total time he rides is four hours, how long did he ride at each speed? 1. Define variables. Let t 1 = time Peter bikes uphill, t = time Peter bikes downhill, and d = distance he rides uphill.. Construct a table We make a table that displays the information we have in a clear manner: TABLE 4.10: Direction Distance (miles) Rate (mph) Time (hours) Uphill d 8 t 1 Downhill 40 d 14 t. Write an equation We know that The time to go uphill is time = distance rate t 1 = d 8 The time to go downhill is We also know that the total time is 4 hours. t = 40 d 14 96

9 Chapter 4. Rational Equations and Functions 4. Solve the equation. Find the lowest common denominator: LCD=56 Multiply all terms by the common denominator: d d = d 40 d + 56 = d d = 4 d = 64 Solve. d 1. miles Answer 5. Check. Uphill: t = hours; Downhill: t = hours. The answer checks out. Shares Eample 8 A group of friends decided to pool together and buy a birthday gift that cost $00. Later 1 of the friends decided not to participate any more. This meant that each person paid $15 more than the original share. How many people were in the group to start? 1. Define variables. Let = the number of friends in the original group. Make a table. We make a table that displays the information we have in a clear manner: TABLE 4.11: Number of People Gift Price Share Amount 00 Original group Later group Write an equation. Since each persons share went up by $15 after 1 people refused to pay, we write the equation: 97

10 4.7. s of Rational Equations 4. Solve the equation. Find the lowest common denominator = LCD = ( 1) Multiply all terms by the LCM. ( 1) Divide out common factors in each term: = ( 1) + ( 1) 15 1 ( 1) = 1 ( 1) 00 + ( 1) Simplify. 00 = 00( 1)+15( 1) Eliminate parentheses. 00 = Collect all terms on one side of the equation. 0 = Divide all terms by = Factor. 0 =( 0)( + 8) Solve. 98

11 Chapter 4. Rational Equations and Functions = 0, = 8 The answer is = 0 people. We discard the negative solution since it does not make sense in the contet of this problem. 5. Check. Originally $00 shared among 0 people is $10 each. After 1 people leave, $00 shared among 8 people is $5 each. So each person pays $15 more. The answer checks out. Review Questions Solve the following equations = = = = = = = = = = = = = = = = = = Juan jogs a certain distance and then walks a certain distance. When he jogs he averages 7 miles/hour. When he walks, he averages.5 miles/hour. If he walks and jogs a total of 6 miles in a total of 1 hour and 1 minutes, how far does he jog and how far does he walk? 0. A boat travels 60 miles downstream in the same time as it takes it to travel 40 miles upstream. The boat s speed in still water is 0 miles/hour. Find the speed of the current. 1. Paul leaves San Diego driving at 50 miles/hour. Two hours later, his mother realizes that he forgot something and drives in the same direction at 70 miles/hour. How long does it take her to catch up to Paul?. On a trip, an airplane flies at a steady speed against the wind. On the return trip the airplane flies with the wind. The airplane takes the same amount of time to fly 00 miles against the wind as it takes to fly 40 miles with the wind. The wind is blowing at 0 miles/hour. What is the speed of the airplane when there is no wind?. A debt of $40 is shared equally by a group of friends. When five of the friends decide not to pay, the share of the other friends goes up by $5. How many friends were in the group originally? 4. A non-profit organization collected $50 in equal donations from their members to share the cost of improving a park. If there were thirty more members, then each member could contribute $0 less. How many members does this organization have? 99

12 4.7. s of Rational Equations Review Answers 1. = = = 1 4. = 1±i = 5 6. = 5 7. = 1 8. = 1 9. = 1, = , , , = 1, = =, = , = = , jogs.6 miles and walks.4 miles (Hint: How do distance and speed related to time?) 0. 4 miles/hour (Hint: How does the current affect the boat s speed upstream? Downstream?) 1. 5 hours (Hint: Who travels for a longer time?). 180 miles/hour (How is the effective of speed of the plane change when flying with the wind? Against the wind?. 1 friends (Hint: Would the original group of friends have paid a higher or lower portion of the debt? By how much did the portion of the share for each person change as a result of five friends not paying? members (Hint: Does having more members in the organization increase or decrease the amount each person would need to contribute? By how much? 00