11-8 Rational Equations
Solve each equation. State any extraneous solutions.
1. SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.
The LCD is x(x + 1).
The solution is x = –2.
Check:
2. SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.
The LCD is 45.
The solution is t = 12.
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Page 1
11-8 Rational Equations
2. SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.
The LCD is 45.
The solution is t = 12.
Check: 3. SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.
The LCD is 5a.
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11-8 Rational Equations
3. SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.
The LCD is 5a.
The solution is a =
.
Check. 4. SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.”
eSolutions
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LCD is
(p – 1).by Cognero
Page 3
11-8 Rational Equations
4. SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.”
The LCD is (p – 1).
The solution is p = 2.
Check. 5. SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.
The LCD is (t + 1)(t – 1).
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11-8 Rational Equations
5. SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.
The LCD is (t + 1)(t – 1).
The solution is t = –3.
Check. 6. SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.”
2
2
2
The factors of (x – 1) are (x – 1)(x + 1). Thus, the LCD of (x – 1) and (x – 1) is (x – 1).
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Page 5
SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.”
11-8 Rational Equations
2
2
2
The factors of (x – 1) are (x – 1)(x + 1). Thus, the LCD of (x – 1) and (x – 1) is (x – 1).
Since x = 1 results in a zero in the denominator of the original equation, it is an extraneous solution. The solution is x
=
.
Check:
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Page 6
Since x = 1 results in a zero in the denominator of the original equation, it is an extraneous solution. The solution is x
=
.
11-8 Rational Equations
Check:
7. WEEDING Maurice can weed the garden in 45 minutes. Olinda can weed the garden in 50 minutes. How long
would it take them to weed the garden if they work together?
SOLUTION: You need to find the rate that each person works and the total time t that it will take them if they work together.
Find the fraction of the job that each person can do in an hour.
45 minutes is
of an hour. 50 minutes is Maurice’s rate is
of an hour.
job per hour. Olinda’s rate is
job per hour.
Since rate · time = fraction of job done, multiply each rate by the time t to represent the amount of the job done by
each person.
To solve, multiply each side of the equation by the LCD to eliminate the denominators. The LCD is 15. eSolutions Manual - Powered by Cognero
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11-8 Rational Equations
7. WEEDING Maurice can weed the garden in 45 minutes. Olinda can weed the garden in 50 minutes. How long
would it take them to weed the garden if they work together?
SOLUTION: You need to find the rate that each person works and the total time t that it will take them if they work together.
Find the fraction of the job that each person can do in an hour.
45 minutes is
of an hour. 50 minutes is Maurice’s rate is
of an hour.
job per hour. Olinda’s rate is
job per hour.
Since rate · time = fraction of job done, multiply each rate by the time t to represent the amount of the job done by
each person.
To solve, multiply each side of the equation by the LCD to eliminate the denominators. The LCD is 15. It would take Maurice and Olinda
hour or about 0.4 hour to weed the garden if they work together.
8. LANDSCAPING Hunter is filling a 3.5-gallon bucket to water plants at a faucet that flows at a rate of 1.75 gallons
a minute. If he were to add a hose that flows at a rate of 1.45 gallons per minute, how many minutes would it take
him to fill the bucket? Round to the nearest tenth.
SOLUTION: You need to find the time it takes to fill the bucket with the faucet or hose And the total time t that it would take
them if they were used at the same time. Hunter can fill the bucket from the faucet in
Hunter can fill the bucket from the hose in or 2 min, so the rate of the faucet is .
or 2.4 min, so the rate of the hose is
.
Since rate · time = fraction of the bucket filled, multiply each rate by t to represent the amount of the bucket filled by
hose or faucet. eSolutions Manual - Powered by Cognero
Page 8
11-8ItRational
Equations
would take
Maurice and Olinda
hour or about 0.4 hour to weed the garden if they work together.
8. LANDSCAPING Hunter is filling a 3.5-gallon bucket to water plants at a faucet that flows at a rate of 1.75 gallons
a minute. If he were to add a hose that flows at a rate of 1.45 gallons per minute, how many minutes would it take
him to fill the bucket? Round to the nearest tenth.
SOLUTION: You need to find the time it takes to fill the bucket with the faucet or hose And the total time t that it would take
them if they were used at the same time. Hunter can fill the bucket from the faucet in
Hunter can fill the bucket from the hose in or 2 min, so the rate of the faucet is .
or 2.4 min, so the rate of the hose is
.
Since rate · time = fraction of the bucket filled, multiply each rate by t to represent the amount of the bucket filled by
hose or faucet. To solve, multiply each side of the equation by the LCD to eliminate the denominators. The integer LCD is 12.
Using both the faucet and the hose, it would take Hunter about 1.1 minutes to fill the 3.5-gallon bucket.
Solve each equation. State any extraneous solutions.
9. SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.
The LCD is n(n – 5).
The Manual
solution
is n = 8.
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by Cognero
Check:
Page 9
11-8 Rational Equations
Using both the faucet and the hose, it would take Hunter about 1.1 minutes to fill the 3.5-gallon bucket.
Solve each equation. State any extraneous solutions.
9. SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.
The LCD is n(n – 5).
The solution is n = 8.
Check:
10. SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.
The LCD is t(t + 2).
The solution is t = 4.
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Page 10
11-8 Rational Equations
10. SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.
The LCD is t(t + 2).
The solution is t = 4.
Check: 11. SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.
The LCD is 12.
The solution is
.
Check:
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11-8 Rational Equations
11. SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.
The LCD is 12.
The solution is
.
Check:
12. SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.
The LCD is 8.
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11-8 Rational Equations
12. SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.
The LCD is 8.
The solution is
.
Check:
13. SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.
The LCD is 15w.
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11-8 Rational Equations
13. SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.
The LCD is 15w.
The solution is w = –13.
Check:
14. SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.
The LCD is (c + 1)(c – 1).
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11-8 Rational Equations
14. SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.
The LCD is (c + 1)(c – 1).
The solution is
.
Check:
15. SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.
The LCD is (x + 1)( x– 1).
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11-8 Rational Equations
15. SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.
The LCD is (x + 1)( x– 1).
The solution is x = 0.
Check:
16. SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.
The LCD is (y + 3)(y – 2).
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11-8 Rational Equations
16. SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.
The LCD is (y + 3)(y – 2).
The solution is y = –4 or –8.
Check: y = –4
y = –8
17. SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.
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The LCD is (a +3).
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11-8 Rational Equations
17. SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.
The LCD is (a +3).
The solution is a = –2 or 3.
Check: a = –2
a=3
18. SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.
eSolutions
Manual - Powered by Cognero
2
The factors of (a – 9) are (a + 3)(a – 3). Thus, the LCD is (a + 3)(a – 3).
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11-8 Rational Equations
18. SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.
2
The factors of (a – 9) are (a + 3)(a – 3). Thus, the LCD is (a + 3)(a – 3).
The solution is Check:
.
19. SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.
The LCD is (n –1).
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11-8 Rational Equations
19. SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.
The LCD is (n –1).
Since n = 1 results in a zero in the denominator of the original equation, it is an extraneous solution. There is no
solution.
20. SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.
2
The factors of (n – n) are n(n – 1). Thus, the LCD is n(n – 1).
Since n = 1 results in a zero in the denominator of the original equation, it is an extraneous solution. There is no
solution.
21. PAINTING It takes Noah 3 hours to paint one side of a fence. It takes Gilberto 5 hours. How long would it take
them if they worked together?
SOLUTION: You need to find the rate that each person works and the total time t that it will take them if they work together.
Find the fraction of the job that each person can do in an hour. eSolutions Manual - Powered by Cognero
Noah’s rate is
job per hour. Gilberto’s rate is
job per hour.
Page 20
11-8Since
Rational
n = 1Equations
results in a zero in the denominator of the original equation, it is an extraneous solution. There is no
solution.
21. PAINTING It takes Noah 3 hours to paint one side of a fence. It takes Gilberto 5 hours. How long would it take
them if they worked together?
SOLUTION: You need to find the rate that each person works and the total time t that it will take them if they work together.
Find the fraction of the job that each person can do in an hour. Noah’s rate is
job per hour. Gilberto’s rate is
job per hour.
Since rate · time = fraction of job done, multiply each rate by the time t to represent the amount of the job done by
each person. To solve, multiply each side of the equation by the LCD to eliminate the denominators. The LCD is 15.
It would take Noah and Gilberto
hours or hours to paint one side of a fence if they work together.
22. DISHWASHING Ron works as a dishwasher and can wash 500 plates in two hours and 15 minutes. Chris can
finish the 500 plates in 3 hours. About how long would it take them to finish all of the plates if they work together?
SOLUTION: You need to find the rate that each person washes the dishes and the total time t that it will take them if they work
together.
Ron’s rate is
job per hour. Chris’s rate is
job per hour.
Since rate · time = fraction of job done, multiply each rate by the time t to represent the amount of the washing job
done by each person.
To solve, multiply each side of the equation by the LCD to eliminate the denominators.The Integer LCD is 9. eSolutions Manual - Powered by Cognero
Page 21
11-8ItRational
Equations
would take
Noah and Gilberto
hours or hours to paint one side of a fence if they work together.
22. DISHWASHING Ron works as a dishwasher and can wash 500 plates in two hours and 15 minutes. Chris can
finish the 500 plates in 3 hours. About how long would it take them to finish all of the plates if they work together?
SOLUTION: You need to find the rate that each person washes the dishes and the total time t that it will take them if they work
together.
Ron’s rate is
job per hour. Chris’s rate is
job per hour.
Since rate · time = fraction of job done, multiply each rate by the time t to represent the amount of the washing job
done by each person.
To solve, multiply each side of the equation by the LCD to eliminate the denominators.The Integer LCD is 9. It would take Ron and Chris
hours or hours to wash 500 plates if they work together.
23. ICE A hotel has two ice machines in its kitchen. How many hours would it take both machines to make 60 pounds
of ice? Round to the nearest tenth.
SOLUTION: You need to find the rate that each machine works and the total time t that it will take them if both machines are
uses.
The rate of ice-machine 1 is
. The rate of ice-machine 2 is
.
Since rate · time = fraction of the job done, multiply each rate by the time t to represent the amount of the ice making
is done by each machine. eSolutions Manual - Powered by Cognero
Page 22
11-8ItRational
Equations
would take
Ron and Chris
hours or hours to wash 500 plates if they work together.
23. ICE A hotel has two ice machines in its kitchen. How many hours would it take both machines to make 60 pounds
of ice? Round to the nearest tenth.
SOLUTION: You need to find the rate that each machine works and the total time t that it will take them if both machines are
uses.
The rate of ice-machine 1 is
. The rate of ice-machine 2 is
.
Since rate · time = fraction of the job done, multiply each rate by the time t to represent the amount of the ice making
is done by each machine. To solve, multiply each side of the equation by the LCD to eliminate the denominators. The LCD is 24.
It would take the two ice machines about 26.2 hours to make 60 pounds of ice.
24. CYCLING Two cyclists travel in opposite directions around a 5.6-mile circular trail. They start at the same time.
The first cyclist completes the trail in 22 minutes and the second cyclist completes the trail in 28 minutes. At what
time do they pass each other?
SOLUTION: You need to find the rate that each person rides. If The first cyclist’s rate is
. The second cyclist’s rate is
, then
.
.
Since rate · time = fraction of distance completed, multiply each rate by the time t to represent the amount of ride
complete by each person. eSolutions
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Then
and
Page 23
.
11-8 Rational Equations
It would take the two ice machines about 26.2 hours to make 60 pounds of ice.
24. CYCLING Two cyclists travel in opposite directions around a 5.6-mile circular trail. They start at the same time.
The first cyclist completes the trail in 22 minutes and the second cyclist completes the trail in 28 minutes. At what
time do they pass each other?
SOLUTION: You need to find the rate that each person rides. If The first cyclist’s rate is
. The second cyclist’s rate is
, then
.
.
Since rate · time = fraction of distance completed, multiply each rate by the time t to represent the amount of ride
complete by each person. Then
and
.
Since the riders are going different ways around the trip, the riders distance should differ by 5.6 miles.
To solve, multiply each side of the equation by the LCD to eliminate the denominators. The LCD is 616.
They will pass each other at 12.32 minutes.
GRAPHING CALCULATOR For each rational function, a) describe the shape of the graph, b) use factoring to simplify the function, and c) determine the zeros of the function.
25. SOLUTION: a. Using a graphing calculator, you can see that the graph is a line.
[-10, 10] scl: 1 by [-10, 10] scl: 1
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b. Page 24
11-8 Rational Equations
They will pass each other at 12.32 minutes.
GRAPHING CALCULATOR For each rational function, a) describe the shape of the graph, b) use factoring to simplify the function, and c) determine the zeros of the function.
25. SOLUTION: a. Using a graphing calculator, you can see that the graph is a line.
[-10, 10] scl: 1 by [-10, 10] scl: 1
b. c. Method 1: Find the zeros using a graphing calculator.
Use the zero option from the 2nd CALC menu. [-10, 10] scl: 1 by [-10, 10] scl: 1
Method 2: Find the zeros algebraically.
The zero of the function is −5.
26. SOLUTION: a. Using
eSolutions
Manual
Powered by
Cognero you can see that the graph is a parabola.
a -graphing
calculator,
Page 25
The zero of the function is −5.
26. 11-8 Rational Equations
SOLUTION: a. Using a graphing calculator, you can see that the graph is a parabola.
[-5, 5] scl: 1 by [-5, 5] scl: 1
b. c. Method 1: Find the zeros using a graphing calculator.
Use the zero option from the 2nd CALC menu. [-5, 5] scl: 1 by [-5, 5] scl: 1
[-5, 5] scl: 1 by [-5, 5] scl: 1
Method 2: Find the zeros algebraically.
The zeros of the function are 0 and 1.
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27. Page 26
11-8 Rational Equations
The zeros of the function are 0 and 1.
27. SOLUTION: a. Using a graphing calculator, you can see that the graph is a parabola.
[-10, 10] scl: 1 by [-10, 10] scl: 1
b.
c. The square of a number cannot be negative, so there are no real zeroes.
28. PAINTING Morgan can paint a standard-sized house in about 5 days. For his latest job, Morgan hires two
assistants. At what rate must these assistants work for Morgan to meet a deadline of two days?
SOLUTION: Morgan can paint 1 house in 5 days, so his rate is
. Let r represent the rate at which the
two assistants work together.
Then, (fraction of the house Morgan paints) + (fraction of the house the assistants paint) equals (1 house).
So, the
assistants
must
paint
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of the house each day for 2 days.
29. AIRPLANES Headwinds push against a plane and reduce its total speed, while tailwinds push on a plane and
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11-8 Rational Equations
The square of a number cannot be negative, so there are no real zeroes.
28. PAINTING Morgan can paint a standard-sized house in about 5 days. For his latest job, Morgan hires two
assistants. At what rate must these assistants work for Morgan to meet a deadline of two days?
SOLUTION: Morgan can paint 1 house in 5 days, so his rate is
. Let r represent the rate at which the
two assistants work together.
Then, (fraction of the house Morgan paints) + (fraction of the house the assistants paint) equals (1 house).
So, the assistants must paint
of the house each day for 2 days.
29. AIRPLANES Headwinds push against a plane and reduce its total speed, while tailwinds push on a plane and
increase its total speed. Let w equal the speed of the wind, r equal the speed set by the pilot, and s equal the total
speed.
a. Write an equation for the total speed with a headwind and an equation for the total speed with a tailwind.
b. Use the rate formula to write an equation for the distance traveled by a plane with a headwind and another
equation for the distance traveled by a plane with a tailwind. Then solve each equation for time instead of distance.
SOLUTION: a. The total speed of the plane facing a headwind is s = r − w. The total speed of the plane being pushed by a
tailwind is s = r + w.
b. d = t(r − w), d = t(r + w);
30. MIXTURES A pitcher of fruit juice has 3 pints of pineapple juice and 2 pints of orange juice. Erin wants to add
more orange juice so that the fruit juice mixture is 60% orange juice. Let x equal the pints of orange juice that she
needs to add.
a. Copy and complete the table below.
b. Write and solve an equation to find the pints of orange juice to add.
SOLUTION: a. The original mixture started with 2 pints of orange juice. Divide 2 pints by total of 2 pints to find the percent of
orange juice in the original mixture. Let x represent the addition orange juice added. Then 2 + x represents the new
amount of orange juice and 5 + x the total pints of juice. eSolutions Manual - Powered by Cognero
Page 28
SOLUTION: a. The total speed of the plane facing a headwind is s = r − w. The total speed of the plane being pushed by a
tailwind is s = r + w.
11-8b.
Rational
d = t(r Equations
− w), d = t(r + w);
30. MIXTURES A pitcher of fruit juice has 3 pints of pineapple juice and 2 pints of orange juice. Erin wants to add
more orange juice so that the fruit juice mixture is 60% orange juice. Let x equal the pints of orange juice that she
needs to add.
a. Copy and complete the table below.
b. Write and solve an equation to find the pints of orange juice to add.
SOLUTION: a. The original mixture started with 2 pints of orange juice. Divide 2 pints by total of 2 pints to find the percent of
orange juice in the original mixture. Let x represent the addition orange juice added. Then 2 + x represents the new
amount of orange juice and 5 + x the total pints of juice. b. Divide 2 + x pints of orange juice by total pints of 5 + x pints to find the percent of orange juice in the final mixture
Erin needs to add 2.5 pt of orange juice.
31. DORMITORIES The number of hours h it takes to clean a dormitory varies inversely with the number of people
cleaning it c and directly with the number of people living there p .
a. Write an equation showing how h, c, and p are related. (Hint: Include the constant k.)
b. It takes 8 hours for 5 people to clean the dormitory when there are 100 people there. How long will it take to
clean the dormitory if there are 10 people cleaning and the number of people living in the dorm stays the same?
SOLUTION: a. b.
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11-8 Rational Equations
Erin needs to add 2.5 pt of orange juice.
31. DORMITORIES The number of hours h it takes to clean a dormitory varies inversely with the number of people
cleaning it c and directly with the number of people living there p .
a. Write an equation showing how h, c, and p are related. (Hint: Include the constant k.)
b. It takes 8 hours for 5 people to clean the dormitory when there are 100 people there. How long will it take to
clean the dormitory if there are 10 people cleaning and the number of people living in the dorm stays the same?
SOLUTION: a. b.
It would take 4 hours to clean the dormitory is there are 10 people cleaning and 100 people living there.
Solve each equation. State any extraneous solutions.
32. SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.
2
2
The factors of (b – 3b) is b(b – 3). Thus, the LCD of (b – 3b) and b is b(b – 3).
The solution is b = 1.
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Check:
Page 30
11-8 Rational Equations
It would take 4 hours to clean the dormitory is there are 10 people cleaning and 100 people living there.
Solve each equation. State any extraneous solutions.
32. SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.
2
2
The factors of (b – 3b) is b(b – 3). Thus, the LCD of (b – 3b) and b is b(b – 3).
The solution is b = 1.
Check:
33. SOLUTION: eSolutions Manual - Powered by Cognero
Page 31
33. 11-8 Rational Equations
SOLUTION: Check each of the solutions.
Since x = 1 results in a zero in the denominator of the original equation, it is an extraneous solution.
eSolutions
- Powered
Cognero extraneous: 1.
The Manual
solutions
are x by
= 0, –4;
Page 32
Since x = 1 results in a zero in the denominator of the original equation, it is an extraneous solution.
The solutions are x = 0, –4; extraneous: 1.
11-8 Rational Equations
34. SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.
3
2
The factors y – 2y . are
3
2
3
2
. Thus the LCD of y and y – 2y is y – 2y .
Since y = 0 results in a zero in the denominator of the original equation, it is an extraneous solution. The solution is y
=
.
Check:
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Page 33
Since y = 0 results in a zero in the denominator of the original equation, it is an extraneous solution. The solution is y
=
.
11-8 Rational Equations
Check:
35. SOLUTION: 2
The factors of (x – 5x) are x(x – 5).Multiply each side of the equation by the LCD to eliminate the denominators.
2
The LCD x, (x – 5) and (x – 5x) is x(x – 5).
eSolutions Manual - Powered by Cognero
Page 34
SOLUTION: 2
The factors of (x – 5x) are x(x – 5).Multiply each side of the equation by the LCD to eliminate the denominators.
11-8 Rational Equations
2
The LCD x, (x – 5) and (x – 5x) is x(x – 5).
The solution is x =
.
Check:
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Manual - Powered
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Solve
36. CHALLENGE SOLUTION: .
Page 35
11-8 Rational Equations
36. CHALLENGE Solve
.
SOLUTION: Multiply each side of the equation by the LCD to eliminate the denominators.”
The LCD is (x + 1)(x – 2).
Check each of the possible solutions: x = –2 x = eSolutions Manual - Powered by Cognero
Page 36
11-8 Rational Equations
x = Therefore, the solutions are x = –2 and
.
37. REASONING How is an excluded value of a rational expression related to an extraneous solution of a
corresponding rational equation? Explain.
SOLUTION: An extraneous solution of a rational equation is an excluded value of one of the expressions in the equation.
For example, consider the rational expression
. The expression is undefined at x = 5. It is an
, the solution is x = 5. However, x = 5 results in
excluded value. If you solve the equation a zero in the denominator of the original equation, it is therefore an extraneous solution. 38. WRITING IN MATH Why should you check solutions of rational equations?
SOLUTION: Sample answer: Multiplying each side of a rational equation by the LCD can result in extraneous solutions.
Therefore, all solutions
should be checked to make sure that they satisfy the original equation.
For example, in the equation below we can eliminate the x – 6 from the numerator and denominator. We will then
get x = 6. However, x cannot be 6 in the original equation.
39. CCSS ARGUMENTS Find a counterexample for the following statement.
The solution of a rational equation can never be zero.
SOLUTION: Find an equation where the solution of a rational equation is zero. Choose any rational expression with an x in the
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Page 37
numerator and anything in the denominator. Set it equal to 0. It will result in a zero solution. For example:
.
get x = 6. However, x cannot be 6 in the original equation.
11-8 Rational Equations
39. CCSS ARGUMENTS Find a counterexample for the following statement.
The solution of a rational equation can never be zero.
SOLUTION: Find an equation where the solution of a rational equation is zero. Choose any rational expression with an x in the
numerator and anything in the denominator. Set it equal to 0. It will result in a zero solution. For example:
.
40. WRITING IN MATH Describe the steps for solving a rational equation that is not a proportion.
SOLUTION: First, find the LCD of the fractions in the equation. Then multiply each side of the equation by the LCD. Simplify and
use the order of operations to solve for the variable.
Consider the following example. 41. It takes Cheng 4 hours to build a fence. If he hires Odell to help him, they can do the job in 3 hours. If Odell built the
same fence alone, how long would it take him?
A hours
B hours
C 8 hours
D 12 hours
SOLUTION: You need to find the rate that each person can mend the fence alone or together and the total time t that it will take
them if they worked together.
Find the fraction of the job that Chen can do in an hour. Chan takes 4 hours to complete the task, so he can do of
the task in 1 hour. Let x represent Odell's hourly rate. Since the rate · time is the fraction of the job done, multiply each by the time 3 hours to represent the amount of the job done by each person. Then eSolutions Manual - Powered by Cognero
Solve to find out Odell's rate. Page 38
11-8 Rational Equations
41. It takes Cheng 4 hours to build a fence. If he hires Odell to help him, they can do the job in 3 hours. If Odell built the
same fence alone, how long would it take him?
A hours
B hours
C 8 hours
D 12 hours
SOLUTION: You need to find the rate that each person can mend the fence alone or together and the total time t that it will take
them if they worked together.
Find the fraction of the job that Chen can do in an hour. Chan takes 4 hours to complete the task, so he can do of
the task in 1 hour. Let x represent Odell's hourly rate. Since the rate · time is the fraction of the job done, multiply each by the time 3 hours to represent the amount of the job done by each person. Then Solve to find out Odell's rate. It would take Odell 12 hours to build the fence alone. So, the correct choice is D.
42. In the 1000-meter race, Zoe finished 35 meters ahead of Taryn and 53 meters ahead of Evan. How far was she
ahead of Evan?
F 18 m
G 35 m
H 53 m
J 88 m
SOLUTION: 53 – 35 = 18
So, the correct choice is F.
43. Twenty gallons of lemonade were poured into two containers of different sizes. Express the amount of lemonade
poured into the smaller container in terms of g, the amount poured into the larger container.
A g + 20
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B 20 + g
C g − 20
D J 88 m
SOLUTION: 11-853
Rational
Equations
– 35 = 18
So, the correct choice is F.
43. Twenty gallons of lemonade were poured into two containers of different sizes. Express the amount of lemonade
poured into the smaller container in terms of g, the amount poured into the larger container.
A g + 20
B 20 + g
C g − 20
D 20 − g
SOLUTION: Total – amount in larger container = amount in smaller container.
20 – g = the amount in the smaller container. So, the correct choice is D.
44. GRIDDED RESPONSE The gym has 2-kilogram and 5-kilogram disks for weight lifting. They have fourteen disks
in all. The total weight of the 2-kilogram disks is the same as the total weight of the 5-kilogram disks. How many 2kilogram disks are there?
SOLUTION: Let a = the number of 2-kg discs and b = the number of 5-kg discs.
There are 10 2-kilogram disks at the gym.
Simplify each expression.
45. SOLUTION: 46. SOLUTION: eSolutions Manual - Powered by Cognero
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11-8 Rational Equations
46. SOLUTION: 47. SOLUTION: 48. SOLUTION: eSolutions Manual - Powered by Cognero
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11-8 Rational Equations
48. SOLUTION: Find the LCM of each pair of polynomials.
49. 2h, 4h
2
SOLUTION: Find the prime factors of each expression.
2h = 2 · h 2
4h = 2 · 2 · h · h use each prime factor, the greatest number of times it appears in either of the factorizations. 2h = 2 · h 2
4h = 2 · 2 · h · h 2
The LCM is 2 · 2 · h · h or 4h .
2
50. 5c , 12c
3
SOLUTION: Find the prime factors of each expression.
2
5c = 5 · c · c 3
12c = 2 · 2 · 3 · c · c · c
Use each prime factor, the greatest number of times it appears in either of the factorizations. 2
5c = 5 · c · c 3
12c = 2 · 2 · 3 · c · c · c
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The LCM is 2 · 2 · 3 · 5 · c · c · c or 60c .
2h = 2 · h 2
4h = 2 · 2 · h · h 2
11-8The
Rational
Equations
LCM is
2 · 2 · h · h or 4h .
2
50. 5c , 12c
3
SOLUTION: Find the prime factors of each expression.
2
5c = 5 · c · c 3
12c = 2 · 2 · 3 · c · c · c
Use each prime factor, the greatest number of times it appears in either of the factorizations. 2
5c = 5 · c · c 3
12c = 2 · 2 · 3 · c · c · c
3
The LCM is 2 · 2 · 3 · 5 · c · c · c or 60c .
51. x − 4, x + 2
SOLUTION: The LCM is (x − 4)(x + 2).
52. p − 7, 2(p − 14)
SOLUTION: The LCM is 2(p − 7)(p − 14).
Look for a pattern in each table of values to determine which kind of model best describes the data.
53. SOLUTION: Find the first differences:
Since the first differences are all equal, the table of values represents a linear function. 54. SOLUTION: Find the first differences. The first differences are not all equal. So, the table of values does not represent a linear function.
Find the second differences and compare.
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Since the first differences are all equal, the table of values represents a linear function. 11-8 Rational Equations
54. SOLUTION: Find the first differences. The first differences are not all equal. So, the table of values does not represent a linear function.
Find the second differences and compare.
The second differences are not all equal. So, the table of values does not represent a quadratic function.
Find the ratios of the y -values and compare.
The rations of successive y-values are equal. Therefore, the table of values can be modeled by an exponential
function.
55. SOLUTION: Find the first differences. The first differences are not all equal. So, the table of values does not represent a linear function.
Find the second differences and compare.
Since, the second differences are all equal, a quadratic functions models the data.
56. eSolutions Manual - Powered by Cognero
SOLUTION: Find the first differences. Page 44
11-8 Rational Equations
Since, the second differences are all equal, a quadratic functions models the data.
56. SOLUTION: Find the first differences. The first differences are all equal. So, the table of values represent a linear function.
57. GENETICS Brown genes B are dominant over blue genes b. A person with genes BB or Bb has brown eyes.
Someone with genes bb has blue eyes. Mrs. Dunn has brown eyes with genes Bb, and Mr. Dunn has blue eyes.
Write an expression for the possible eye coloring of their children. Then find the probability that a child would have
blue eyes.
SOLUTION: First, make a chart of the possible outcomes.
There are 4 possible outcomes: two that come out with genes Bb, and two that come out with genes bb. This could
2
be expressed by the equation 0.5Bb + 0.5b . Since B is the dominant gene, there are two outcomes that produce
brown eyes and two outcomes that produce blue eyes.
Therefore, the probability that a child would have blue eyes is
Solve each inequality. Check your solution.
58. SOLUTION: The solution is {b|b ≤ 50}.
Verify the inequality by substituting the value 40 for b. . eSolutions Manual - Powered by Cognero
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2
be expressed by the equation 0.5Bb + 0.5b . Since B is the dominant gene, there are two outcomes that produce
brown eyes and two outcomes that produce blue eyes.
11-8Therefore,
Rational Equations
the probability that a child would have blue eyes is
Solve each inequality. Check your solution.
58. SOLUTION: The solution is {b|b ≤ 50}.
Verify the inequality by substituting the value 40 for b. . 59. SOLUTION: The solution is {r|r > 49}.
Verify by substituting the value 70 for r. 60. SOLUTION: eSolutions Manual - Powered by Cognero
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11-8 Rational Equations
60. SOLUTION: The solution is {y|y ≥ −24}.
Verify by substituting the value –8 for y . Determine the probability of each event if you randomly select a marble from a bag containing 9 red
marbles, 6 blue marbles, and 5 yellow marbles.
61. P(blue)
SOLUTION: 62. P(red)
SOLUTION: 63. P(not yellow)
SOLUTION: eSolutions Manual - Powered by Cognero
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11-8 Rational Equations
63. P(not yellow)
SOLUTION: eSolutions Manual - Powered by Cognero
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