11-8 Rational Equations
Solve each equation. State any extraneous
solutions.
2. 1. SOLUTION: Multiply each side of the equation by the LCD to
eliminate the denominators.
SOLUTION: Multiply each side of the equation by the LCD to
eliminate the denominators.
The LCD is 45.
The LCD is x(x + 1).
The solution is t = 12.
The solution is x = –2.
Check:
Check: 3. 2. SOLUTION: Multiply each side of the equation by the LCD to
eliminate the denominators.
The LCD is 45.
SOLUTION: Multiply each side of the equation by the LCD to
eliminate the denominators.
The LCD is 5a.
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The solution is t = 12.
Check: Page 1
11-8 Rational Equations
3. 4. SOLUTION: Multiply each side of the equation by the LCD to
eliminate the denominators.
The LCD is 5a.
SOLUTION: Multiply each side of the equation by the LCD to
eliminate the denominators.”
The LCD is (p – 1).
The solution is p = 2.
The solution is a =
Check. .
Check. 5. SOLUTION: Multiply each side of the equation by the LCD to
eliminate the denominators.
The LCD is (t + 1)(t – 1).
4. SOLUTION: Multiply each side of the equation by the LCD to
eliminate
denominators.”
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The LCD is (p – 1).
The solution is t = –3.
Check. Page 2
eliminate the denominators.”
2
11-8 Rational Equations
The factors of (x – 1) are (x – 1)(x + 1). Thus, the
2
2
LCD of (x – 1) and (x – 1) is (x – 1).
5. SOLUTION: Multiply each side of the equation by the LCD to
eliminate the denominators.
The LCD is (t + 1)(t – 1).
Since x = 1 results in a zero in the denominator of the
original equation, it is an extraneous solution. The
solution is x =
The solution is t = –3.
Check. Check:
.
6. SOLUTION: Multiply each side of the equation by the LCD to
eliminate the denominators.”
2
The factors of (x – 1) are (x – 1)(x + 1). Thus, the
2
2
LCD of (x – 1) and (x – 1) is (x – 1).
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?
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Page 3
SOLUTION: You need to find the rate that each person works and
It would take Maurice and Olinda
11-8 Rational Equations
hour or about 0.4 hour to weed the garden if they work together.
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.
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
or 2 min, so the rate of the faucet Find the fraction of the job that each person can do in
an hour.
45 minutes is
of an hour. 50 minutes is of an hour.
is
Maurice’s rate is
is
job per hour. Olinda’s rate
.
Hunter can fill the bucket from the hose in
or 2.4 min, so the rate of the hose
job per hour.
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. 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 integer
LCD is 12.
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
Using both the faucet and the hose, it would take
Hunter about 1.1 minutes to fill the 3.5-gallon bucket.
hour or about Solve each equation. State any extraneous
solutions.
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 Manual
flows at
a rate of
gallons per minute, how
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many minutes would it take him to fill the bucket?
Round to the nearest tenth.
9. SOLUTION: Multiply each side of the equation by the LCD toPage 4
eliminate the denominators.
11-8Using
Rational
bothEquations
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.
10. 9. SOLUTION: Multiply each side of the equation by the LCD to
eliminate the denominators.
The LCD is t(t + 2).
SOLUTION: Multiply each side of the equation by the LCD to
eliminate the denominators.
The LCD is n(n – 5).
The solution is t = 4.
The solution is n = 8.
Check: Check:
11. 10. SOLUTION: Multiply each side of the equation by the LCD to
eliminate the denominators.
The LCD is t(t + 2).
SOLUTION: Multiply each side of the equation by the LCD to
eliminate the denominators.
The LCD is 12.
The solution is t = 4.
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The solution is
Check: Check:
.
Page 5
11-8 Rational Equations
11. 12. SOLUTION: Multiply each side of the equation by the LCD to
eliminate the denominators.
The LCD is 12.
SOLUTION: Multiply each side of the equation by the LCD to
eliminate the denominators.
The LCD is 8.
The solution is
.
The solution is
Check:
.
Check:
12. SOLUTION: Multiply each side of the equation by the LCD to
eliminate the denominators.
The LCD is 8.
13. SOLUTION: Multiply each side of the equation by the LCD to
eliminate the denominators.
The LCD is 15w.
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Page 6
11-8 Rational Equations
13. 14. SOLUTION: Multiply each side of the equation by the LCD to
eliminate the denominators.
The LCD is 15w.
SOLUTION: Multiply each side of the equation by the LCD to
eliminate the denominators.
The LCD is (c + 1)(c – 1).
The solution is w = –13.
Check:
The solution is
.
Check:
14. SOLUTION: Multiply each side of the equation by the LCD to
eliminate the denominators.
The LCD is (c + 1)(c – 1).
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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Page 7
11-8 Rational Equations
15. 16. SOLUTION: Multiply each side of the equation by the LCD to
eliminate the denominators.
The LCD is (x + 1)( x– 1).
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
The solution is x = 0.
Check:
y = –8
16. SOLUTION: Multiply each side of the equation by the LCD to
eliminate the denominators.
The LCD is (y + 3)(y – 2).
17. The solution is y = –4 or –8.
Check: y = –4
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SOLUTION: Multiply each side of the equation by the LCD to
eliminate the denominators.
The LCD is (a +3).
Page 8
11-8 Rational Equations
18. 17. SOLUTION: Multiply each side of the equation by the LCD to
eliminate the denominators.
SOLUTION: Multiply each side of the equation by the LCD to
eliminate the denominators.
The LCD is (a +3).
2
The factors of (a – 9) are (a + 3)(a – 3). Thus, the
LCD is (a + 3)(a – 3).
The solution is Check:
The solution is a = –2 or 3.
Check: a = –2
.
a=3
19. SOLUTION: Multiply each side of the equation by the LCD to
eliminate the denominators.
The LCD is (n –1).
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
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LCD is (a + 3)(a – 3).
Page 9
Since n = 1 results in a zero in the denominator of the
original equation, it is an extraneous solution. There is
no solution.
11-8 Rational Equations
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?
19. SOLUTION: Multiply each side of the equation by the LCD to
eliminate the denominators.
The LCD is (n –1).
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. Since n = 1 results in a zero in the denominator of the
original equation, it is an extraneous solution. There is
no solution.
To solve, multiply each side of the equation by the
LCD to eliminate the denominators. The LCD is 15.
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).
It would take Noah and Gilberto
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.
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Find the fraction of the job that each person can do in
an hour. 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
Page 10
they work together.
Ron’s rate is
job per hour. Chris’s rate is
It would take Noah and Gilberto
hours or hours to paint one side of a fence if they work 11-8 Rational Equations
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?
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 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. 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
machine 2 is
. The rate of ice-
.
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 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.
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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
Page 11
trail in 28 minutes. At what time do they pass each
other?
They will pass each other at 12.32 minutes.
11-8ItRational
Equations
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?
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.
SOLUTION: You need to find the rate that each person rides. If , then
.
The first cyclist’s rate is
rate is
. The second cyclist’s
.
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.
[-10, 10] scl: 1 by [-10, 10] scl: 1
b. c. 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
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graph is a line.
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 a graphing calculator, you can see thatPage
the 12
graph is a parabola.
Method 2: Find the zeros algebraically.
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.
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.
[-5, 5] scl: 1 by [-5, 5] scl: 1
b. [-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. [-5, 5] scl: 1 by [-5, 5] scl: 1
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?
[-5, 5] scl: 1 by [-5, 5] scl: 1
Method 2: Find the zeros algebraically.
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eSolutions
The zeros of the function are 0 and 1.
27. 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).
Page 13
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.
11-8The
Rational
squareEquations
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).
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. 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
to add.
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a. Copy and complete the table below.
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: Page 14
a. 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?
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
The factors of (b – 3b) is b(b – 3). Thus, the LCD
2
of (b – 3b) and b is b(b – 3).
SOLUTION: a. b.
The solution is b = 1.
Check:
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.
33. SOLUTION: 2
The factors of (b – 3b) is b(b – 3). Thus, the LCD
2
of (b – 3b) and b is b(b – 3).
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Check each of the solutions.
Page 15
The solutions are x = 0, –4; extraneous: 1.
11-8 Rational Equations
34. Check each of the solutions.
SOLUTION: Multiply each side of the equation by the LCD to
eliminate the denominators.
3
2
The factors y – 2y . are
3
2
. Thus the
3
2
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:
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.
34. SOLUTION: Multiply each side of the equation by the LCD to
eliminate the denominators.
3
2
The factors y – 2y . are
3 by Cognero
2
3
eSolutions Manual - Powered
. Thus the
2
LCD of y and y – 2y is y – 2y .
35. SOLUTION: 2
The factors of (x – 5x) are x(x – 5).Multiply each
side of the equation by the LCD to eliminate thePage 16
2
denominators. The LCD x, (x – 5) and (x – 5x) is x
(x – 5).
SOLUTION: Multiply each side of the equation by the LCD to
eliminate the denominators.”
11-8 Rational Equations
The LCD is (x + 1)(x – 2).
35. SOLUTION: 2
The factors of (x – 5x) are x(x – 5).Multiply each
side of the equation by the LCD to eliminate the
2
denominators. The LCD x, (x – 5) and (x – 5x) is x
(x – 5).
Check each of the possible solutions: x = –2 x = The solution is x =
.
Check:
36. CHALLENGE Solve
Therefore, the solutions are x = –2 and
.
.
SOLUTION: Multiply each side of the equation by the LCD to
eliminate the denominators.”
The LCD is (x + 1)(x – 2).
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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.
Page 17
For example, consider the rational expression
. The expression is undefined at
11-8Therefore,
Rational Equations
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 excluded value. If you solve the equation
, the solution is x
= 5. However, x = 5 results in a zero in the
denominator of the original equation, it is therefore an
extraneous 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. 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.
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.
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.
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Then multiply each side of the equation by the LCD.
Simplify and use the order of operations to solve for
the variable.
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. Page 18
It would take Odell 12 hours to build the fence alone.
So, the correct choice is D.
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
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.
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. 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.
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. 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 2kilogram disks is the same as the total weight of the
5-kilogram disks. How many 2- kilogram disks are
there?
SOLUTION: Let a = the number of 2-kg discs and b = the number
of 5-kg discs.
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
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SOLUTION: 53 – 35 = 18
So, the correct choice is F.
There are 10 2-kilogram disks at the gym.
Simplify each expression.
45. SOLUTION: Page 19
11-8 Rational Equations
There are 10 2-kilogram disks at the gym.
Simplify each expression.
48. 45. SOLUTION: SOLUTION: Find the LCM of each pair of polynomials.
2
49. 2h, 4h
SOLUTION: Find the prime factors of each expression.
46. 2h = 2 · h 2
SOLUTION: 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
47. 3
SOLUTION: Find the prime factors of each expression.
SOLUTION: 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
48. 3
The LCM is 2 · 2 · 3 · 5 · c · c · c or 60c .
51. x − 4, x + 2
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SOLUTION: SOLUTION: The LCM is (x − 4)(x + 2).
Page 20
2
5c = 5 · c · c 3
12c = 2 · 2 · 3 · c · c · c
Since the first differences are all equal, the table of
values represents a linear function. 11-8 Rational Equations
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).
54. SOLUTION: Find the first differences. 52. p − 7, 2(p − 14)
SOLUTION: The LCM is 2(p − 7)(p − 14).
The first differences are not all equal. So, the table
of values does not represent a linear function.
Find the second differences and compare.
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:
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.
Since the first differences are all equal, the table of
values represents a linear function. The rations of successive y-values are equal.
Therefore, the table of values can be modeled by an
exponential 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.
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.
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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.
The rations of successive y-values are equal.
Page 21
The rations of successive y-values are equal.
Therefore, the table of values can be modeled by an
function.
11-8exponential
Rational Equations
The first differences are all equal. So, the table of values represent a linear function.
55. SOLUTION: Find the first differences. 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.
The first differences are not all equal. So, the table
of values does not represent a linear function.
Find the second differences and compare.
There are 4 possible outcomes: two that come out
with genes Bb, and two that come out with genes
bb. This could be expressed by the equation 0.5Bb +
2
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
Since, the second differences are all equal, a
quadratic functions models the data.
blue eyes is
56. SOLUTION: Find the first differences. Solve each inequality. Check your solution.
58. SOLUTION: 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.
The solution is {b|b ≤ 50}.
Verify the inequality by substituting the value 40 for
b. . 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 be expressed by the equation 0.5Bb +
2
0.5b . Since B is the dominant gene, there are two
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Page 22
59. SOLUTION: 11-8 Rational Equations
59. SOLUTION: 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: The solution is {r|r > 49}.
62. P(red)
Verify by substituting the value 70 for r. SOLUTION: 63. P(not yellow)
SOLUTION: 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.
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61. P(blue)
SOLUTION: Page 23