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Solve each equation. Check your solution.
1.
SOLUTION:
Check:
8-6 Solving Rational Equations and Inequalities The solution is 11.
Solve each equation. Check your solution.
2.
1.
SOLUTION:
SOLUTION:
Check:
Check:
The solution is 9.
The solution is 11.
3.
2.
SOLUTION:
SOLUTION:
Check:
eSolutions Manual - Powered by Cognero Check: Page1
The solution is 9.
3.
The solution is 7.
SOLUTION:
4.
SOLUTION:
Check:
Check:
The solution is 7.
The solution is 3.
4.
5.
SOLUTION:
SOLUTION:
Check:
Check:
The solution is 3. The solution is 8.
5.
6.
SOLUTION:
SOLUTION:
Check:
Check:
The solution is 8.
6.
The solution is x = 5.
SOLUTION:
7.
SOLUTION:
Check:
Check:
The solution is x = 5.
The solution is 14.
7.
8.
SOLUTION:
SOLUTION:
Check:
Check:
The solution is 14.
The solution is 14.
8.
CCSS STRUCTURE
9. Sara has 10 pounds of dried
SOLUTION: fruit selling for $6.25 per pound. She wants to know
how many pounds of mixed nuts selling for $4.50 per
pound she needs to make a trail mix selling for $5 per
pound.
a. Let m = the number of pounds of mixed nuts.
Complete the following table.
b. Write a rational equation using the last column of
the table.
Check: c. Solve the equation to determine how many pounds
of mixed nuts are needed.
SOLUTION:
a.
b.
The solution is 14.
c.
CCSS STRUCTURE
9. Sara has 10 pounds of dried
fruit selling for $6.25 per pound. She wants to know
how many pounds of mixed nuts selling for $4.50 per
pound she needs to make a trail mix selling for $5 per
pound.
a. Let m = the number of pounds of mixed nuts. Therefore, 25 pounds of mixed nuts are needed.
Complete the following table.
10. DISTANCEAlicia’s average speed riding her bike
is 11.5 miles per hour. She takes a round trip of 40
miles. It takes her 1 hour and 20 minutes with the
wind and 2 hours and 30 minutes against the wind.
b. Write a rational equation using the last column of a. Write an expression for Alicia’s time with the
the table. wind.
c. Solve the equation to determine how many pounds b. Write an expression for Alicia’s time against the
of mixed nuts are needed. wind.
c. How long does it take to complete the trip?
SOLUTION:
a. d. Write and solve the rational equation to determine
the speed of the wind.
SOLUTION:
a. Let x be the speed of the wind.
The expression for Alicia’s time with the wind is
b.
.
b. The expression for Alicia’s time against the wind
c.
is .
c.
Therefore, 25 pounds of mixed nuts are needed.
d.
10. DISTANCEAlicia’s average speed riding her bike
is 11.5 miles per hour. She takes a round trip of 40
miles. It takes her 1 hour and 20 minutes with the
wind and 2 hours and 30 minutes against the wind.
a. Write an expression for Alicia’s time with the
wind.
b. Write an expression for Alicia’s time against the
wind.
c. How long does it take to complete the trip?
d. Write and solve the rational equation to determine
the speed of the wind.
The speed of the wind is 3.5 mph.
SOLUTION:
a. Let x be the speed of the wind.
The expression for Alicia’s time with the wind is 11. WORK Kendal and Chandi wax cars. Kendal can
. wax a particular car in 60 minutes and Chandi can
wax the same car in 80 minutes. They plan on
waxing the same car together and want to know how
b. The expression for Alicia’s time against the wind long it will take.
is .
a. How much will Kendal complete in 1 minute?
c. b. How much will Kendal complete in x minutes?
c. How much will Chandi complete in 1 minute?
d.
d. How much will Chandi complete in x minutes?
e.Write a rational equation representing Kendal and
Chandi working together on the car.
f. Solve the equation to determine how long it will
take them to finish the car.
SOLUTION:
a.
b.
c.
The speed of the wind is 3.5 mph.
d.
11. WORKKendal and Chandi wax cars. Kendal can
wax a particular car in 60 minutes and Chandi can
wax the same car in 80 minutes. They plan on
waxing the same car together and want to know how e.
long it will take.
a. How much will Kendal complete in 1 minute? f.
b. How much will Kendal complete in x minutes?
c. How much will Chandi complete in 1 minute?
d. How much will Chandi complete in x minutes?
e.Write a rational equation representing Kendal and
Chandi working together on the car.
f. Solve the equation to determine how long it will
take them to finish the car. It will take them about 34.3 minutes to finish the car.
SOLUTION: Solve each inequality. Check your solutions.
a.
12.
b.
SOLUTION:
The excluded value for this inequality is 0.
c.
Solve the related equation .
d.
e.
f.
Divide the real line in to three intervals as shown.
It will take them about 34.3 minutes to finish the car.
Solve each inequality. Check your solutions. Test x = –1.
12.
SOLUTION:
The excluded value for this inequality is 0.
Solve the related equation .
Test x = 1.
Test x = 2.
Divide the real line in to three intervals as shown.
Test x = –1.
Therefore, the solution is 0 < x < 1.15.
13.
SOLUTION:
Test x = 1. The excluded value for this inequality is 0.
Solve the related equation .
Test x = 2.
Divide the real line in to three intervals as shown.
Therefore, the solution is 0 < x < 1.15. Test c = –1.
13.
SOLUTION:
The excluded value for this inequality is 0.
Solve the related equation .
Test c = 0.5.
Test c = 1.
Divide the real line in to three intervals as shown.
Test c = –1.
Therefore, the solution is c < 0 or .
14.
Test c = 0.5. SOLUTION:
The excluded value for this inequality is y = 0.
Solve the related equation .
Test c = 1.
Divide the real line in to three intervals as shown.
Test y = –1.
Therefore, the solution is c < 0 or .
14.
SOLUTION:
The excluded value for this inequality is y = 0.
Test .
Solve the related equation .
Test y = 2.
Divide the real line in to three intervals as shown.
Test y = –1.
Therefore, the solution is .
15.
Test .
SOLUTION:
The excluded value of this inequality is b = 0.
Solve the related equation .
Test y = 2.
Divide the real line in to three intervals as shown.
Therefore, the solution is . Test b = −1.
15.
SOLUTION:
The excluded value of this inequality is b = 0.
Solve the related equation . Test b = 1.
Test b = 3.
Divide the real line in to three intervals as shown.
Test b = −1.
Therefore, the solution is .
Solve each equation. Check your solutions.
16.
SOLUTION:
Test b = 1.
Test b = 3. Check:
Therefore, the solution is . The solution is 9.
Solve each equation. Check your solutions.
17.
16.
SOLUTION:
SOLUTION:
Check:
Check:
The solution is 2.
The solution is 9.
18.
17.
SOLUTION:
SOLUTION:
Check: Check:
The solution is 2. The solution is 7.
19.
18.
SOLUTION:
SOLUTION:
Check:
Check:
The solution is 7. The solution is 1.
20.
19.
SOLUTION:
SOLUTION:
Check: Use the quadratic formula.
The solution is 1.
Check: x =
20.
SOLUTION:
Check: x =
Use the quadratic formula.
Therefore, the solution set is {2, –12}
21.
SOLUTION:
Check: x =
Check: x = Use the Quadratic formula to solve .
There is no real solution for the quadratic equation
. Therefore, the solution for the given
rational equation is .
Therefore, the solution set is {2, –12}
22. CHEMISTRYHow many milliliters of a 20% acid
solution must be added to 40 milliliters of a 75% acid
solution to create a 30% acid solution?
21.
SOLUTION:
Let x milliliters of a 20% acid solution is added to 40
SOLUTION: milliliters of a 75% acid solution.
Use the Quadratic formula to solve .
Check:
There is no real solution for the quadratic equation
. Therefore, the solution for the given
rational equation is .
22. CHEMISTRYHow many milliliters of a 20% acid
solution must be added to 40 milliliters of a 75% acid Therefore, 180 milliliters of a 20% acid solution must
solution to create a 30% acid solution?
be added to 40 milliliters of a 75% acid solution to
create a 30% acid solution.
SOLUTION:
Let x milliliters of a 20% acid solution is added to 40
milliliters of a 75% acid solution. 23. GROCERIES Ellen bought 3 pounds of bananas for
$0.90 per pound. How many pounds of apples
costing $1.25 per pound must she purchase so that
the total cost for fruit is $1 per pound?
SOLUTION:
Let Ellen bought x pounds of apples.
Check: She needs to purchase 1.2 pounds of apples.
24. BUILDINGBryan’s volunteer group can build a
garage in 12 hours. Sequoia’s group can build it in 16
hours. How long would it take them if they worked
together?
SOLUTION:
The rate for Bryan’s volunteer group is .
Therefore, 180 milliliters of a 20% acid solution must
be added to 40 milliliters of a 75% acid solution to
create a 30% acid solution.
The rate for Sequoia’s group is .
23. GROCERIES Ellen bought 3 pounds of bananas for
$0.90 per pound. How many pounds of apples Let their combined rate is .
costing $1.25 per pound must she purchase so that
the total cost for fruit is $1 per pound?
SOLUTION:
Let Ellen bought x pounds of apples.
Therefore, it would take about 6.86 hours to build a
garage if they worked together.
She needs to purchase 1.2 pounds of apples.
Solve each inequality. Check your solutions.
24. BUILDINGBryan’s volunteer group can build a
garage in 12 hours. Sequoia’s group can build it in 16
hours. How long would it take them if they worked 25.
together?
SOLUTION:
SOLUTION: The excluded value for this inequality is x = 0.
The rate for Bryan’s volunteer group is .
The rate for Sequoia’s group is .
Let their combined rate is .
Divide the real line in to three intervals as shown.
Therefore, it would take about 6.86 hours to build a
garage if they worked together.
Test x = −1.
Solve each inequality. Check your solutions.
25.
SOLUTION:
The excluded value for this inequality is x = 0. Test x = 1.
Test x = 2.
Divide the real line in to three intervals as shown. The solution for the inequality is x < 0 or x > 1.75.
26.
Test x = −1.
SOLUTION:
The excluded value for this inequality is a = 0.
Test x = 1.
Test x = 2.
Divide the inequality in to three intervals as shown.
The solution for the inequality is x < 0 or x > 1.75. Test a = −1.
26.
SOLUTION:
The excluded value for this inequality is a = 0.
Test a = 1.
Test a = 2.
Divide the inequality in to three intervals as shown. Therefore, the solution set is 0 < a < 1.1.
27.
Test a = −1.
SOLUTION:
The excluded values for this inequality is x = −2 and
x = 2.
Solve the related equation .
Test a = 1.
Test a = 2.
Divide the real line in to four intervals as shown.
Therefore, the solution set is 0 < a < 1.1.
Test x = −4.
27.
SOLUTION:
The excluded values for this inequality is x = −2 and
x = 2.
Solve the related equation .
Test x = 0.
Divide the real line in to four intervals as shown. Test x = 4.
Test x = −4.
Test x = 16.
Test x = 0.
Therefore, the solution set for the inequality is x < −2
or 2 < x < 14.
28.
SOLUTION:
Test x = 4. The excluded value for this inequality is x = 3 and x =
–4.
Solve the related equation .
Test x = 16.
There exists no real solution for the quadratic
equation .
Therefore, the solution set for the inequality is x < −2 Divide the real line in to three intervals as shown.
or 2 < x < 14.
28. Test x = −5.
SOLUTION:
The excluded value for this inequality is x = 3 and x =
–4.
Solve the related equation .
Test x = 0.
Test x = 5.
There exists no real solution for the quadratic
equation .
Divide the real line in to three intervals as shown.
Test x = −5.
The solution set is 4 < x < 3.
–
29.
SOLUTION:
The excluded value for this inequality is x = 4.
Solve the related equation .
Test x = 0.
Test x = 5.
Solve the quadratic equation using the Quadratic
formula.
The solution set is 4 < x < 3.
–
29.
SOLUTION:
The excluded value for this inequality is x = 4.
Solve the related equation .
Divide the real line in to 4 intervals as shown.
Test x = 6.
–
Solve the quadratic equation using the Quadratic
formula.
Test x = 0.
Test x = 5.
Divide the real line in to 4 intervals as shown.
Test x = 6.
Test x = 6.
–
The solution set for the inequality is x < 5 or
–
.
Test x = 0.
30.
SOLUTION:
The excluded values for this inequality are x = 2
–
and x = 1.
Test x = 5.
Solve the related equation .
Test x = 6.
The solution set for the inequality is x < 5 or
–
.
Divide the real line in to 5 intervals as shown.
30.
Test x = 6.
–
SOLUTION:
The excluded values for this inequality are x = 2
–
and x = 1.
Solve the related equation .
Test x = 4.
–
Test x = 0.
Divide the real line in to 5 intervals as shown.
Test x = 6. Test .
–
Test x = 4.
–
Test x = 4.
Test x = 0.
Test .
The solution set for the inequality is x < 5 or 2 < x
– −
< 1 or x > 2.
AIR TRAVELIt takes a plane 20 hours to fly to its
31.
destination against the wind. The return trip takes 16
hours. If the plane’s average speed in still air is 500
miles per hour, what is the average speed of the wind
during the flight?
SOLUTION:
Test x = 4.
The average speed of the wind during the flight is
about 55.56 miles per hour.
FINANCIAL LITERACY
32. Judie wants to invest
$10,000 in two different accounts. The risky account
earns 9% interest, while the other account earns 5%
interest. She wants to earn $750 interest for the year.
The solution set for the inequality is x < 5 or 2 < x
– − Of tables, graphs, or equations, choose the best
< 1 or x > 2. representation needed and determine how much
should be invested in each account.
AIR TRAVELIt takes a plane 20 hours to fly to its
31.
destination against the wind. The return trip takes 16
SOLUTION:
hours. If the plane’s average speed in still air is 500 Judie invest x dollars in the account earns 9%
miles per hour, what is the average speed of the wind interest and (10000 x) dollars in the account earns
during the flight? −
5% interest.
SOLUTION:
Thus, Judie should invest $6250 at 9% account and
$3750 at 5% account.
MULTIPLE REPRESENTATIONSConsider
33.
The average speed of the wind during the flight is
about 55.56 miles per hour. a. ALGEBRAICSolve the equation for x. Were
any values of x extraneous?
FINANCIAL LITERACY
32. Judie wants to invest
b. GRAPHICALGraph
$10,000 in two different accounts. The risky account and
earns 9% interest, while the other account earns 5% x < 5.
interest. She wants to earn $750 interest for the year. on the same graph for 0 <
Of tables, graphs, or equations, choose the best
representation needed and determine how much
should be invested in each account. c. ANALYTICALFor what value(s) of x do they
intersect? Do they intersect where x is extraneous
for the original equation?
SOLUTION:
Judie invest x dollars in the account earns 9% d. VERBALUse this knowledge to describe how
interest and (10000 x) dollars in the account earns you can use a graph to determine whether an
− apparent solution of a rational equation is extraneous.
5% interest.
SOLUTION:
a.
Thus, Judie should invest $6250 at 9% account and
$3750 at 5% account.
MULTIPLE REPRESENTATIONSConsider
33.
a. ALGEBRAICSolve the equation for x. Were
any values of x extraneous? Check: x = 1
b. GRAPHICALGraph
and
x < 5.
on the same graph for 0 <
x = 3 is the excluded value for the equation.
c. ANALYTICALFor what value(s) of x do they Therefore, x = 3 is the extraneous solution and x = 1
intersect? Do they intersect where x is extraneous is the solution for the equation.
for the original equation?
b.
d. VERBALUse this knowledge to describe how
you can use a graph to determine whether an
apparent solution of a rational equation is extraneous.
SOLUTION:
a.
c. Two graphs intersect at x = 1 and they do not
intersect at the extraneous solution x = 3.]
d. Graph both sides of the equation. Where the
graphs intersect, there is a solution. If they do not,
then the possible solution is extraneous.
Solve each equation. Check your solutions.
Check: x = 1
34.
SOLUTION:
x = 3 is the excluded value for the equation.
Therefore, x = 3 is the extraneous solution and x = 1
is the solution for the equation.
b.
Check:
c. Two graphs intersect at x = 1 and they do not
intersect at the extraneous solution x = 3.]
d. Graph both sides of the equation. Where the
graphs intersect, there is a solution. If they do not,
then the possible solution is extraneous.
The solution is y = 1.
Solve each equation. Check your solutions. −
34.
35.
SOLUTION:
SOLUTION:
Check:
Check:
The solution set is .
OPEN ENDEDGive an example of a rational
36.
The solution is y = 1. equation that can be solved by multiplying each side
−
of the equation by 4(x + 3)(x – 4).
SOLUTION:
35. Sample answer:
SOLUTION:
CHALLENGESolve
37.
SOLUTION:
Check:
The solution set is .
OPEN ENDEDGive an example of a rational
36.
equation that can be solved by multiplying each side
of the equation by 4(x + 3)(x 4).
–
Therefore, the solution is all real numbers except 5,
SOLUTION: 5, and 0.
Sample answer: −
CCSS TOOLS
38. While using the table feature on the
graphing calculator to explore
the
ERROR
values 2 and 3 say . Explain its meaning.
– “ ”
CHALLENGE
37. Solve
SOLUTION:
Sample answer:
SOLUTION:
The denominator will equal 0 when x = 2 or x = 3.
−
The values 2 and 3 are undefined values. On the
−
graph of f (x) there would be vertical asymptotes at
these values.
Why should you check
WRITING IN MATH
39.
solutions of rational equations and inequalities?
SOLUTION:
Sample answer: Multiplying each side of a rational
equation or inequality by the LCD can result in
extraneous solutions. Therefore, you should check all
solutions to make sure that they satisfy the original
equation or inequality.
Therefore, the solution is all real numbers except 5, Nine pounds of mixed nuts containing 55% peanuts
5, and 0. 40.
− were mixed with 6 pounds of another kind of mixed
nuts that contain 40% peanuts. What percent of the
new mixture is peanuts?
CCSS TOOLS
38. While using the table feature on the
graphing calculator to explore A 58%
the
ERROR
values 2 and 3 say . Explain its meaning.
– “ ” B 51%
C 49%
SOLUTION:
Sample answer:
D47%
SOLUTION:
Let the new mixture contains x percent of peanuts.
The denominator will equal 0 when x = 2 or x = 3.
−
The values 2 and 3 are undefined values. On the
−
graph of f (x) there would be vertical asymptotes at
these values.
Why should you check
WRITING IN MATH
39.
solutions of rational equations and inequalities?
So, the new mixture contains 0.49 or 49% percent of
SOLUTION: peanuts. The correct choice is C.
Sample answer: Multiplying each side of a rational
equation or inequality by the LCD can result in
extraneous solutions. Therefore, you should check all Working alone, Dato can dig a 10-foot by 10-foot
solutions to make sure that they satisfy the original 41.
hole in five hours. Pedro can dig the same hole in six
equation or inequality. hours. How long would it take them if they worked
together?
Nine pounds of mixed nuts containing 55% peanuts
40.
were mixed with 6 pounds of another kind of mixed F1.5 hours
nuts that contain 40% peanuts. What percent of the
new mixture is peanuts? G2.34 hours
A 58% H 2.52 hours
B 51% J 2.73 hours
C 49%
SOLUTION:
D 47%
SOLUTION:
Let the new mixture contains x percent of peanuts.
It would take about 2.73 hours to dig the hole if they
worked together. The correct choice is J.
So, the new mixture contains 0.49 or 49% percent of An aircraft carrier made a trip to Guam and back.
peanuts. The correct choice is C. 42.
The trip there took three hours and the trip back took
four hours. It averaged 6 kilometers per hour on the
Working alone, Dato can dig a 10-foot by 10-foot return trip. Find the average speed of the trip to
41. Guam.
hole in five hours. Pedro can dig the same hole in six
hours. How long would it take them if they worked
together? A 6 km/h
B 8 km/h
F1.5 hours
C 10 km/h
G2.34 hours
D 12 km/h
H 2.52 hours
J 2.73 hours
SOLUTION:
SOLUTION:
It would take about 2.73 hours to dig the hole if they
worked together. The correct choice is J.
42. An aircraft carrier made a trip to Guam and back.
The trip there took three hours and the trip back took
four hours. It averaged 6 kilometers per hour on the
return trip. Find the average speed of the trip to
Guam.
A 6 km/h
B 8 km/h
C 10 km/h
D 12 km/h
SOLUTION:
Solve each equation. Check your solution.
1.
SOLUTION:
Check:
The solution is 11.
2.
SOLUTION:
Solve each equation. Check your solution.
1.
SOLUTION:
Check:
Check:
The solution is 9.
3.
The solution is 11. SOLUTION:
2.
SOLUTION:
Check:
Check:
The solution is 9. The solution is 7.
8-6 Solving Rational Equations and Inequalities
3. 4.
SOLUTION:
SOLUTION:
Check:
Check:
The solution is 3.
5.
The solution is 7.
SOLUTION:
4.
SOLUTION:
eSolutions Manual - Powered by Cognero Page2
Check:
Check:
The solution is 8.
6.
The solution is 3.
SOLUTION:
5.
SOLUTION:
Check: Check:
The solution is 8. The solution is x = 5.
7.
6.
SOLUTION:
SOLUTION:
Check:
Check:
The solution is 14.
The solution is x = 5. 8.
SOLUTION:
7.
SOLUTION:
Check:
Check:
The solution is 14.
The solution is 14.
CCSS STRUCTURE
9. Sara has 10 pounds of dried
fruit selling for $6.25 per pound. She wants to know
how many pounds of mixed nuts selling for $4.50 per
pound she needs to make a trail mix selling for $5 per
8. pound.
a. Let m = the number of pounds of mixed nuts.
SOLUTION: Complete the following table.
b. Write a rational equation using the last column of
the table.
c. Solve the equation to determine how many pounds
of mixed nuts are needed.
SOLUTION:
a.
Check:
b.
c.
The solution is 14.
CCSS STRUCTURE
9. Sara has 10 pounds of dried Therefore, 25 pounds of mixed nuts are needed.
fruit selling for $6.25 per pound. She wants to know
how many pounds of mixed nuts selling for $4.50 per
pound she needs to make a trail mix selling for $5 per
pound. 10. DISTANCEAlicia’s average speed riding her bike
is 11.5 miles per hour. She takes a round trip of 40
a. Let m = the number of pounds of mixed nuts. miles. It takes her 1 hour and 20 minutes with the
Complete the following table. wind and 2 hours and 30 minutes against the wind.
a. Write an expression for Alicia’s time with the
wind.
b. Write an expression for Alicia’s time against the
wind.
b. Write a rational equation using the last column of
the table. c. How long does it take to complete the trip?
d. Write and solve the rational equation to determine
c. Solve the equation to determine how many pounds the speed of the wind.
of mixed nuts are needed.
SOLUTION:
SOLUTION: a.
a. Let x be the speed of the wind.
The expression for Alicia’s time with the wind is
.
b. The expression for Alicia’s time against the wind
is .
b.
c.
c.
d.
Therefore, 25 pounds of mixed nuts are needed.
10. DISTANCEAlicia’s average speed riding her bike
is 11.5 miles per hour. She takes a round trip of 40
miles. It takes her 1 hour and 20 minutes with the
wind and 2 hours and 30 minutes against the wind.
a. Write an expression for Alicia’s time with the
wind.
b. Write an expression for Alicia’s time against the
wind.
The speed of the wind is 3.5 mph.
c. How long does it take to complete the trip?
d. Write and solve the rational equation to determine 11. WORK Kendal and Chandi wax cars. Kendal can
the speed of the wind. wax a particular car in 60 minutes and Chandi can
wax the same car in 80 minutes. They plan on
waxing the same car together and want to know how
SOLUTION: long it will take.
a. Let x be the speed of the wind.
The expression for Alicia’s time with the wind is a. How much will Kendal complete in 1 minute?
.
b. How much will Kendal complete in x minutes?
b. The expression for Alicia’s time against the wind
c. How much will Chandi complete in 1 minute?
is .
d. How much will Chandi complete in x minutes?
c.
e.Write a rational equation representing Kendal and
Chandi working together on the car.
d. f. Solve the equation to determine how long it will
take them to finish the car.
SOLUTION:
a.
b.
c.
d.
The speed of the wind is 3.5 mph. e.
11. WORK Kendal and Chandi wax cars. Kendal can f.
wax a particular car in 60 minutes and Chandi can
wax the same car in 80 minutes. They plan on
waxing the same car together and want to know how
long it will take.
a. How much will Kendal complete in 1 minute?
b. How much will Kendal complete in x minutes?
c. How much will Chandi complete in 1 minute?
d. How much will Chandi complete in x minutes? It will take them about 34.3 minutes to finish the car.
e.Write a rational equation representing Kendal and
Chandi working together on the car. Solve each inequality. Check your solutions.
f. Solve the equation to determine how long it will
take them to finish the car. 12.
SOLUTION:
SOLUTION:
a. The excluded value for this inequality is 0.
Solve the related equation .
b.
c.
d.
e.
f.
Divide the real line in to three intervals as shown.
Test x = –1.
It will take them about 34.3 minutes to finish the car.
Solve each inequality. Check your solutions.
12.
Test x = 1.
SOLUTION:
The excluded value for this inequality is 0.
Solve the related equation .
Test x = 2.
Divide the real line in to three intervals as shown.
Therefore, the solution is 0 < x < 1.15.
Test x = –1.
13.
SOLUTION:
The excluded value for this inequality is 0.
Solve the related equation .
Test x = 1.
Test x = 2.
Divide the real line in to three intervals as shown.
Test c = –1.
Therefore, the solution is 0 < x < 1.15.
13.
Test c = 0.5.
SOLUTION:
The excluded value for this inequality is 0.
Solve the related equation .
Test c = 1.
Divide the real line in to three intervals as shown.
Therefore, the solution is c < 0 or .
Test c = –1.
14.
SOLUTION:
The excluded value for this inequality is y = 0.
Solve the related equation .
Test c = 0.5.
Test c = 1.
Divide the real line in to three intervals as shown.
Test y = –1.
Therefore, the solution is c < 0 or .
14. Test .
SOLUTION:
The excluded value for this inequality is y = 0.
Solve the related equation .
Test y = 2.
Divide the real line in to three intervals as shown.
Test y = –1. Therefore, the solution is .
15.
SOLUTION:
The excluded value of this inequality is b = 0.
Solve the related equation .
Test .
Test y = 2.
Divide the real line in to three intervals as shown.
Test b = −1.
Therefore, the solution is .
15.
Test b = 1.
SOLUTION:
The excluded value of this inequality is b = 0.
Solve the related equation .
Test b = 3.
Divide the real line in to three intervals as shown.
Therefore, the solution is .
Solve each equation. Check your solutions.
Test b = −1.
16.
SOLUTION:
Test b = 1.
Check:
Test b = 3.
The solution is 9.
17.
Therefore, the solution is .
SOLUTION:
Solve each equation. Check your solutions.
16.
SOLUTION:
Check:
Check:
The solution is 2.
18.
SOLUTION:
The solution is 9.
17.
SOLUTION:
Check:
Check:
The solution is 7.
19.
The solution is 2.
SOLUTION:
18.
SOLUTION:
Check:
Check:
The solution is 1.
20.
The solution is 7.
SOLUTION:
19.
SOLUTION:
Use the quadratic formula.
Check:
Check: x =
The solution is 1.
20.
Check: x =
SOLUTION:
Therefore, the solution set is {2, –12}
21.
Use the quadratic formula.
SOLUTION:
Check: x =
Use the Quadratic formula to solve .
Check: x = There is no real solution for the quadratic equation
. Therefore, the solution for the given
rational equation is .
22. CHEMISTRYHow many milliliters of a 20% acid
solution must be added to 40 milliliters of a 75% acid
solution to create a 30% acid solution?
Therefore, the solution set is {2, –12} SOLUTION:
Let x milliliters of a 20% acid solution is added to 40
milliliters of a 75% acid solution.
21.
SOLUTION:
Check:
Use the Quadratic formula to solve .
There is no real solution for the quadratic equation
. Therefore, the solution for the given Therefore, 180 milliliters of a 20% acid solution must
rational equation is . be added to 40 milliliters of a 75% acid solution to
create a 30% acid solution.
22. CHEMISTRYHow many milliliters of a 20% acid 23. GROCERIES Ellen bought 3 pounds of bananas for
solution must be added to 40 milliliters of a 75% acid $0.90 per pound. How many pounds of apples
solution to create a 30% acid solution?
costing $1.25 per pound must she purchase so that
the total cost for fruit is $1 per pound?
SOLUTION:
Let x milliliters of a 20% acid solution is added to 40
milliliters of a 75% acid solution. SOLUTION:
Let Ellen bought x pounds of apples.
She needs to purchase 1.2 pounds of apples.
24. BUILDINGBryan’s volunteer group can build a
garage in 12 hours. Sequoia’s group can build it in 16
hours. How long would it take them if they worked
Check: together?
SOLUTION:
The rate for Bryan’s volunteer group is .
The rate for Sequoia’s group is .
Therefore, 180 milliliters of a 20% acid solution must Let their combined rate is .
be added to 40 milliliters of a 75% acid solution to
create a 30% acid solution.
23. GROCERIES Ellen bought 3 pounds of bananas for
$0.90 per pound. How many pounds of apples
costing $1.25 per pound must she purchase so that
the total cost for fruit is $1 per pound?
SOLUTION:
Let Ellen bought x pounds of apples.
Therefore, it would take about 6.86 hours to build a
garage if they worked together.
Solve each inequality. Check your solutions.
25.
She needs to purchase 1.2 pounds of apples.
SOLUTION:
24. BUILDINGBryan’s volunteer group can build a The excluded value for this inequality is x = 0.
garage in 12 hours. Sequoia’s group can build it in 16
hours. How long would it take them if they worked
together?
SOLUTION:
The rate for Bryan’s volunteer group is .
The rate for Sequoia’s group is .
Let their combined rate is .
Divide the real line in to three intervals as shown.
Test x = −1.
Therefore, it would take about 6.86 hours to build a
garage if they worked together.
Solve each inequality. Check your solutions.
Test x = 1.
25.
SOLUTION:
The excluded value for this inequality is x = 0.
Test x = 2.
The solution for the inequality is x < 0 or x > 1.75.
26.
Divide the real line in to three intervals as shown.
SOLUTION:
The excluded value for this inequality is a = 0.
Test x = −1.
Test x = 1.
Divide the inequality in to three intervals as shown.
Test x = 2.
Test a = −1.
The solution for the inequality is x < 0 or x > 1.75.
26.
Test a = 1.
SOLUTION:
The excluded value for this inequality is a = 0.
Test a = 2.
Therefore, the solution set is 0 < a < 1.1.
27.
Divide the inequality in to three intervals as shown.
SOLUTION:
The excluded values for this inequality is x = −2 and
x = 2.
Test a = −1.
Solve the related equation .
Test a = 1.
Test a = 2. Divide the real line in to four intervals as shown.
Test x = −4.
Therefore, the solution set is 0 < a < 1.1.
27.
SOLUTION:
The excluded values for this inequality is x = −2 and
x = 2.
Solve the related equation . Test x = 0.
Test x = 4.
Divide the real line in to four intervals as shown.
Test x = −4. Test x = 16.
Therefore, the solution set for the inequality is x < −2
or 2 < x < 14.
Test x = 0.
28.
SOLUTION:
The excluded value for this inequality is x = 3 and x =
–4.
Solve the related equation .
Test x = 4.
Test x = 16.
There exists no real solution for the quadratic
equation .
Divide the real line in to three intervals as shown.
Test x = −5.
Therefore, the solution set for the inequality is x < −2
or 2 < x < 14.
28.
SOLUTION:
The excluded value for this inequality is x = 3 and x =
–4.
Solve the related equation .
Test x = 0.
Test x = 5.
There exists no real solution for the quadratic
equation .
Divide the real line in to three intervals as shown.
The solution set is 4 < x < 3.
–
Test x = −5. 29.
SOLUTION:
The excluded value for this inequality is x = 4.
Solve the related equation .
Test x = 0.
Solve the quadratic equation using the Quadratic
formula.
Test x = 5.
The solution set is 4 < x < 3.
–
29.
Divide the real line in to 4 intervals as shown.
SOLUTION:
The excluded value for this inequality is x = 4.
Test x = 6.
Solve the related equation . –
Test x = 0.
Solve the quadratic equation using the Quadratic
formula.
Test x = 5.
Test x = 6.
Divide the real line in to 4 intervals as shown.
Test x = 6.
– The solution set for the inequality is x < 5 or
–
.
30.
SOLUTION:
The excluded values for this inequality are x = 2
Test x = 0. –
and x = 1.
Solve the related equation .
Test x = 5.
Test x = 6.
Divide the real line in to 5 intervals as shown.
The solution set for the inequality is x < 5 or
–
.
Test x = 6.
–
30.
SOLUTION:
The excluded values for this inequality are x = 2
–
and x = 1.
Solve the related equation .
Test x = 4.
–
Test x = 0.
Test .
Divide the real line in to 5 intervals as shown.
Test x = 6.
–
Test x = 4.
–
Test x = 4.
Test x = 0.
The solution set for the inequality is x < 5 or 2 < x
– −
< 1 or x > 2.
AIR TRAVELIt takes a plane 20 hours to fly to its
31.
Test . destination against the wind. The return trip takes 16
hours. If the plane’s average speed in still air is 500
miles per hour, what is the average speed of the wind
during the flight?
SOLUTION:
The average speed of the wind during the flight is
about 55.56 miles per hour.
Test x = 4.
FINANCIAL LITERACY
32. Judie wants to invest
$10,000 in two different accounts. The risky account
earns 9% interest, while the other account earns 5%
interest. She wants to earn $750 interest for the year.
Of tables, graphs, or equations, choose the best
representation needed and determine how much
should be invested in each account.
SOLUTION:
The solution set for the inequality is x < 5 or 2 < x
– − Judie invest x dollars in the account earns 9%
< 1 or x > 2. interest and (10000 x) dollars in the account earns
−
5% interest.
AIR TRAVELIt takes a plane 20 hours to fly to its
31.
destination against the wind. The return trip takes 16
hours. If the plane’s average speed in still air is 500
miles per hour, what is the average speed of the wind
during the flight?
SOLUTION:
Thus, Judie should invest $6250 at 9% account and
$3750 at 5% account.
MULTIPLE REPRESENTATIONSConsider
33.
a. ALGEBRAICSolve the equation for x. Were
any values of x extraneous?
b. GRAPHICALGraph
and
The average speed of the wind during the flight is x < 5.
about 55.56 miles per hour. on the same graph for 0 <
c. ANALYTICALFor what value(s) of x do they
FINANCIAL LITERACY
32. Judie wants to invest intersect? Do they intersect where x is extraneous
$10,000 in two different accounts. The risky account for the original equation?
earns 9% interest, while the other account earns 5%
interest. She wants to earn $750 interest for the year.
Of tables, graphs, or equations, choose the best d. VERBALUse this knowledge to describe how
representation needed and determine how much you can use a graph to determine whether an
should be invested in each account. apparent solution of a rational equation is extraneous.
SOLUTION:
SOLUTION: a.
Judie invest x dollars in the account earns 9%
interest and (10000 − x) dollars in the account earns
5% interest.
Thus, Judie should invest $6250 at 9% account and
$3750 at 5% account.
Check: x = 1
MULTIPLE REPRESENTATIONSConsider
33.
a. ALGEBRAICSolve the equation for x. Were
any values of x extraneous?
x = 3 is the excluded value for the equation.
Therefore, x = 3 is the extraneous solution and x = 1
b. GRAPHICALGraph
and is the solution for the equation.
x < 5.
on the same graph for 0 < b.
c. ANALYTICALFor what value(s) of x do they
intersect? Do they intersect where x is extraneous
for the original equation?
d. VERBALUse this knowledge to describe how
you can use a graph to determine whether an
apparent solution of a rational equation is extraneous.
SOLUTION:
a.
c. Two graphs intersect at x = 1 and they do not
intersect at the extraneous solution x = 3.]
d. Graph both sides of the equation. Where the
graphs intersect, there is a solution. If they do not,
then the possible solution is extraneous.
Solve each equation. Check your solutions.
34.
SOLUTION:
Check: x = 1
x = 3 is the excluded value for the equation.
Therefore, x = 3 is the extraneous solution and x = 1
is the solution for the equation.
b.
Check:
c. Two graphs intersect at x = 1 and they do not
intersect at the extraneous solution x = 3.]
The solution is y = 1.
−
d. Graph both sides of the equation. Where the
graphs intersect, there is a solution. If they do not,
then the possible solution is extraneous.
35.
Solve each equation. Check your solutions.
SOLUTION:
34.
SOLUTION:
Check:
Check:
The solution set is .
OPEN ENDEDGive an example of a rational
36.
equation that can be solved by multiplying each side
of the equation by 4(x + 3)(x 4).
–
SOLUTION:
Sample answer:
The solution is y = 1.
−
35.
CHALLENGESolve
37.
SOLUTION:
SOLUTION:
Check:
The solution set is .
Therefore, the solution is all real numbers except 5,
5, and 0.
−
OPEN ENDEDGive an example of a rational
36.
equation that can be solved by multiplying each side
of the equation by 4(x + 3)(x 4).
–
CCSS TOOLS
38. While using the table feature on the
graphing calculator to explore
the
SOLUTION:
ERROR
values 2 and 3 say . Explain its meaning.
Sample answer: – “ ”
SOLUTION:
Sample answer:
CHALLENGESolve
37.
The denominator will equal 0 when x = 2 or x = 3.
−
The values 2 and 3 are undefined values. On the
−
SOLUTION: graph of f (x) there would be vertical asymptotes at
these values.
Why should you check
WRITING IN MATH
39.
solutions of rational equations and inequalities?
SOLUTION:
Sample answer: Multiplying each side of a rational
equation or inequality by the LCD can result in
extraneous solutions. Therefore, you should check all
solutions to make sure that they satisfy the original
equation or inequality.
40. Nine pounds of mixed nuts containing 55% peanuts
were mixed with 6 pounds of another kind of mixed
nuts that contain 40% peanuts. What percent of the
new mixture is peanuts?
A 58%
Therefore, the solution is all real numbers except 5,
5, and 0.
− B 51%
CCSS TOOLS C
38. While using the table feature on the 49%
graphing calculator to explore
the D 47%
ERROR
values 2 and 3 say . Explain its meaning.
– “ ”
SOLUTION:
Let the new mixture contains x percent of peanuts.
SOLUTION:
Sample answer:
The denominator will equal 0 when x = 2 or x = 3.
−
The values 2 and 3 are undefined values. On the
−
graph of f (x) there would be vertical asymptotes at
these values.
So, the new mixture contains 0.49 or 49% percent of
peanuts. The correct choice is C.
Why should you check
WRITING IN MATH
39.
solutions of rational equations and inequalities? Working alone, Dato can dig a 10-foot by 10-foot
41.
hole in five hours. Pedro can dig the same hole in six
hours. How long would it take them if they worked
SOLUTION: together?
Sample answer: Multiplying each side of a rational
equation or inequality by the LCD can result in
extraneous solutions. Therefore, you should check all F1.5 hours
solutions to make sure that they satisfy the original
G2.34 hours
equation or inequality.
Nine pounds of mixed nuts containing 55% peanuts H 2.52 hours
40.
were mixed with 6 pounds of another kind of mixed
nuts that contain 40% peanuts. What percent of the J 2.73 hours
new mixture is peanuts?
SOLUTION:
A58%
B 51%
C 49%
D 47%
SOLUTION:
Let the new mixture contains x percent of peanuts.
It would take about 2.73 hours to dig the hole if they
worked together. The correct choice is J.
42. An aircraft carrier made a trip to Guam and back.
The trip there took three hours and the trip back took
four hours. It averaged 6 kilometers per hour on the
return trip. Find the average speed of the trip to
Guam.
So, the new mixture contains 0.49 or 49% percent of
peanuts. The correct choice is C. A 6 km/h
B 8 km/h
Working alone, Dato can dig a 10-foot by 10-foot
41.
hole in five hours. Pedro can dig the same hole in six C10 km/h
hours. How long would it take them if they worked
together? D 12 km/h
F1.5 hours
SOLUTION:
G2.34 hours
H 2.52 hours
J 2.73 hours
SOLUTION:
It would take about 2.73 hours to dig the hole if they
worked together. The correct choice is J.
42. An aircraft carrier made a trip to Guam and back.
The trip there took three hours and the trip back took
four hours. It averaged 6 kilometers per hour on the
return trip. Find the average speed of the trip to
Guam.
A 6 km/h
B 8 km/h
C 10 km/h
D 12 km/h
SOLUTION:
Solve each equation. Check your solution.
1.
SOLUTION:
Check:
The solution is 11.
2.
Solve each equation. Check your solution.
1.
SOLUTION:
SOLUTION:
Check:
Check:
The solution is 9.
The solution is 11.
3.
2.
SOLUTION:
SOLUTION:
Check:
Check:
The solution is 9.
3.
The solution is 7.
SOLUTION:
4.
SOLUTION:
Check:
Check:
The solution is 7. The solution is 3.
4.
5.
SOLUTION: SOLUTION:
Check:
Check:
The solution is 3. The solution is 8.
8-6 Solving Rational Equations and Inequalities
5. 6.
SOLUTION: SOLUTION:
Check:
Check:
The solution is 8.
6.
The solution is x = 5.
SOLUTION:
7.
SOLUTION:
eSolutions Manual - Powered by Cognero Page3
Check:
Check:
The solution is x = 5.
The solution is 14.
7.
8.
SOLUTION:
SOLUTION:
Check:
Check:
The solution is 14.
The solution is 14.
8.
CCSS STRUCTURE
9. Sara has 10 pounds of dried
fruit selling for $6.25 per pound. She wants to know
SOLUTION: how many pounds of mixed nuts selling for $4.50 per
pound she needs to make a trail mix selling for $5 per
pound.
a. Let m = the number of pounds of mixed nuts.
Complete the following table.
b. Write a rational equation using the last column of
the table.
c. Solve the equation to determine how many pounds
Check: of mixed nuts are needed.
SOLUTION:
a.
b.
The solution is 14.
c.
CCSS STRUCTURE
9. Sara has 10 pounds of dried
fruit selling for $6.25 per pound. She wants to know
how many pounds of mixed nuts selling for $4.50 per
pound she needs to make a trail mix selling for $5 per
pound.
Therefore, 25 pounds of mixed nuts are needed.
a. Let m = the number of pounds of mixed nuts.
Complete the following table.
10. DISTANCEAlicia’s average speed riding her bike
is 11.5 miles per hour. She takes a round trip of 40
miles. It takes her 1 hour and 20 minutes with the
wind and 2 hours and 30 minutes against the wind.
b. Write a rational equation using the last column of a. Write an expression for Alicia’s time with the
the table. wind.
b. Write an expression for Alicia’s time against the
c. Solve the equation to determine how many pounds wind.
of mixed nuts are needed.
c. How long does it take to complete the trip?
SOLUTION:
a. d. Write and solve the rational equation to determine
the speed of the wind.
SOLUTION:
a. Let x be the speed of the wind.
The expression for Alicia’s time with the wind is
b. .
b. The expression for Alicia’s time against the wind
c. is .
c.
Therefore, 25 pounds of mixed nuts are needed. d.
10. DISTANCEAlicia’s average speed riding her bike
is 11.5 miles per hour. She takes a round trip of 40
miles. It takes her 1 hour and 20 minutes with the
wind and 2 hours and 30 minutes against the wind.
a. Write an expression for Alicia’s time with the
wind.
b. Write an expression for Alicia’s time against the
wind.
c. How long does it take to complete the trip?
d. Write and solve the rational equation to determine
the speed of the wind.
The speed of the wind is 3.5 mph.
SOLUTION:
a. Let x be the speed of the wind. 11. WORK Kendal and Chandi wax cars. Kendal can
The expression for Alicia’s time with the wind is wax a particular car in 60 minutes and Chandi can
. wax the same car in 80 minutes. They plan on
waxing the same car together and want to know how
b. long it will take.
The expression for Alicia’s time against the wind
is . a. How much will Kendal complete in 1 minute?
b. How much will Kendal complete in x minutes?
c.
c. How much will Chandi complete in 1 minute?
d. d. How much will Chandi complete in x minutes?
e.Write a rational equation representing Kendal and
Chandi working together on the car.
f. Solve the equation to determine how long it will
take them to finish the car.
SOLUTION:
a.
b.
c.
The speed of the wind is 3.5 mph.
d.
11. WORK Kendal and Chandi wax cars. Kendal can
wax a particular car in 60 minutes and Chandi can
wax the same car in 80 minutes. They plan on e.
waxing the same car together and want to know how
long it will take.
f.
a. How much will Kendal complete in 1 minute?
b. How much will Kendal complete in x minutes?
c. How much will Chandi complete in 1 minute?
d. How much will Chandi complete in x minutes?
e.Write a rational equation representing Kendal and
Chandi working together on the car.
f. Solve the equation to determine how long it will It will take them about 34.3 minutes to finish the car.
take them to finish the car.
Solve each inequality. Check your solutions.
SOLUTION:
a.
12.
b.
SOLUTION:
The excluded value for this inequality is 0.
c.
Solve the related equation .
d.
e.
f.
Divide the real line in to three intervals as shown.
It will take them about 34.3 minutes to finish the car.
Solve each inequality. Check your solutions. Test x = –1.
12.
SOLUTION:
The excluded value for this inequality is 0.
Solve the related equation .
Test x = 1.
Test x = 2.
Divide the real line in to three intervals as shown.
Test x = –1.
Therefore, the solution is 0 < x < 1.15.
13.
SOLUTION:
The excluded value for this inequality is 0.
Test x = 1.
Solve the related equation .
Test x = 2.
Divide the real line in to three intervals as shown.
Test c = –1.
Therefore, the solution is 0 < x < 1.15.
13.
SOLUTION:
The excluded value for this inequality is 0.
Solve the related equation . Test c = 0.5.
Test c = 1.
Divide the real line in to three intervals as shown.
Test c = –1.
Therefore, the solution is c < 0 or .
14.
SOLUTION:
Test c = 0.5. The excluded value for this inequality is y = 0.
Solve the related equation .
Test c = 1.
Divide the real line in to three intervals as shown.
Test y = –1.
Therefore, the solution is c < 0 or .
14.
SOLUTION:
The excluded value for this inequality is y = 0.
Test .
Solve the related equation .
Test y = 2.
Divide the real line in to three intervals as shown.
Test y = –1.
Therefore, the solution is .
15.
Test .
SOLUTION:
The excluded value of this inequality is b = 0.
Solve the related equation .
Test y = 2.
Divide the real line in to three intervals as shown.
Test b = −1.
Therefore, the solution is .
15.
SOLUTION:
The excluded value of this inequality is b = 0.
Test b = 1.
Solve the related equation .
Test b = 3.
Divide the real line in to three intervals as shown.
Test b = −1. Therefore, the solution is .
Solve each equation. Check your solutions.
16.
SOLUTION:
Test b = 1.
Test b = 3. Check:
The solution is 9.
Therefore, the solution is .
Solve each equation. Check your solutions. 17.
16.
SOLUTION:
SOLUTION:
Check:
Check:
The solution is 2.
The solution is 9.
18.
SOLUTION:
17.
SOLUTION:
Check:
Check:
The solution is 7.
The solution is 2.
19.
18.
SOLUTION:
SOLUTION:
Check:
Check:
The solution is 1.
The solution is 7.
20.
19.
SOLUTION:
SOLUTION:
Use the quadratic formula.
Check:
The solution is 1.
Check: x =
20.
SOLUTION:
Check: x =
Use the quadratic formula. Therefore, the solution set is {2, –12}
21.
SOLUTION:
Check: x =
Check: x = Use the Quadratic formula to solve .
There is no real solution for the quadratic equation
. Therefore, the solution for the given
rational equation is .
Therefore, the solution set is {2, –12}
22. CHEMISTRYHow many milliliters of a 20% acid
solution must be added to 40 milliliters of a 75% acid
solution to create a 30% acid solution?
21.
SOLUTION:
Let x milliliters of a 20% acid solution is added to 40
milliliters of a 75% acid solution.
SOLUTION:
Use the Quadratic formula to solve .
Check:
There is no real solution for the quadratic equation
. Therefore, the solution for the given
rational equation is .
22. CHEMISTRYHow many milliliters of a 20% acid
solution must be added to 40 milliliters of a 75% acid Therefore, 180 milliliters of a 20% acid solution must
solution to create a 30% acid solution? be added to 40 milliliters of a 75% acid solution to
create a 30% acid solution.
SOLUTION:
Let x milliliters of a 20% acid solution is added to 40 23. GROCERIES Ellen bought 3 pounds of bananas for
milliliters of a 75% acid solution. $0.90 per pound. How many pounds of apples
costing $1.25 per pound must she purchase so that
the total cost for fruit is $1 per pound?
SOLUTION:
Let Ellen bought x pounds of apples.
She needs to purchase 1.2 pounds of apples.
Check:
24. BUILDINGBryan’s volunteer group can build a
garage in 12 hours. Sequoia’s group can build it in 16
hours. How long would it take them if they worked
together?
SOLUTION:
The rate for Bryan’s volunteer group is .
Therefore, 180 milliliters of a 20% acid solution must
be added to 40 milliliters of a 75% acid solution to
create a 30% acid solution. The rate for Sequoia’s group is .
23. GROCERIES Ellen bought 3 pounds of bananas for Let their combined rate is .
$0.90 per pound. How many pounds of apples
costing $1.25 per pound must she purchase so that
the total cost for fruit is $1 per pound?
SOLUTION:
Let Ellen bought x pounds of apples.
Therefore, it would take about 6.86 hours to build a
garage if they worked together.
She needs to purchase 1.2 pounds of apples.
Solve each inequality. Check your solutions.
24. BUILDINGBryan’s volunteer group can build a
garage in 12 hours. Sequoia’s group can build it in 16
25.
hours. How long would it take them if they worked
together?
SOLUTION:
The excluded value for this inequality is x = 0.
SOLUTION:
The rate for Bryan’s volunteer group is .
The rate for Sequoia’s group is .
Let their combined rate is .
Divide the real line in to three intervals as shown.
Therefore, it would take about 6.86 hours to build a
garage if they worked together. Test x = −1.
Solve each inequality. Check your solutions.
25.
SOLUTION: Test x = 1.
The excluded value for this inequality is x = 0.
Test x = 2.
The solution for the inequality is x < 0 or x > 1.75.
Divide the real line in to three intervals as shown.
26.
Test x = −1.
SOLUTION:
The excluded value for this inequality is a = 0.
Test x = 1.
Test x = 2.
Divide the inequality in to three intervals as shown.
Test a = −1.
The solution for the inequality is x < 0 or x > 1.75.
26.
SOLUTION:
The excluded value for this inequality is a = 0.
Test a = 1.
Test a = 2.
Divide the inequality in to three intervals as shown. Therefore, the solution set is 0 < a < 1.1.
27.
Test a = −1.
SOLUTION:
The excluded values for this inequality is x = −2 and
x = 2.
Solve the related equation .
Test a = 1.
Test a = 2.
Divide the real line in to four intervals as shown.
Therefore, the solution set is 0 < a < 1.1.
Test x = −4.
27.
SOLUTION:
The excluded values for this inequality is x = −2 and
x = 2.
Solve the related equation .
Test x = 0.
Test x = 4.
Divide the real line in to four intervals as shown.
Test x = −4.
Test x = 16.
Test x = 0.
Therefore, the solution set for the inequality is x < −2
or 2 < x < 14.
28.
SOLUTION:
The excluded value for this inequality is x = 3 and x =
Test x = 4. –4.
Solve the related equation .
Test x = 16.
There exists no real solution for the quadratic
equation .
Divide the real line in to three intervals as shown.
Therefore, the solution set for the inequality is x < −2
or 2 < x < 14.
Test x = −5.
28.
SOLUTION:
The excluded value for this inequality is x = 3 and x =
–4.
Solve the related equation .
Test x = 0.
Test x = 5.
There exists no real solution for the quadratic
equation .
Divide the real line in to three intervals as shown.
Test x = −5.
The solution set is 4 < x < 3.
–
29.
SOLUTION:
The excluded value for this inequality is x = 4.
Solve the related equation .
Test x = 0.
Test x = 5.
Solve the quadratic equation using the Quadratic
formula.
The solution set is 4 < x < 3.
–
29.
SOLUTION:
The excluded value for this inequality is x = 4.
Solve the related equation . Divide the real line in to 4 intervals as shown.
Test x = 6.
–
Solve the quadratic equation using the Quadratic
formula.
Test x = 0.
Test x = 5.
Test x = 6.
Divide the real line in to 4 intervals as shown.
Test x = 6.
–
The solution set for the inequality is x < 5 or
–
.
Test x = 0.
30.
SOLUTION:
The excluded values for this inequality are x = 2
–
and x = 1.
Test x = 5.
Solve the related equation .
Test x = 6.
The solution set for the inequality is x < 5 or
–
.
Divide the real line in to 5 intervals as shown.
30.
Test x = 6.
–
SOLUTION:
The excluded values for this inequality are x = 2
–
and x = 1.
Solve the related equation .
Test x = 4.
–
Test x = 0.
Divide the real line in to 5 intervals as shown.
Test .
Test x = 6.
–
Test x = 4.
–
Test x = 4.
Test x = 0.
The solution set for the inequality is x < 5 or 2 < x
Test . – −
< 1 or x > 2.
AIR TRAVELIt takes a plane 20 hours to fly to its
31.
destination against the wind. The return trip takes 16
hours. If the plane’s average speed in still air is 500
miles per hour, what is the average speed of the wind
during the flight?
SOLUTION:
Test x = 4.
The average speed of the wind during the flight is
about 55.56 miles per hour.
FINANCIAL LITERACY
32. Judie wants to invest
$10,000 in two different accounts. The risky account
earns 9% interest, while the other account earns 5%
interest. She wants to earn $750 interest for the year.
Of tables, graphs, or equations, choose the best
The solution set for the inequality is x < 5 or 2 < x
– − representation needed and determine how much
< 1 or x > 2. should be invested in each account.
AIR TRAVELIt takes a plane 20 hours to fly to its
31.
destination against the wind. The return trip takes 16 SOLUTION:
hours. If the plane s average speed in still air is 500 Judie invest x dollars in the account earns 9%
’ interest and (10000 x) dollars in the account earns
miles per hour, what is the average speed of the wind −
during the flight? 5% interest.
SOLUTION:
Thus, Judie should invest $6250 at 9% account and
$3750 at 5% account.
MULTIPLE REPRESENTATIONSConsider
33.
The average speed of the wind during the flight is a. ALGEBRAICSolve the equation for x. Were
about 55.56 miles per hour. any values of x extraneous?
b. GRAPHICALGraph
and
FINANCIAL LITERACY
32. Judie wants to invest
$10,000 in two different accounts. The risky account x < 5.
earns 9% interest, while the other account earns 5% on the same graph for 0 <
interest. She wants to earn $750 interest for the year.
Of tables, graphs, or equations, choose the best
representation needed and determine how much c. ANALYTICALFor what value(s) of x do they
should be invested in each account. intersect? Do they intersect where x is extraneous
for the original equation?
d. VERBALUse this knowledge to describe how
SOLUTION:
Judie invest x dollars in the account earns 9% you can use a graph to determine whether an
interest and (10000 x) dollars in the account earns apparent solution of a rational equation is extraneous.
−
5% interest.
SOLUTION:
a.
Thus, Judie should invest $6250 at 9% account and
$3750 at 5% account.
MULTIPLE REPRESENTATIONSConsider
33.
a. ALGEBRAICSolve the equation for x. Were Check: x = 1
any values of x extraneous?
b. GRAPHICALGraph
and
x < 5.
on the same graph for 0 <
x = 3 is the excluded value for the equation.
c. ANALYTICALFor what value(s) of x do they Therefore, x = 3 is the extraneous solution and x = 1
intersect? Do they intersect where x is extraneous is the solution for the equation.
for the original equation?
b.
d. VERBALUse this knowledge to describe how
you can use a graph to determine whether an
apparent solution of a rational equation is extraneous.
SOLUTION:
a.
c. Two graphs intersect at x = 1 and they do not
intersect at the extraneous solution x = 3.]
d. Graph both sides of the equation. Where the
graphs intersect, there is a solution. If they do not,
then the possible solution is extraneous.
Solve each equation. Check your solutions.
Check: x = 1
34.
SOLUTION:
x = 3 is the excluded value for the equation.
Therefore, x = 3 is the extraneous solution and x = 1
is the solution for the equation.
b.
Check:
c. Two graphs intersect at x = 1 and they do not
intersect at the extraneous solution x = 3.]
d. Graph both sides of the equation. Where the
graphs intersect, there is a solution. If they do not,
then the possible solution is extraneous.
The solution is y = 1.
−
Solve each equation. Check your solutions.
34.
35.
SOLUTION:
SOLUTION:
Check:
Check:
The solution set is .
OPEN ENDEDGive an example of a rational
36.
equation that can be solved by multiplying each side
The solution is y = 1.
− of the equation by 4(x + 3)(x 4).
–
SOLUTION:
Sample answer:
35.
SOLUTION:
CHALLENGESolve
37.
SOLUTION:
Check:
The solution set is .
OPEN ENDEDGive an example of a rational
36.
equation that can be solved by multiplying each side
of the equation by 4(x + 3)(x 4).
–
Therefore, the solution is all real numbers except 5,
5, and 0.
−
SOLUTION:
Sample answer:
CCSS TOOLS
38. While using the table feature on the
graphing calculator to explore the
ERROR
values 2 and 3 say . Explain its meaning.
– “ ”
CHALLENGESolve
37.
SOLUTION:
Sample answer:
SOLUTION:
The denominator will equal 0 when x = 2 or x = 3.
−
The values 2 and 3 are undefined values. On the
−
graph of f (x) there would be vertical asymptotes at
these values.
Why should you check
WRITING IN MATH
39.
solutions of rational equations and inequalities?
SOLUTION:
Sample answer: Multiplying each side of a rational
equation or inequality by the LCD can result in
extraneous solutions. Therefore, you should check all
solutions to make sure that they satisfy the original
equation or inequality.
Nine pounds of mixed nuts containing 55% peanuts
Therefore, the solution is all real numbers except 5, 40.
5, and 0. were mixed with 6 pounds of another kind of mixed
− nuts that contain 40% peanuts. What percent of the
new mixture is peanuts?
CCSS TOOLS
38. While using the table feature on the A 58%
graphing calculator to explore the
ERROR B 51%
values 2 and 3 say . Explain its meaning.
– “ ”
C 49%
SOLUTION:
Sample answer: D 47%
SOLUTION:
Let the new mixture contains x percent of peanuts.
The denominator will equal 0 when x = 2 or x = 3.
−
The values 2 and 3 are undefined values. On the
−
graph of f (x) there would be vertical asymptotes at
these values.
Why should you check
WRITING IN MATH
39.
solutions of rational equations and inequalities?
So, the new mixture contains 0.49 or 49% percent of
peanuts. The correct choice is C.
SOLUTION:
Sample answer: Multiplying each side of a rational
equation or inequality by the LCD can result in Working alone, Dato can dig a 10-foot by 10-foot
extraneous solutions. Therefore, you should check all 41.
solutions to make sure that they satisfy the original hole in five hours. Pedro can dig the same hole in six
hours. How long would it take them if they worked
equation or inequality. together?
40. Nine pounds of mixed nuts containing 55% peanuts F1.5 hours
were mixed with 6 pounds of another kind of mixed
nuts that contain 40% peanuts. What percent of the
new mixture is peanuts? G2.34 hours
H 2.52 hours
A 58%
J 2.73 hours
B 51%
C 49% SOLUTION:
D 47%
SOLUTION:
Let the new mixture contains x percent of peanuts.
It would take about 2.73 hours to dig the hole if they
worked together. The correct choice is J.
An aircraft carrier made a trip to Guam and back.
So, the new mixture contains 0.49 or 49% percent of 42.
peanuts. The correct choice is C. The trip there took three hours and the trip back took
four hours. It averaged 6 kilometers per hour on the
return trip. Find the average speed of the trip to
Working alone, Dato can dig a 10-foot by 10-foot Guam.
41.
hole in five hours. Pedro can dig the same hole in six
hours. How long would it take them if they worked A 6 km/h
together?
B
8 km/h
F1.5 hours
C
10 km/h
G2.34 hours
D
12 km/h
H 2.52 hours
J 2.73 hours SOLUTION:
SOLUTION:
It would take about 2.73 hours to dig the hole if they
worked together. The correct choice is J.
42. An aircraft carrier made a trip to Guam and back.
The trip there took three hours and the trip back took
four hours. It averaged 6 kilometers per hour on the
return trip. Find the average speed of the trip to
Guam.
A 6 km/h
B 8 km/h
C 10 km/h
D 12 km/h
SOLUTION:
Solve each equation. Check your solution.
1.
SOLUTION:
Check:
The solution is 11.
2.
Solve each equation. Check your solution.
SOLUTION:
1.
SOLUTION:
Check:
Check:
The solution is 9.
The solution is 11. 3.
SOLUTION:
2.
SOLUTION:
Check: Check:
The solution is 9.
3. The solution is 7.
SOLUTION:
4.
SOLUTION:
Check: Check:
The solution is 3.
The solution is 7.
5.
4.
SOLUTION:
SOLUTION:
Check:
Check:
The solution is 8.
The solution is 3.
6.
5.
SOLUTION:
SOLUTION:
Check:
Check:
The solution is 8.
The solution is x = 5.
6.
7.
SOLUTION:
SOLUTION:
Check: Check:
The solution is x = 5. The solution is 14.
8-6 Solving Rational Equations and Inequalities
7. 8.
SOLUTION: SOLUTION:
Check:
Check:
The solution is 14.
The solution is 14.
8.
CCSS STRUCTURE
9. Sara has 10 pounds of dried
fruit selling for $6.25 per pound. She wants to know
how many pounds of mixed nuts selling for $4.50 per
SOLUTION: pound she needs to make a trail mix selling for $5 per
pound.
a. Let m = the number of pounds of mixed nuts.
Complete the following table.
b. Write a rational equation using the last column of
eSolutions Manual - Powered by Cognero Page4
the table.
c. Solve the equation to determine how many pounds
of mixed nuts are needed.
Check:
SOLUTION:
a.
b.
The solution is 14.
c.
CCSS STRUCTURE
9. Sara has 10 pounds of dried
fruit selling for $6.25 per pound. She wants to know
how many pounds of mixed nuts selling for $4.50 per
pound she needs to make a trail mix selling for $5 per
pound. Therefore, 25 pounds of mixed nuts are needed.
a. Let m = the number of pounds of mixed nuts.
Complete the following table.
10. DISTANCEAlicia’s average speed riding her bike
is 11.5 miles per hour. She takes a round trip of 40
miles. It takes her 1 hour and 20 minutes with the
wind and 2 hours and 30 minutes against the wind.
a. Write an expression for Alicia’s time with the
wind.
b. Write a rational equation using the last column of
the table. b. Write an expression for Alicia’s time against the
wind.
c. Solve the equation to determine how many pounds
of mixed nuts are needed.
c. How long does it take to complete the trip?
d. Write and solve the rational equation to determine
SOLUTION: the speed of the wind.
a.
SOLUTION:
a. Let x be the speed of the wind.
The expression for Alicia’s time with the wind is
.
b.
b. The expression for Alicia’s time against the wind
is .
c.
c.
d.
Therefore, 25 pounds of mixed nuts are needed.
10. DISTANCEAlicia’s average speed riding her bike
is 11.5 miles per hour. She takes a round trip of 40
miles. It takes her 1 hour and 20 minutes with the
wind and 2 hours and 30 minutes against the wind.
a. Write an expression for Alicia’s time with the
wind.
b. Write an expression for Alicia’s time against the
wind.
c. How long does it take to complete the trip?
d. Write and solve the rational equation to determine
the speed of the wind. The speed of the wind is 3.5 mph.
SOLUTION: 11. WORK Kendal and Chandi wax cars. Kendal can
a. Let x be the speed of the wind. wax a particular car in 60 minutes and Chandi can
The expression for Alicia’s time with the wind is wax the same car in 80 minutes. They plan on
. waxing the same car together and want to know how
long it will take.
b. The expression for Alicia’s time against the wind a. How much will Kendal complete in 1 minute?
is .
b. How much will Kendal complete in x minutes?
c.
c. How much will Chandi complete in 1 minute?
d. How much will Chandi complete in x minutes?
d.
e.Write a rational equation representing Kendal and
Chandi working together on the car.
f. Solve the equation to determine how long it will
take them to finish the car.
SOLUTION:
a.
b.
c.
The speed of the wind is 3.5 mph. d.
11. WORK Kendal and Chandi wax cars. Kendal can
wax a particular car in 60 minutes and Chandi can e.
wax the same car in 80 minutes. They plan on
waxing the same car together and want to know how
long it will take. f.
a. How much will Kendal complete in 1 minute?
b. How much will Kendal complete in x minutes?
c. How much will Chandi complete in 1 minute?
d. How much will Chandi complete in x minutes?
e.Write a rational equation representing Kendal and
Chandi working together on the car.
It will take them about 34.3 minutes to finish the car.
f. Solve the equation to determine how long it will
take them to finish the car.
Solve each inequality. Check your solutions.
SOLUTION:
a. 12.
b. SOLUTION:
The excluded value for this inequality is 0.
Solve the related equation .
c.
d.
e.
f.
Divide the real line in to three intervals as shown.
It will take them about 34.3 minutes to finish the car.
Test x = –1.
Solve each inequality. Check your solutions.
12.
SOLUTION:
The excluded value for this inequality is 0.
Solve the related equation . Test x = 1.
Test x = 2.
Divide the real line in to three intervals as shown.
Test x = –1.
Therefore, the solution is 0 < x < 1.15.
13.
SOLUTION:
The excluded value for this inequality is 0.
Test x = 1. Solve the related equation .
Test x = 2.
Divide the real line in to three intervals as shown.
Test c = –1.
Therefore, the solution is 0 < x < 1.15.
13.
SOLUTION:
The excluded value for this inequality is 0.
Test c = 0.5.
Solve the related equation .
Test c = 1.
Divide the real line in to three intervals as shown.
Test c = –1.
Therefore, the solution is c < 0 or .
14.
SOLUTION:
The excluded value for this inequality is y = 0.
Test c = 0.5.
Solve the related equation .
Test c = 1.
Divide the real line in to three intervals as shown.
Test y = –1.
Therefore, the solution is c < 0 or .
14.
SOLUTION: Test .
The excluded value for this inequality is y = 0.
Solve the related equation .
Test y = 2.
Divide the real line in to three intervals as shown.
Test y = –1.
Therefore, the solution is .
15.
Test . SOLUTION:
The excluded value of this inequality is b = 0.
Solve the related equation .
Test y = 2.
Divide the real line in to three intervals as shown.
Test b = −1.
Therefore, the solution is .
15.
SOLUTION:
The excluded value of this inequality is b = 0. Test b = 1.
Solve the related equation .
Test b = 3.
Divide the real line in to three intervals as shown.
Therefore, the solution is .
Test b = −1.
Solve each equation. Check your solutions.
16.
SOLUTION:
Test b = 1.
Check:
Test b = 3.
The solution is 9.
Therefore, the solution is .
17.
Solve each equation. Check your solutions.
SOLUTION:
16.
SOLUTION:
Check:
Check:
The solution is 2.
18.
The solution is 9.
SOLUTION:
17.
SOLUTION:
Check:
Check:
The solution is 7.
The solution is 2.
19.
18.
SOLUTION:
SOLUTION:
Check:
Check:
The solution is 1.
The solution is 7.
20.
19.
SOLUTION:
SOLUTION:
Use the quadratic formula.
Check:
The solution is 1.
Check: x =
20.
SOLUTION:
Check: x =
Therefore, the solution set is {2, –12}
Use the quadratic formula.
21.
SOLUTION:
Check: x =
Use the Quadratic formula to solve .
Check: x =
There is no real solution for the quadratic equation
. Therefore, the solution for the given
rational equation is .
22. CHEMISTRYHow many milliliters of a 20% acid
Therefore, the solution set is {2, –12} solution must be added to 40 milliliters of a 75% acid
solution to create a 30% acid solution?
SOLUTION:
21.
Let x milliliters of a 20% acid solution is added to 40
milliliters of a 75% acid solution.
SOLUTION:
Use the Quadratic formula to solve .
Check:
There is no real solution for the quadratic equation
. Therefore, the solution for the given
rational equation is .
Therefore, 180 milliliters of a 20% acid solution must
22. CHEMISTRYHow many milliliters of a 20% acid be added to 40 milliliters of a 75% acid solution to
solution must be added to 40 milliliters of a 75% acid create a 30% acid solution.
solution to create a 30% acid solution?
23. GROCERIES Ellen bought 3 pounds of bananas for
SOLUTION:
Let x milliliters of a 20% acid solution is added to 40 $0.90 per pound. How many pounds of apples
milliliters of a 75% acid solution. costing $1.25 per pound must she purchase so that
the total cost for fruit is $1 per pound?
SOLUTION:
Let Ellen bought x pounds of apples.
She needs to purchase 1.2 pounds of apples.
Check:
24. BUILDINGBryan’s volunteer group can build a
garage in 12 hours. Sequoia’s group can build it in 16
hours. How long would it take them if they worked
together?
SOLUTION:
The rate for Bryan’s volunteer group is .
Therefore, 180 milliliters of a 20% acid solution must The rate for Sequoia’s group is .
be added to 40 milliliters of a 75% acid solution to
create a 30% acid solution.
23. GROCERIES Ellen bought 3 pounds of bananas for Let their combined rate is .
$0.90 per pound. How many pounds of apples
costing $1.25 per pound must she purchase so that
the total cost for fruit is $1 per pound?
SOLUTION:
Let Ellen bought x pounds of apples.
Therefore, it would take about 6.86 hours to build a
garage if they worked together.
She needs to purchase 1.2 pounds of apples. Solve each inequality. Check your solutions.
25.
24. BUILDINGBryan’s volunteer group can build a
garage in 12 hours. Sequoia’s group can build it in 16
hours. How long would it take them if they worked
together? SOLUTION:
The excluded value for this inequality is x = 0.
SOLUTION:
The rate for Bryan’s volunteer group is .
The rate for Sequoia’s group is .
Let their combined rate is .
Divide the real line in to three intervals as shown.
Therefore, it would take about 6.86 hours to build a Test x = −1.
garage if they worked together.
Solve each inequality. Check your solutions.
25.
Test x = 1.
SOLUTION:
The excluded value for this inequality is x = 0.
Test x = 2.
The solution for the inequality is x < 0 or x > 1.75.
Divide the real line in to three intervals as shown.
26.
SOLUTION:
Test x = −1. The excluded value for this inequality is a = 0.
Test x = 1.
Test x = 2.
Divide the inequality in to three intervals as shown.
Test a = −1.
The solution for the inequality is x < 0 or x > 1.75.
26.
SOLUTION:
The excluded value for this inequality is a = 0.
Test a = 1.
Test a = 2.
Therefore, the solution set is 0 < a < 1.1.
Divide the inequality in to three intervals as shown.
27.
Test a = −1. SOLUTION:
The excluded values for this inequality is x = −2 and
x = 2.
Solve the related equation .
Test a = 1.
Test a = 2.
Divide the real line in to four intervals as shown.
Test x = −4.
Therefore, the solution set is 0 < a < 1.1.
27.
SOLUTION:
The excluded values for this inequality is x = −2 and
x = 2.
Solve the related equation .
Test x = 0.
Test x = 4.
Divide the real line in to four intervals as shown.
Test x = −4.
Test x = 16.
Test x = 0.
Therefore, the solution set for the inequality is x < −2
or 2 < x < 14.
28.
SOLUTION:
The excluded value for this inequality is x = 3 and x =
–4.
Test x = 4.
Solve the related equation .
Test x = 16.
There exists no real solution for the quadratic
equation .
Divide the real line in to three intervals as shown.
Therefore, the solution set for the inequality is x < −2
or 2 < x < 14.
Test x = −5.
28.
SOLUTION:
The excluded value for this inequality is x = 3 and x =
–4.
Solve the related equation .
Test x = 0.
Test x = 5.
There exists no real solution for the quadratic
equation .
Divide the real line in to three intervals as shown.
The solution set is 4 < x < 3.
Test x = −5. –
29.
SOLUTION:
The excluded value for this inequality is x = 4.
Solve the related equation .
Test x = 0.
Solve the quadratic equation using the Quadratic
Test x = 5. formula.
The solution set is 4 < x < 3.
–
29.
SOLUTION:
The excluded value for this inequality is x = 4.
Divide the real line in to 4 intervals as shown.
Solve the related equation .
Test x = 6.
–
Solve the quadratic equation using the Quadratic
formula.
Test x = 0.
Test x = 5.
Test x = 6.
Divide the real line in to 4 intervals as shown.
Test x = 6.
–
The solution set for the inequality is x < 5 or
–
.
30.
Test x = 0.
SOLUTION:
The excluded values for this inequality are x = 2
–
and x = 1.
Test x = 5. Solve the related equation .
Test x = 6.
The solution set for the inequality is x < 5 or
–
. Divide the real line in to 5 intervals as shown.
30.
Test x = 6.
–
SOLUTION:
The excluded values for this inequality are x = 2
–
and x = 1.
Solve the related equation .
Test x = 4.
–
Test x = 0.
Divide the real line in to 5 intervals as shown.
Test .
Test x = 6.
–
Test x = 4.
–
Test x = 4.
Test x = 0.
The solution set for the inequality is x < 5 or 2 < x
– −
< 1 or x > 2.
Test .
AIR TRAVELIt takes a plane 20 hours to fly to its
31.
destination against the wind. The return trip takes 16
hours. If the plane’s average speed in still air is 500
miles per hour, what is the average speed of the wind
during the flight?
SOLUTION:
Test x = 4.
The average speed of the wind during the flight is
about 55.56 miles per hour.
FINANCIAL LITERACY
32. Judie wants to invest
$10,000 in two different accounts. The risky account
earns 9% interest, while the other account earns 5%
interest. She wants to earn $750 interest for the year.
Of tables, graphs, or equations, choose the best
representation needed and determine how much
The solution set for the inequality is x < 5 or 2 < x
– − should be invested in each account.
< 1 or x > 2.
SOLUTION:
AIR TRAVELIt takes a plane 20 hours to fly to its Judie invest x dollars in the account earns 9%
31.
destination against the wind. The return trip takes 16 interest and (10000 − x) dollars in the account earns
hours. If the plane s average speed in still air is 500 5% interest.
’
miles per hour, what is the average speed of the wind
during the flight?
SOLUTION:
Thus, Judie should invest $6250 at 9% account and
$3750 at 5% account.
MULTIPLE REPRESENTATIONSConsider
33.
a. ALGEBRAICSolve the equation for x. Were
any values of x extraneous?
The average speed of the wind during the flight is
about 55.56 miles per hour.
b. GRAPHICALGraph
and
FINANCIAL LITERACY
32. Judie wants to invest x < 5.
$10,000 in two different accounts. The risky account on the same graph for 0 <
earns 9% interest, while the other account earns 5%
interest. She wants to earn $750 interest for the year. c. ANALYTICALFor what value(s) of x do they
Of tables, graphs, or equations, choose the best intersect? Do they intersect where x is extraneous
representation needed and determine how much for the original equation?
should be invested in each account.
d. VERBALUse this knowledge to describe how
you can use a graph to determine whether an
SOLUTION: apparent solution of a rational equation is extraneous.
Judie invest x dollars in the account earns 9%
interest and (10000 x) dollars in the account earns
−
5% interest.
SOLUTION:
a.
Thus, Judie should invest $6250 at 9% account and
$3750 at 5% account.
MULTIPLE REPRESENTATIONSConsider
33.
Check: x = 1
a. ALGEBRAICSolve the equation for x. Were
any values of x extraneous?
b. GRAPHICALGraph
and
x < 5.
on the same graph for 0 < x = 3 is the excluded value for the equation.
Therefore, x = 3 is the extraneous solution and x = 1
is the solution for the equation.
c. ANALYTICALFor what value(s) of x do they
intersect? Do they intersect where x is extraneous
for the original equation? b.
d. VERBALUse this knowledge to describe how
you can use a graph to determine whether an
apparent solution of a rational equation is extraneous.
SOLUTION:
a.
c. Two graphs intersect at x = 1 and they do not
intersect at the extraneous solution x = 3.]
d. Graph both sides of the equation. Where the
graphs intersect, there is a solution. If they do not,
then the possible solution is extraneous.
Solve each equation. Check your solutions.
34.
Check: x = 1
SOLUTION:
x = 3 is the excluded value for the equation.
Therefore, x = 3 is the extraneous solution and x = 1
is the solution for the equation.
b.
Check:
c. Two graphs intersect at x = 1 and they do not
intersect at the extraneous solution x = 3.]
d. Graph both sides of the equation. Where the
graphs intersect, there is a solution. If they do not,
then the possible solution is extraneous.
The solution is y = 1.
−
Solve each equation. Check your solutions.
35.
34.
SOLUTION:
SOLUTION:
Check:
Check:
The solution set is .
OPEN ENDEDGive an example of a rational
36.
equation that can be solved by multiplying each side
of the equation by 4(x + 3)(x 4).
–
The solution is y = 1.
−
SOLUTION:
Sample answer:
35.
SOLUTION:
CHALLENGESolve
37.
SOLUTION:
Check:
The solution set is .
OPEN ENDEDGive an example of a rational
36.
equation that can be solved by multiplying each side
of the equation by 4(x + 3)(x 4).
– Therefore, the solution is all real numbers except 5,
5, and 0.
−
SOLUTION:
Sample answer:
CCSS TOOLS
38. While using the table feature on the
graphing calculator to explore the
ERROR
values 2 and 3 say . Explain its meaning.
– “ ”
SOLUTION:
CHALLENGESolve
37.
Sample answer:
SOLUTION:
The denominator will equal 0 when x = 2 or x = 3.
−
The values 2 and 3 are undefined values. On the
−
graph of f (x) there would be vertical asymptotes at
these values.
Why should you check
WRITING IN MATH
39.
solutions of rational equations and inequalities?
SOLUTION:
Sample answer: Multiplying each side of a rational
equation or inequality by the LCD can result in
extraneous solutions. Therefore, you should check all
solutions to make sure that they satisfy the original
equation or inequality.
40. Nine pounds of mixed nuts containing 55% peanuts
were mixed with 6 pounds of another kind of mixed
Therefore, the solution is all real numbers except 5, nuts that contain 40% peanuts. What percent of the
5, and 0.
− new mixture is peanuts?
A 58%
CCSS TOOLS
38. While using the table feature on the
graphing calculator to explore B 51%
the
ERROR
values 2 and 3 say . Explain its meaning.
– “ ” C 49%
D 47%
SOLUTION:
Sample answer:
SOLUTION:
Let the new mixture contains x percent of peanuts.
The denominator will equal 0 when x = 2 or x = 3.
−
The values 2 and 3 are undefined values. On the
−
graph of f (x) there would be vertical asymptotes at
these values.
Why should you check
WRITING IN MATH
39.
solutions of rational equations and inequalities?
So, the new mixture contains 0.49 or 49% percent of
peanuts. The correct choice is C.
SOLUTION:
Sample answer: Multiplying each side of a rational Working alone, Dato can dig a 10-foot by 10-foot
equation or inequality by the LCD can result in 41.
extraneous solutions. Therefore, you should check all hole in five hours. Pedro can dig the same hole in six
solutions to make sure that they satisfy the original hours. How long would it take them if they worked
together?
equation or inequality.
F1.5 hours
Nine pounds of mixed nuts containing 55% peanuts
40.
were mixed with 6 pounds of another kind of mixed G2.34 hours
nuts that contain 40% peanuts. What percent of the
new mixture is peanuts? H 2.52 hours
A 58% J 2.73 hours
B 51%
SOLUTION:
C 49%
D 47%
SOLUTION:
Let the new mixture contains x percent of peanuts.
It would take about 2.73 hours to dig the hole if they
worked together. The correct choice is J.
42. An aircraft carrier made a trip to Guam and back.
The trip there took three hours and the trip back took
So, the new mixture contains 0.49 or 49% percent of four hours. It averaged 6 kilometers per hour on the
peanuts. The correct choice is C. return trip. Find the average speed of the trip to
Guam.
Working alone, Dato can dig a 10-foot by 10-foot
41. A 6 km/h
hole in five hours. Pedro can dig the same hole in six
hours. How long would it take them if they worked
together? B 8 km/h
C 10 km/h
F1.5 hours
D 12 km/h
G2.34 hours
H 2.52 hours
SOLUTION:
J 2.73 hours
SOLUTION:
It would take about 2.73 hours to dig the hole if they
worked together. The correct choice is J.
42. An aircraft carrier made a trip to Guam and back.
The trip there took three hours and the trip back took
four hours. It averaged 6 kilometers per hour on the
return trip. Find the average speed of the trip to
Guam.
A 6 km/h
B 8 km/h
C 10 km/h
D 12 km/h
SOLUTION:
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