Download 12.7Solve Rational Equations

12.7
Solve Rational Equations
You simplified rational expressions.
Before
You will solve rational equations.
Now
So you can calculate a hockey statistic, as in Ex. 31.
Why?
Key Vocabulary
• rational equation
• cross product, p. 168
• extraneous
solution, p. 730
• least common
denominator
(LCD) of rational
expressions, p. 813
A rational equation is an equation that contains one or more
rational expressions. One method for solving a rational equation
is to use the cross products property. You can use this method
when both sides of the equation are single rational expressions.
EXAMPLE 1
Use the cross products property
6
x
Solve }
5}
. Check your solution.
x14
6
x14
2
x
2
}5}
REVIEW CROSS
PRODUCTS
Write original equation.
12 5 x2 1 4x
For help with using the
cross products property,
see p. 168.
Cross products property
2
0 5 x 1 4x 2 12
Subtract 12 from each side.
0 5 (x 1 6)(x 2 2)
Factor polynomial.
x1650
or x 2 2 5 0
x 5 26 or
Zero-product property
x52
Solve for x.
c The solutions are 26 and 2.
CHECK
If x 5 26:
26
6
}0}
26 1 4
2
23 5 23 ✓
✓
GUIDED PRACTICE
If x 5 2:
6
2
}0}
214
2
151✓
for Example 1
Solve the equation. Check your solution.
y
3
5
1. }
5}
y22
z
7
2
2. }
5}
z15
USING THE LCD Given an equation with fractional coefficients such as
3
2
1
} x 1 } 5 }, you can multiply each side by the least common
4
3
6
denominator (LCD), 12. The equation becomes 8x 1 2 5 9, which you
may find easier to solve than the original equation. You can use this
method to solve a rational equation.
820
Chapter 12 Rational Equations and Functions
EXAMPLE 2
Multiply by the LCD
x
1
2
Solve }
1}
5}
. Check your solution.
x22
x22
5
x
x22
1
5
2
x22
Write original equation.
2
x22
Multiply by LCD, 5(x 2 2).
}1}5}
x
x22
1
5
} p 5(x 2 2) 1 } p 5(x 2 2) 5 } p 5(x 2 2)
x p 5(x 2 2)
5(x 2 2)
2 p 5(x 2 2)
}1}5 }
5
x22
x22
Multiply and divide out
common factors.
5x 1 x 2 2 5 10
Simplify.
6x 2 2 5 10
Combine like terms.
6x 5 12
Add 2 to each side.
x52
AVOID ERRORS
Be sure to identify the
excluded values for the
rational expressions in
the original equation.
Divide each side by 6.
x
2
The solution appears to be 2, but the expressions }
and }
are
x22
undefined when x 5 2. So, 2 is an extraneous solution.
x22
c There is no solution.
EXAMPLE 3
Factor to find the LCD
3
8
Solve }
1 1 5 }}
. Check your solution.
2
x27
x 2 9x 1 14
Solution
Write each denominator in factored form. The LCD is (x 2 2)(x 2 7).
3
x27
8
(x 2 2)(x 2 7)
} 1 1 5 }}
8
(x 2 2)(x 2 7)
3
x27
} p (x 2 2)(x 2 7) 1 1 p (x 2 2)(x 2 7) 5 }} p (x 2 2)(x 2 7)
3(x 2 2)(x 2 7)
8(x 2 2)(x 2 7)
}} 1 (x 2 2)(x 2 7) 5 }}
x27
(x 2 2)(x 2 7)
3(x 2 2) 1 (x 2 2 9x 1 14) 5 8
x 2 2 6x 1 8 5 8
x 2 2 6x 5 0
x(x 2 6) 5 0
x 5 0 or x 2 6 5 0
x 5 0 or
x56
c The solutions are 0 and 6.
If x 5 0:
CHECK
8
0 2 9 p 0 1 14
} 1 1 0 }}
2
3
027
4
7
4
7
}5} ✓
If x 5 6:
8
6 2 9 p 6 1 14
1 1 0 }}
}
2
2
3
627
22 5 22 ✓
12.7 Solve Rational Equations
821
EXAMPLE 4
Solve a multi-step problem
PAINT MIXING You have an 8 pint mixture of paint that is made up of equal
amounts of yellow paint and blue paint. To create a certain shade of green,
you need a paint mixture that is 80% yellow. How many pints of yellow paint
do you need to add to the mixture?
ANOTHER WAY
Solution
For an alternative
method for solving
the problem in
Example 4, turn to
page 827 for the
Problem Solving
Workshop.
Because the amount of yellow paint equals the amount of blue paint, the
mixture has 4 pints of yellow paint. Let p represent the number of pints of
yellow paint that you need to add.
STEP 1 Write a verbal model. Then write an equation.
Pints of
yellow paint
in mixture
Pints of
yellow paint
needed
1
Pints of
paint in
mixture
5
Desired
percent yellow
in mixture
5
0.8
Pints of
yellow paint
needed
1
41p
81p
}
STEP 2 Solve the equation .
41p
} 5 0.8
81p
Write equation.
4 1 p 5 0.8(8 1 p)
Cross products property
4 1 p 5 6.4 1 0.8p
Distributive property
0.2p 5 2.4
p 5 12
Rewrite equation.
Solve for p.
c You need to add 12 pints of yellow paint.
41p
} 5 0.8
81p
CHECK
4 1 12 0
0.8
}
8 1 12
} 0 0.8
16
20
0.8 5 0.8 ✓
✓
GUIDED PRACTICE
Write original equation.
Substitute 12 for p.
Simplify numerator and denominator.
Write fraction as decimal. Solution checks.
for Examples 2, 3, and 4
Solve the equation. Check your solution.
1
212
a
3. }
1}
5}
a14
3
a14
22
n
4. }
2 1 5 }}
2
n 2 11
n 2 5n 2 66
5. WHAT IF? In Example 4, suppose you need a paint mixture that is 75%
yellow. How many pints of yellow paint do you need to add to the mixture?
822
Chapter 12 Rational Equations and Functions
12.7
EXERCISES
HOMEWORK
KEY
5 WORKED-OUT SOLUTIONS
on p. WS1 for Exs. 7, 15, and 33
★ 5 STANDARDIZED TEST PRACTICE
Exs. 2, 24, 28, and 35
SKILL PRACTICE
3
7
1. VOCABULARY The equation } 5 }
x 1 4 is an example of a(n) ? .
x21
2.
EXAMPLE 1
on p. 820
for Exs. 3–13, 24
★ WRITING Describe two methods for solving a rational equation. Which
method can you use to solve any kind of rational equation? Explain.
SOLVING EQUATIONS Solve the equation. Check your solution.
r
s
10
2
5
3
3. }
4. }
5}
5. }
t5}
r 5}
20
t26
s 2 13 10
25
c21
c13
m14
15
w
9. }
5}
2
n23
n26
3
m21
2m
7. }
5}
2
6. }
5}
2y
x
x24
24
11. } 5 }
y
y23
2x
10. }
5}
w11
42x
n11
n15
8. } 5 }
ERROR ANALYSIS Describe and correct the error in solving the equation.
3
3
x11
11
12. }
5}
13. 4x
}5}
5
2x
2x 1 2
8x 2 1
x11
2x 1 2
3
2x
}5}
(x 1 1)2x 5 3(2x 1 2)
2x 1 2x 5 6x 1 6
2
2x 2 4x 2 6 5 0
2
2(x 2 3)(x 1 1) 5 0
x2350
or
x53
or
x1150
x 5 21
4x 1 1
8x 2 1
3
}5}
5
5(4x 1 1) 5 3(8x 2 1)
20x 1 1 5 24x 2 3
1 5 4x 2 3
4 5 4x
15x
The solution is 1.
The solutions are 3 and 21.
EXAMPLES
2 and 3
on p. 821
for Exs. 14–23
SOLVING EQUATIONS Solve the equation. Check your solution.
3
21
6x
z
14. }
115}
15. }
235}
z17
x 2 11
z17
x 2 11
a 1 10
17
16. a
}215}
2a 1 8
a14
3m
m24
22m 1 2
m 2 6m 1 8
m
18. }
2 } 5 }}
2
m22
26
p 2 3p 1 2
2
3
20. }
2}
5}
2
p21
p21
r12
22. }
5}
2
2
r 1 6r 2 7
24.
8
r 1 3r 2 4
b2 2 3
b 1 12b 1 27
1
17. }
1 2 5 }}
2
b13
n14
n21
n 21
q
2q 2 27
5
21. }
5}1}
q23
q14
q2 1 q 2 12
12
3n
19. }
5}
1}
2
n11
4 2 5s
s22
9
23. }
5}
2
s 24
★ OPEN – ENDED
Write a rational equation that can be solved using
the cross products property. Then solve the equation.
12.7 Solve Rational Equations
823
x
2
25. REASONING Consider the equation }
x2a5}
x 2 a where a is a real number.
For what value(s) of a does the equation have exactly one solution? no
solution? Explain your answers.
26. USING ANOTHER METHOD Another way to solve a rational equation is
to write each side of the equation as a single rational expression and
then use the cross products property. Use this method to solve the
x22
2
2x 2 1
4
x
equation }
1 } 5 }.
x11
27. SOLVING SYSTEMS OF EQUATIONS Consider the following system:
y 5 3x 1 1
25
x23
y5} 26
a. Solve the system algebraically.
b. Check your solution by graphing the equations.
28.
★ MULTIPLE CHOICE Let a be a real number. How many solutions does
2a
x 2a
2
1
the equation }
have?
2
2
x2a5}1}
x1a
A Zero
B One
C Two
x1a
x111a
D Infinitely many
x
x11
29. REASONING Is the expression } ever equivalent to } for some
nonzero value of a? Justify your answer algebraically.
30. CHALLENGE Let a and b be real numbers. The solutions of the equation
30
x12
ax 1 b 5 } 2 1 are 28 and 8. What are the values of a and b? Explain
your answer.
PROBLEM SOLVING
EXAMPLE 4
on p. 822
for Exs. 31–34
31. ICE HOCKEY In ice hockey, a goalie’s save percentage
(in decimal form) is the number of shots blocked by
a goalie divided by the number of shots made by an
opposing team. Suppose a goalie has blocked
160 out of 200 shots. How many consecutive shots
does the goalie need to block in order to raise
the save percentage to 0.840?
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
32. RUNNING TIMES You are running a 6000 meter
charity race. Your average speed in the first half of the race
is 50 meters per minute faster than your average speed in the second
half. You finish the race in 27 minutes. What is your average speed in the
second half of the race?
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
824
5 WORKED-OUT SOLUTIONS
on p. WS1
★ 5 STANDARDIZED
TEST PRACTICE
33. CLEANING SOLUTIONS You have a cleaning solution that consists of
2 cups of vinegar and 7 cups of water. You need a cleaning solution that
consists of 5 parts water and 1 part vinegar in order to clean windows.
How many cups of water do you need to add to your cleaning solution so
that you can use it to clean windows?
34. MULTI-STEP PROBLEM Working together, a painter and
an assistant can paint a certain room in 2 hours. The
painter can paint the room alone in half the time it takes
the assistant to paint the room alone. Let t represent the
time (in hours) that the painter can paint the room alone.
a. Copy and complete the table.
Person
Fraction
of room
painted
each hour
Time
(hours)
Fraction
of room
painted
Painter
}
1
t
2
?
Assistant
?
2
?
b. Explain why the sum of the expressions in the fourth column of the
table must be 1.
c. Write a rational equation that you can use to find the time that the
painter takes to paint the room alone. Then solve the equation.
d. How long does the assistant take to paint the room alone?
35.
★
EXTENDED RESPONSE You and your sister can rake a neighbor’s front
lawn together in 30 minutes. Your sister takes 1.5 times as long as you to
rake the lawn by herself.
a. Solve Write an equation that you can use to find the time t (in
minutes) you take to rake the lawn by yourself. Then solve the
equation.
b. Compare With more experience, both of you can now rake the lawn
together in 20 minutes, and your sister can rake the lawn alone in
the same amount of time as you. Tell how you would change the
equation in part (a) in order to describe this situation. Then solve
the equation.
c. Explain Explain why your solution of the equation in part (b) makes
sense. Then justify your explanation algebraically for any given
amount of time that both of you rake the lawn together.
36. TELEVISION The average time t (in minutes) that a person in the United
States watched television per day during the period 1950–2000 can be
modeled by
1 8.85x
t 5 265
}
1 1 0.0114x
where x is the number of years since 1950.
a. Approximate the year in which a person watched television for an
average of 6 hours per day.
b. About how many years had passed when the average time a person
spent watching television per day increased from 5 hours to 7 hours?
12.7 Solve Rational Equations
825
37. SCIENCE Atmospheric pressure, measured in pounds per square inch
(psi), is the pressure exerted on an object by the weight of the atmosphere
above the object. The atmospheric pressure p (in psi) can be modeled by
14.55(56,267 2 a)
55,545 1 a
p 5 }}
where a is the altitude (in feet). Is the change in altitude greater when the
atmospheric pressure changes from 10 psi to 9 psi or from 8 psi to 7 psi?
Explain your answer.
38. CHALLENGE Butterfat makes up about 1% of the volume of milk in 1%
milk. Butterfat can make up no more than 0.2% of the volume of milk
in skim milk. A container holds 15 fluid ounces of 1% milk. How many
fluid ounces of butterfat must be removed in order for the milk to be
considered skim milk? Round your answer to the nearest hundredth.
MIXED REVIEW
PREVIEW
Prepare for
Lesson 13.1 in
Exs. 39–44.
Write the fraction as a decimal and as a percent. Round decimals to the
nearest thousandth. Round percents to the nearest tenth of a percent. (p. 916)
7
10
1
39. }
1
40. }
41. }
24
42. }
25
43. }
44. }
46. 6[4 2 (16 2 14)2 ] (p. 8)
47. Ï 289 (p. 110)
32
49. }
(p. 495)
27
42
50. }
(p. 495)
5
4
8
7
2
30
25
Evaluate the expression.
45. (92 2 7) 4 2 (p. 8)
}
48. 6Ï 1600 (p. 110)
1 18 2
51. 2}
3
(p. 495)
}
4
3
27
3 10
52. 1.61
(p. 512)
}
23
53. (1.2 3 106 )2 (p. 512)
2.3 3 10
QUIZ for Lessons 12.5–12.7
Find the product or quotient. (p. 802)
4x 3
15
y2
3y2 1 6y
2. }
4}
2
5
1. }
p}
2
8x
y24
y 2 16
Find the sum or difference. (p. 812)
5a 2 1
a 1 11
8a
3. }
2}
a 1 11
n21
n 1 5n 1 6
6n
4. }
1}
2
n13
Solve the equation. Check your solution. (p. 820)
z
z23
2z
5. }
5}
z15
32x
24
2x
6. }
2
x 1}5}
x11
x 1x
7. BATTING AVERAGES A softball player’s batting average is the number of
hits divided by the number of times at bat. A softball player has a batting
average of .200 after 90 times at bat. How many consecutive hits does the
player need in order to raise the batting average to .250? (p. 820)
826
EXTRA PRACTICE for Lesson 12.7, p. 949
ONLINE QUIZ at classzone.com
Using
ALTERNATIVE METHODS
LESSON 12.7
Another Way to Solve Example 4, page 822
MULTIPLE REPRESENTATIONS In Example 4 on page 822, you saw how to
solve a problem about mixing paint by using a rational equation. You can
also solve the problem by using a table or by reinterpreting the problem.
PROBLEM
PAINT MIXING You have an 8 pint mixture of paint that is made up of
equal amounts of yellow paint and blue paint. To create a certain shade
of green, you need a paint mixture that is 80% yellow. How many pints
of yellow paint do you need to add to the mixture?
METHOD 1
Use a Table One alternative approach is to use a table.
The mixture has 8 pints of paint. Because the mixture has an equal
amount of yellow paint and blue paint, the mixture has 8 4 2 5 4 pints of
yellow paint.
STEP 1 Make a table that shows the percent of the mixture that is yellow paint
after you add various amounts of yellow paint.
A mixture with 6 pints of yellow
paint is the result of adding
2 pints of yellow paint to the
mixture.
This amount of yellow paint
gives you the percent yellow
you want.
Yellow paint
(pints)
Paint in mixture
(pints)
Percent of mixture
that is yellow paint
4
8
} 5 50%
6
10
} 5 60%
8
12
}
8
12
ø 67%
10
14
}
10
14
ø 71%
12
16
} 5 75%
14
18
}
14
18
ø 78%
15
19
}
15
19
ø 79%
16
20
} 5 80%
4
8
6
10
12
16
16
20
STEP 2 Find the number of pints of yellow paint needed. Subtract the number
of pints of yellow paint already in the mixture from the total number
of pints of yellow paint you have: 16 2 4 5 12.
c You need to add 12 pints of yellow paint.
Using Alternative Methods
827
METHOD 2
Reinterpret Problem Another alternative approach is to reinterpret
the problem.
STEP 1 Reinterpret the problem. A mixture with 80% yellow paint means
4
1
that }
of the mixture is yellow and }
of the mixture is blue. So, the
5
5
ratio of yellow paint to blue paint needs to be 4 : 1. You need 4 times
as many pints of yellow paint as pints of blue paint.
STEP 2 Write a verbal model. Then write an equation. Let p represent the
number of pints of yellow paint that you need to add.
Pints of yellow
paint already 1
in mixture
Pints of yellow
paint you need
to add
1
4
p
Pints of
5 4 p blue paint
in mixture
5 4 p
4
STEP 3 Solve the equation.
41p54p4
Write equation.
4 1 p 5 16
Multiply.
p 5 12
Subtract 4 from each side.
c You need to add 12 pints of yellow paint to the mixture.
P R AC T I C E
1. INVESTING Jill has $10,000 in various
investments, including $1000 in a mutual
fund. Jill wants the amount in the mutual
fund to make up 20% of the amount in all of
her investments. How much money should
she add to the mutual fund? Solve this
problem using two different methods.
3. BASKETBALL A basketball player has made
40% of 30 free throw attempts so far. How
many consecutive free throws must the
player make in order to increase the percent
of free throw attempts made to 50%? Solve
this problem using two different methods.
4. WHAT IF? In Exercise 3, suppose the
2. ERROR ANALYSIS Describe and correct the
error in solving Exercise 1.
Amount in
mutual fund
Amount in all
investments
Percent in
mutual fund
1000
1400
1800
2250
10,000
10,400
10,800
11,250
10%
About 13%
About 17%
20%
Jill needs to add $2250 to her mutual fund.
828
Chapter 12 Rational Equations and Functions
basketball player instead wants to increase
the percent of free throw attempts made to
60%. How many consecutive free throws
must the player make?
5. SNOW SHOVELING You and your friend are
shoveling snow out of a driveway. You can
shovel the snow alone in 50 minutes. Both
of you can shovel the snow in 30 minutes
when working together. How many minutes
will your friend take to shovel the snow
alone? Solve this problem using two different
methods.
MIXED REVIEW of Problem Solving
STATE TEST PRACTICE
classzone.com
Lessons 12.5–12.7
1. MULTI-STEP PROBLEM For the period
1991–2002, the average total revenue T (in
dollars per admission) that a movie theater
earned and the average revenue C (in dollars
per admission) from concessions in the
United States can be modeled by
2
0.018x 1 5.4
1 2 0.0011x
2
0.013x 1 1.1
0.0011x 1 1
T 5 }}2 and C 5 }}
2
where x is the number of years since 1991.
a. Write a model that gives the percent p
(in decimal form) of the average total
revenue per admission that came from
concessions as a function of x.
b. About what percent of the average total
revenue came from concessions in 2001?
2. SHORT RESPONSE The diagram of the
truck shows the distance between the first
axle and the last axle for each of two groups
of consecutive axles.
3. MULTI-STEP PROBLEM A rower travels
5 miles upstream (against the current) and
5 miles downstream (with the current). The
speed of the current is 1 mile per hour.
a. Write an equation that gives the total
travel time t (in hours) as a function of the
rower’s average speed r (in miles per hour)
in still water.
b. Find the total travel time if the rower’s
average speed in still water is 7 miles per
hour.
4. GRIDDED ANSWER You take 7 minutes to fill
your washing machine tub using only the
cold water valve. You take 4 minutes to fill
the tub using both the cold water valve and
the hot water valve. How many minutes will
you take to fill the tub using only the hot
water valve?
5. OPEN - ENDED Describe a real-world situation
that can be modeled by the equation
115 1 x
} 5 0.75. Explain what the solution of
170 1 x
the equation means in this situation.
6. EXTENDED RESPONSE The number D (in
thousands) of all college degrees earned and
the number M (in thousands) of master’s
degrees earned in the United States during
the period 1984–2001 can be modeled by
The maximum weight W (to the nearest
500 pounds) that a truck on a highway can
carry on a group of consecutive axles is given
by the formula
1
d
n21
W 5 500 } 1 12n 1 36
2
1800 1 17x 2
1 1 0.0062x
280 1 2.5x 2
1 1 0.0040x
D 5 }}2 and M 5 }}2
where x is the number of years since 1984.
a. Write a model that gives the percent p
(in decimal form) of all college degrees
earned that were master’s degrees.
where d is the distance (in feet) between the
first axle and the last axle of the group and n
is the number of axles in the group.
b. Approximate the percent of all college
a. Rewrite the expression on the right
c. Graph the equation in part (a) on a
side of the equation as a single rational
expression. Then find the maximum
weight that the truck shown can carry on
axles 1–3.
b. Can the truck carry 65,500 pounds on
axles 2–5? Explain your answer.
degrees earned that were master’s degrees
in 2000.
graphing calculator. Describe how the
percent of college degrees that were
master’s degrees changed during the
period. Can you use the graph to describe
how the number of master’s degrees
changed during the period? Explain.
Mixed Review of Problem Solving
829