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Solve each inequality. Then graph the solution
set on a number line.
1.
ANSWER:
2.
ANSWER:
3.
ANSWER:
4.
ANSWER:
5.
ANSWER:
6.
ANSWER:
1-5 Solving Inequalities
Solve each inequality. Then graph the solution
7.
set on a number line.
1.
ANSWER:
ANSWER:
2.
8.
ANSWER:
ANSWER:
3. YARD WORK
9. Tara is delivering bags of mulch.
ANSWER: Each bag weighs 48 pounds, and the push cart
weighs 65 pounds. If her flat-bed truck is capable of
hauling 2000 pounds, how many bags of mulch can
Tara safely take on each trip?
ANSWER:
4. 40 bags
ANSWER: Solve each inequality. Then graph the solution
set on a number line.
10.
ANSWER:
5.
ANSWER:
11.
ANSWER:
6.
ANSWER:
12.
ANSWER:
7.
ANSWER:
13.
ANSWER:
eSolutions Manual - Powered by Cognero Page1
8.
14.
ANSWER:
ANSWER:
YARD WORK
9. Tara is delivering bags of mulch.
Each bag weighs 48 pounds, and the push cart
weighs 65 pounds. If her flat-bed truck is capable of
hauling 2000 pounds, how many bags of mulch can 15.
Tara safely take on each trip?
ANSWER:
ANSWER:
40 bags
Solve each inequality. Then graph the solution
set on a number line.
16.
10.
ANSWER:
ANSWER:
17.
11.
ANSWER:
ANSWER:
18.
12.
ANSWER:
ANSWER:
19.
13.
ANSWER:
ANSWER:
20.
14.
ANSWER:
ANSWER:
21.
15.
ANSWER:
ANSWER:
GYMNASTICS In a gymnastics competition, an
22.
16. athlete s final score is calculated by taking 75% of
’
ANSWER: the average technical score and adding 25% of the
artistic score. All scores are out of 10, and one
gymnast has a 7.6 average technical score. What
artistic score does the gymnast need to have a final
score of at least 8.0?
17.
ANSWER:
ANSWER: 9.2
Define a variable and write an inequality for
each problem. Then solve.
23. Twelve less than the product of three and a number
is less than 21.
18.
ANSWER:
ANSWER: 3x 12 < 21; x < 11
–
24. The quotient of three times a number and 4 is at least
16.
–
19.
ANSWER:
ANSWER:
25. The difference of 5 times a number and 6 is greater
than the number.
ANSWER:
20.
ANSWER: The quotient of the sum of 3 and a number and 6 is
26.
less than –2.
ANSWER:
21.
HIKINGDanielle can hike 3 miles in an hour, but
27.
she has to take a one-hour break for lunch and a
ANSWER: one-hour break for dinner. If Danielle wants to hike
at least 18 miles, solve to determine
how many hours the hike should take.
GYMNASTICS In a gymnastics competition, an ANSWER:
22. at least 8 hours
athlete’s final score is calculated by taking 75% of
the average technical score and adding 25% of the Solve each inequality. Then graph the solution
artistic score. All scores are out of 10, and one
gymnast has a 7.6 average technical score. What set on a number line.
artistic score does the gymnast need to have a final 28.
score of at least 8.0?
ANSWER:
ANSWER:
9.2
Define a variable and write an inequality for
each problem. Then solve.
Twelve less than the product of three and a number 29.
23.
is less than 21.
ANSWER:
ANSWER:
3x – 12 < 21; x < 11
24. The quotient of three times a number and 4 is at least
16.
–
30.
ANSWER:
ANSWER:
25. The difference of 5 times a number and 6 is greater
than the number.
31.
ANSWER:
ANSWER:
The quotient of the sum of 3 and a number and 6 is
26.
less than –2.
ANSWER:
32.
ANSWER:
HIKINGDanielle can hike 3 miles in an hour, but
27.
she has to take a one-hour break for lunch and a
one-hour break for dinner. If Danielle wants to hike
at least 18 miles, solve to determine
33.
how many hours the hike should take.
ANSWER:
ANSWER:
at least 8 hours
Solve each inequality. Then graph the solution
set on a number line.
34.
28.
ANSWER:
ANSWER:
29.
35.
ANSWER:
ANSWER:
30.
36.
ANSWER:
ANSWER:
31.
ANSWER:
MONEYJin is selling advertising space in Central
37.
City Magazine to local businesses. Jin earns 3%
commission for every advertisement he sells plus a
salary of $250 a week. If the average amount of
money that a business spends on an advertisement is
32. $500, how many advertisements must he sell each
week to make a salary of at least $700 that week?
ANSWER: a. Write an inequality to describe this situation.
b. Solve the inequality and interpret the solution.
ANSWER:
a.
33. b.
He must sell at least 30 advertisements.
ANSWER:
Define a variable and write an inequality for
each problem. Then solve.
38. One third of the sum of 5 times a number and 3 is
less than one fourth the sum of six times that number
and 5.
34.
ANSWER:
ANSWER:
39. The sum of one third a number and 4 is at most the
sum of twice that number and 12.
ANSWER:
35.
ANSWER:
SENSE-MAKINGThe sides of square ABCD are
40.
extended to form rectangle DEFG. If the perimeter
of the rectangle is at least twice the perimeter of the
square, what is the maximum length of a side of
square ABCD?
36.
ANSWER:
MONEYJin is selling advertising space in Central
37.
City Magazine to local businesses. Jin earns 3%
commission for every advertisement he sells plus a ANSWER:
salary of $250 a week. If the average amount of 9 in.
money that a business spends on an advertisement is
MARATHONSJamie wants to be able to run at
$500, how many advertisements must he sell each 41.
week to make a salary of at least $700 that week? least the standard marathon distance of 26.2 miles. A
a. Write an inequality to describe this situation. good rule for training is that runners generally have
b. Solve the inequality and interpret the solution. enough endurance to finish a race that is up to 3
times his or her average daily distance.
a. If the length of her current daily run is 5 miles,
ANSWER: write an inequality to find the amount by which she
a. needs to increase her daily run to have enough
b. endurance to finish a marathon.
He must sell at least 30 advertisements.
b. Solve the inequality and interpret the solution.
Define a variable and write an inequality for
ANSWER:
each problem. Then solve.
38. One third of the sum of 5 times a number and 3 is a.
less than one fourth the sum of six times that number b
and 5. . In order to have enough endurance to
run a marathon, Jamie should increase the distance
of her average daily run by at least 3.73 miles.
ANSWER:
MODELINGThe costs for renting a car from Ace
42.
Car Rental and from Basic Car Rental are shown in
The sum of one third a number and 4 is at most the the table. For what mileage does Basic have the
39. better deal? Use the inequality
sum of twice that number and 12. . Explain why this inequality
ANSWER: works.
SENSE-MAKINGThe sides of square ABCD are
40.
extended to form rectangle DEFG. If the perimeter
of the rectangle is at least twice the perimeter of the
square, what is the maximum length of a side of
square ABCD?
ANSWER:
Basic has the better deal as long as you are traveling
more than 80 miles. Yes, this is the correct inequality
to use. Sample explanation: It works because the
inequality finds the mileage at which Ace’s charge is
greater than Basic’s charge.
MULTIPLE REPRESENTATIONSIn this
43.
exercise, you will explore graphing inequalities on a
coordinate plane.
a. TABULAR
Organize the following into a table.
ANSWER:
9 in. Substitute 5 points into the inequality .
State whether the resulting statement is true or
MARATHONSJamie wants to be able to run at
41. false.
least the standard marathon distance of 26.2 miles. A
good rule for training is that runners generally have b. GRAPHICAL
enough endurance to finish a race that is up to 3 Graph . Also graph
times his or her average daily distance. the 5 points from the table. Label all points that
a. If the length of her current daily run is 5 miles, resulted in a true statement with a T. Label all points
write an inequality to find the amount by which she that resulted in a false statement with an F.
c. VERBAL
needs to increase her daily run to have enough Describe the pattern produced by the
endurance to finish a marathon. points you have labeled. Make a conjecture about
b. Solve the inequality and interpret the solution. which points on the coordinate plane would result in
true and false statements.
ANSWER:
a. ANSWER:
b a. Sample answer:
. In order to have enough endurance to
run a marathon, Jamie should increase the distance
of her average daily run by at least 3.73 miles.
MODELINGThe costs for renting a car from Ace
42.
Car Rental and from Basic Car Rental are shown in
the table. For what mileage does Basic have the
better deal? Use the inequality
. Explain why this inequality
works.
b. Sample answer:
ANSWER:
Basic has the better deal as long as you are traveling
more than 80 miles. Yes, this is the correct inequality
to use. Sample explanation: It works because the
inequality finds the mileage at which Ace’s charge is
greater than Basic’s charge.
MULTIPLE REPRESENTATIONSIn this
43.
exercise, you will explore graphing inequalities on a
coordinate plane. c. Sample answer: The points on or above the line
a. TABULAR result in true statements, and the points below the
Organize the following into a table. line result in false statements. This is true for all
Substitute 5 points into the inequality . points on the coordinate plane.
State whether the resulting statement is true or
CHALLENGEIf , then
false. 44. and
. What is ?
b. GRAPHICAL
Graph . Also graph
the 5 points from the table. Label all points that
resulted in a true statement with a T. Label all points ANSWER:
that resulted in a false statement with an F. (a + b) < 4
c. VERBAL
Describe the pattern produced by the
ERROR ANALYSISMadlynn and Emilie were
points you have labeled. Make a conjecture about 45.
which points on the coordinate plane would result in comparing their homework. Is either of them
true and false statements. correct? Explain your reasoning.
ANSWER:
a. Sample answer:
ANSWER:
No; sample answer: Madlynn reversed the inequality
sign when she added 1 to each side. Emilie did not
reverse the inequality sign at all.
REASONINGDetermine whether the following
46.
statement is sometimes, always, or never true.
Explain your reasoning.
The opposite of the absolute value of a negative
number is less than the opposite of that number.
b. Sample answer: ANSWER:
Sample answer: Always; the opposite of the absolute
value of a negative number will always be a negative
value, while the opposite of a negative number will
always be a positive value. A negative value will
always be less than a positive value.
CHALLENGEGiven
47. with sides
and , determine
the values of x such that
exists.
ANSWER:
Using the Triangle Inequality Theorem, we know
c. Sample answer: The points on or above the line that the sum of the lengths of any 2 sides of a
result in true statements, and the points below the triangle must be greater than the length of the
line result in false statements. This is true for all remaining side. This generates 3 inequalities to
points on the coordinate plane. examine.
CHALLENGEIf , then
44. and
. What is ?
ANSWER:
(a + b) < 4
ERROR ANALYSISMadlynn and Emilie were In order for all 3 conditions to be true, x must be
45.
comparing their homework. Is either of them greater than 0.2.
correct? Explain your reasoning.
OPEN ENDEDWrite an inequality for which the
48.
solution is all real numbers in the form
. Explain how you know this.
ANSWER:
Sample answer: ; This has a
ANSWER: solution set of all real numbers because it simplifies
No; sample answer: Madlynn reversed the inequality to or . This indicates that for
sign when she added 1 to each side. Emilie did not any real value of x the inequality is equivalent to
reverse the inequality sign at all. , that is the left side will always be 1 greater than the
right side.
REASONINGDetermine whether the following
46.
statement is sometimes, always, or never true.
WRITING IN MATHWhy does the inequality
Explain your reasoning. 49.
symbol need to be reversed when multiplying or
The opposite of the absolute value of a negative dividing by a negative number?
number is less than the opposite of that number.
ANSWER:
ANSWER: Sample answer: When one number is greater than
Sample answer: Always; the opposite of the absolute another number, it is either more positive or less
value of a negative number will always be a negative negative than that number. When these numbers are
value, while the opposite of a negative number will multiplied by a negative value, their roles are
always be a positive value. A negative value will reversed. That is, the number that was more positive
always be less than a positive value. is now more negative than the other number. Thus, it
is now less than that number and the inequality
CHALLENGEGiven
47. with sides symbol needs to be reversed.
and , determine
SHORT RESPONSERogelio found a cookie
the values of x such that 50.
exists.
recipe that requires cup of sugar and 2 cups of
ANSWER:
Using the Triangle Inequality Theorem, we know flour. How many cups of sugar would he need if he
that the sum of the lengths of any 2 sides of a used 6 cups of flour?
triangle must be greater than the length of the
remaining side. This generates 3 inequalities to ANSWER:
examine.
STATISTICS The mean score for Samantha s first
51. ’
six algebra quizzes was 88. If she scored a 95 on her
next quiz, what will her mean score be for all 7
quizzes?
A C 91
89
B D 92
In order for all 3 conditions to be true, x must be 90
greater than 0.2.
ANSWER:
A
OPEN ENDEDWrite an inequality for which the
48.
solution is all real numbers in the form SAT/ACT The average of five numbers is 9. The
. Explain how you know this. 52.
average of 7 other numbers is 8. What is the average
of all 12 numbers?
ANSWER:
Sample answer: ; This has a F
solution set of all real numbers because it simplifies G
to or . This indicates that for
any real value of x the inequality is equivalent to H
, that is the left side will always be 1 greater than the
right side. J
WRITING IN MATHWhy does the inequality
49.
symbol need to be reversed when multiplying or K
dividing by a negative number?
ANSWER:
ANSWER: F
Sample answer: When one number is greater than
another number, it is either more positive or less What is the complete solution of the equation
negative than that number. When these numbers are 53.
multiplied by a negative value, their roles are ?
reversed. That is, the number that was more positive A x = 8; x = 12
is now more negative than the other number. Thus, it B x = 8; x = 12
is now less than that number and the inequality –
C x = 8; x = 12
symbol needs to be reversed. – –
D x = 8; x = 12
–
SHORT RESPONSERogelio found a cookie
50.
ANSWER:
recipe that requires D
cup of sugar and 2 cups of
flour. How many cups of sugar would he need if he Solve each equation. Check your solutions.
used 6 cups of flour?
54.
ANSWER:
ANSWER:
STATISTICS The mean score for Samantha s first
51. ’ 55.
six algebra quizzes was 88. If she scored a 95 on her
next quiz, what will her mean score be for all 7 ANSWER:
quizzes?
A C 91
89
B D 92
90
56.
ANSWER:
A
ANSWER:
SAT/ACT The average of five numbers is 9. The
52.
average of 7 other numbers is 8. What is the average
ASTRONOMYPluto travels in a path that is not
of all 12 numbers? 57.
circular. Pluto’s farthest distance from the Sun is
F 4539 million miles, and its closest distance is 2756
G million miles. Write an equation that can be solved to
find the minimum and maximum distances from the
Sun to Pluto.
H
ANSWER:
J
POPULATIONIn 2005, the population of Bay City
K 58.
was 19,611. For each of the next five years, the
population decreased by an average of 715 people
ANSWER:
F per year.
a. What was the population in 2010?
What is the complete solution of the equation b. If the population continues to decline at the same
53. rate as from 2005 to 2010, what would you expect
? the population to be in 2025?
A x = 8; x = 12
B x = 8; x = 12 ANSWER:
– a. 16,036
C x = 8; x = 12
– – b. 5311
D x = 8; x = 12
–
GEOMETRY
ANSWER: 59. The formula for the surface area of a
D cylinder is .
Solve each equation. Check your solutions. a. Use the Distributive Property to rewrite the
formula by factoring out the greatest common factor
54. of the two terms.
b. Find the surface area for a cylinder with radius 3
ANSWER: centimeters and height 10 centimeters using both
formulas. Leave the answer in terms of .
c. Which formula do you prefer? Explain your
reasoning.
55.
ANSWER:
ANSWER: a.
b.
c. b
Sample answer: The formula in part is quicker.
56.
CONSTRUCTIONThe Sawyers are adding a
60.
family room to their house. The dimensions of the
ANSWER: room are 26 feet by 28 feet. Show how to use the
Distributive Property to mentally calculate the area
of the room.
ASTRONOMYPluto travels in a path that is not
57.
circular. Pluto’s farthest distance from the Sun is
ANSWER:
4539 million miles, and its closest distance is 2756
million miles. Write an equation that can be solved to
find the minimum and maximum distances from the Solve each equation. Check your solutions.
Sun to Pluto.
61.
ANSWER:
ANSWER:
POPULATIONIn 2005, the population of Bay City
58.
was 19,611. For each of the next five years, the
population decreased by an average of 715 people 62.
per year.
a. What was the population in 2010? ANSWER:
b. If the population continues to decline at the same
rate as from 2005 to 2010, what would you expect
the population to be in 2025? 63.
ANSWER:
ANSWER:
a. 16,036
b. 5311
GEOMETRY The formula for the surface area of a
59.
64.
cylinder is .
a. Use the Distributive Property to rewrite the ANSWER:
formula by factoring out the greatest common factor
of the two terms.
b. Find the surface area for a cylinder with radius 3
centimeters and height 10 centimeters using both 65.
formulas. Leave the answer in terms of . ANSWER:
c. Which formula do you prefer? Explain your
reasoning.
66.
ANSWER:
a.
ANSWER:
b.
c. b
Sample answer: The formula in part is quicker.
CONSTRUCTIONThe Sawyers are adding a
60.
family room to their house. The dimensions of the
room are 26 feet by 28 feet. Show how to use the
Distributive Property to mentally calculate the area
of the room.
ANSWER:
Solve each equation. Check your solutions.
61.
ANSWER:
62.
ANSWER:
63.
ANSWER:
64.
ANSWER:
65.
ANSWER:
66.
ANSWER:
Solve each inequality. Then graph the solution
set on a number line.
1.
ANSWER:
2.
ANSWER:
3.
ANSWER:
4.
ANSWER:
5.
ANSWER:
6.
ANSWER:
7.
ANSWER:
Solve each inequality. Then graph the solution
set on a number line.
1.
8.
ANSWER:
ANSWER:
2.
YARD WORK
9. Tara is delivering bags of mulch.
ANSWER: Each bag weighs 48 pounds, and the push cart
weighs 65 pounds. If her flat-bed truck is capable of
hauling 2000 pounds, how many bags of mulch can
Tara safely take on each trip?
ANSWER:
3.
40 bags
ANSWER:
Solve each inequality. Then graph the solution
set on a number line.
10.
ANSWER:
4.
ANSWER:
11.
5.
ANSWER:
ANSWER:
12.
6. ANSWER:
ANSWER:
13.
ANSWER:
7.
ANSWER:
14.
ANSWER:
8.
ANSWER:
15.
ANSWER:
YARD WORK
9. Tara is delivering bags of mulch.
Each bag weighs 48 pounds, and the push cart
weighs 65 pounds. If her flat-bed truck is capable of
hauling 2000 pounds, how many bags of mulch can
Tara safely take on each trip? 16.
ANSWER:
ANSWER:
40 bags
Solve each inequality. Then graph the solution
set on a number line.
17.
10.
ANSWER:
ANSWER:
18.
11.
ANSWER:
ANSWER:
19.
12.
ANSWER:
ANSWER:
1-5 Solving Inequalities
13.
20.
ANSWER:
ANSWER:
14.
21.
ANSWER:
ANSWER:
15. GYMNASTICS In a gymnastics competition, an
22.
athlete’s final score is calculated by taking 75% of
ANSWER: the average technical score and adding 25% of the
artistic score. All scores are out of 10, and one
gymnast has a 7.6 average technical score. What
artistic score does the gymnast need to have a final
score of at least 8.0?
16.
ANSWER:
ANSWER: 9.2
Define a variable and write an inequality for
each problem. Then solve.
23. Twelve less than the product of three and a number
17. is less than 21.
ANSWER:
ANSWER:
3x 12 < 21; x < 11
–
24. The quotient of three times a number and 4 is at least
16.
–
18.
ANSWER:
ANSWER:
25. The difference of 5 times a number and 6 is greater
than the number.
19.
ANSWER:
ANSWER:
26. The quotient of the sum of 3 and a number and 6 is
less than 2.
–
ANSWER:
20.
eSolutions Manual - Powered by Cognero Page2
ANSWER:
HIKINGDanielle can hike 3 miles in an hour, but
27.
she has to take a one-hour break for lunch and a
one-hour break for dinner. If Danielle wants to hike
at least 18 miles, solve to determine
how many hours the hike should take.
21.
ANSWER:
at least 8 hours
ANSWER:
Solve each inequality. Then graph the solution
set on a number line.
28.
GYMNASTICS In a gymnastics competition, an ANSWER:
22.
athlete’s final score is calculated by taking 75% of
the average technical score and adding 25% of the
artistic score. All scores are out of 10, and one
gymnast has a 7.6 average technical score. What
artistic score does the gymnast need to have a final
29.
score of at least 8.0?
ANSWER:
ANSWER:
9.2
Define a variable and write an inequality for
each problem. Then solve.
23. Twelve less than the product of three and a number
is less than 21.
30.
ANSWER:
3x – 12 < 21; x < 11
ANSWER:
24. The quotient of three times a number and 4 is at least
16.
–
ANSWER:
31.
ANSWER:
25. The difference of 5 times a number and 6 is greater
than the number.
ANSWER:
32.
ANSWER:
The quotient of the sum of 3 and a number and 6 is
26.
less than –2.
ANSWER:
33.
ANSWER:
HIKINGDanielle can hike 3 miles in an hour, but
27.
she has to take a one-hour break for lunch and a
one-hour break for dinner. If Danielle wants to hike
at least 18 miles, solve to determine
how many hours the hike should take.
34.
ANSWER:
at least 8 hours ANSWER:
Solve each inequality. Then graph the solution
set on a number line.
28.
ANSWER:
35.
ANSWER:
29.
ANSWER:
36.
ANSWER:
30.
ANSWER:
MONEYJin is selling advertising space in Central
37.
City Magazine to local businesses. Jin earns 3%
commission for every advertisement he sells plus a
salary of $250 a week. If the average amount of
money that a business spends on an advertisement is
31. $500, how many advertisements must he sell each
week to make a salary of at least $700 that week?
ANSWER:
a. Write an inequality to describe this situation.
b. Solve the inequality and interpret the solution.
ANSWER:
a.
32.
b.
He must sell at least 30 advertisements.
ANSWER:
Define a variable and write an inequality for
each problem. Then solve.
38. One third of the sum of 5 times a number and 3 is
less than one fourth the sum of six times that number
33. and 5.
ANSWER:
ANSWER:
39. The sum of one third a number and 4 is at most the
sum of twice that number and 12.
34.
ANSWER:
ANSWER:
SENSE-MAKINGThe sides of square ABCD are
40.
extended to form rectangle DEFG. If the perimeter
of the rectangle is at least twice the perimeter of the
square, what is the maximum length of a side of
35. square ABCD?
ANSWER:
36.
ANSWER:
ANSWER:
9 in.
MARATHONSJamie wants to be able to run at
41.
least the standard marathon distance of 26.2 miles. A
MONEYJin is selling advertising space in Central good rule for training is that runners generally have
37.
City Magazine to local businesses. Jin earns 3% enough endurance to finish a race that is up to 3
commission for every advertisement he sells plus a times his or her average daily distance.
salary of $250 a week. If the average amount of a. If the length of her current daily run is 5 miles,
money that a business spends on an advertisement is write an inequality to find the amount by which she
$500, how many advertisements must he sell each needs to increase her daily run to have enough
week to make a salary of at least $700 that week? endurance to finish a marathon.
a. Write an inequality to describe this situation. b. Solve the inequality and interpret the solution.
b. Solve the inequality and interpret the solution.
ANSWER:
ANSWER: a.
a. b
. In order to have enough endurance to
b. run a marathon, Jamie should increase the distance
He must sell at least 30 advertisements.
of her average daily run by at least 3.73 miles.
Define a variable and write an inequality for
each problem. Then solve. MODELINGThe costs for renting a car from Ace
42.
38. One third of the sum of 5 times a number and 3 is Car Rental and from Basic Car Rental are shown in
less than one fourth the sum of six times that number the table. For what mileage does Basic have the
and 5. better deal? Use the inequality
. Explain why this inequality
ANSWER: works.
39. The sum of one third a number and 4 is at most the
sum of twice that number and 12.
ANSWER:
ANSWER:
Basic has the better deal as long as you are traveling
SENSE-MAKINGThe sides of square ABCD are more than 80 miles. Yes, this is the correct inequality
40.
extended to form rectangle DEFG. If the perimeter to use. Sample explanation: It works because the
of the rectangle is at least twice the perimeter of the inequality finds the mileage at which Ace’s charge is
square, what is the maximum length of a side of greater than Basic’s charge.
square ABCD?
MULTIPLE REPRESENTATIONSIn this
43.
exercise, you will explore graphing inequalities on a
coordinate plane.
a. TABULAR
Organize the following into a table.
Substitute 5 points into the inequality .
State whether the resulting statement is true or
false.
b. GRAPHICAL
Graph . Also graph
ANSWER: the 5 points from the table. Label all points that
9 in. resulted in a true statement with a T. Label all points
that resulted in a false statement with an F.
MARATHONSJamie wants to be able to run at
41.
c. VERBAL
least the standard marathon distance of 26.2 miles. A Describe the pattern produced by the
good rule for training is that runners generally have points you have labeled. Make a conjecture about
enough endurance to finish a race that is up to 3 which points on the coordinate plane would result in
times his or her average daily distance. true and false statements.
a. If the length of her current daily run is 5 miles,
write an inequality to find the amount by which she ANSWER:
needs to increase her daily run to have enough a. Sample answer:
endurance to finish a marathon.
b. Solve the inequality and interpret the solution.
ANSWER:
a.
b
. In order to have enough endurance to
run a marathon, Jamie should increase the distance
of her average daily run by at least 3.73 miles.
MODELINGThe costs for renting a car from Ace
42.
Car Rental and from Basic Car Rental are shown in
the table. For what mileage does Basic have the
better deal? Use the inequality
. Explain why this inequality b. Sample answer:
works.
ANSWER:
Basic has the better deal as long as you are traveling
more than 80 miles. Yes, this is the correct inequality
to use. Sample explanation: It works because the c. Sample answer: The points on or above the line
inequality finds the mileage at which Ace s charge is result in true statements, and the points below the
’ line result in false statements. This is true for all
greater than Basic’s charge. points on the coordinate plane.
MULTIPLE REPRESENTATIONSIn this
43.
exercise, you will explore graphing inequalities on a 44. CHALLENGEIf and , then
coordinate plane. . What is ?
a. TABULAR
Organize the following into a table.
Substitute 5 points into the inequality . ANSWER:
State whether the resulting statement is true or (a + b) < 4
false.
45. ERROR ANALYSISMadlynn and Emilie were
b. GRAPHICAL comparing their homework. Is either of them
Graph . Also graph correct? Explain your reasoning.
the 5 points from the table. Label all points that
resulted in a true statement with a T. Label all points
that resulted in a false statement with an F.
c. VERBAL
Describe the pattern produced by the
points you have labeled. Make a conjecture about
which points on the coordinate plane would result in
true and false statements.
ANSWER:
No; sample answer: Madlynn reversed the inequality
ANSWER: sign when she added 1 to each side. Emilie did not
a. Sample answer: reverse the inequality sign at all.
46. REASONINGDetermine whether the following
statement is sometimes, always, or never true.
Explain your reasoning.
The opposite of the absolute value of a negative
number is less than the opposite of that number.
ANSWER:
Sample answer: Always; the opposite of the absolute
value of a negative number will always be a negative
value, while the opposite of a negative number will
always be a positive value. A negative value will
always be less than a positive value.
b. Sample answer:
47. CHALLENGEGiven with sides
and , determine
the values of x such that exists.
ANSWER:
Using the Triangle Inequality Theorem, we know
that the sum of the lengths of any 2 sides of a
triangle must be greater than the length of the
remaining side. This generates 3 inequalities to
examine.
c. Sample answer: The points on or above the line
result in true statements, and the points below the
line result in false statements. This is true for all
points on the coordinate plane.
CHALLENGEIf , then
44. and
. What is ? In order for all 3 conditions to be true, x must be
greater than 0.2.
OPEN ENDEDWrite an inequality for which the
48.
ANSWER: solution is all real numbers in the form
(a + b) < 4 . Explain how you know this.
ERROR ANALYSISMadlynn and Emilie were
45.
comparing their homework. Is either of them ANSWER:
correct? Explain your reasoning. Sample answer: ; This has a
solution set of all real numbers because it simplifies
to or . This indicates that for
any real value of x the inequality is equivalent to
, that is the left side will always be 1 greater than the
right side.
WRITING IN MATHWhy does the inequality
49.
ANSWER: symbol need to be reversed when multiplying or
No; sample answer: Madlynn reversed the inequality dividing by a negative number?
sign when she added 1 to each side. Emilie did not
reverse the inequality sign at all.
ANSWER:
Sample answer: When one number is greater than
REASONINGDetermine whether the following
46. another number, it is either more positive or less
statement is sometimes, always, or never true. negative than that number. When these numbers are
Explain your reasoning. multiplied by a negative value, their roles are
The opposite of the absolute value of a negative reversed. That is, the number that was more positive
number is less than the opposite of that number. is now more negative than the other number. Thus, it
is now less than that number and the inequality
ANSWER: symbol needs to be reversed.
Sample answer: Always; the opposite of the absolute
value of a negative number will always be a negative
SHORT RESPONSERogelio found a cookie
value, while the opposite of a negative number will 50.
always be a positive value. A negative value will recipe that requires
always be less than a positive value. cup of sugar and 2 cups of
flour. How many cups of sugar would he need if he
CHALLENGEGiven used 6 cups of flour?
47. with sides
and , determine
the values of x such that ANSWER:
exists.
ANSWER:
Using the Triangle Inequality Theorem, we know
STATISTICSThe mean score for Samantha s first
that the sum of the lengths of any 2 sides of a 51. ’
triangle must be greater than the length of the six algebra quizzes was 88. If she scored a 95 on her
remaining side. This generates 3 inequalities to next quiz, what will her mean score be for all 7
examine. quizzes?
A C
89 91
B D 92
90
ANSWER:
A
SAT/ACT The average of five numbers is 9. The
52.
average of 7 other numbers is 8. What is the average
In order for all 3 conditions to be true, x must be of all 12 numbers?
greater than 0.2.
F
OPEN ENDEDWrite an inequality for which the
48. G
solution is all real numbers in the form
. Explain how you know this. H
ANSWER:
Sample answer: ; This has a J
solution set of all real numbers because it simplifies
to . This indicates that for K
or
any real value of x the inequality is equivalent to
, that is the left side will always be 1 greater than the ANSWER:
right side. F
WRITING IN MATHWhy does the inequality 53. What is the complete solution of the equation
49.
symbol need to be reversed when multiplying or ?
dividing by a negative number? A x = 8; x = 12
B x = 8; x = 12
ANSWER: –
C x = 8; x = 12
Sample answer: When one number is greater than – –
another number, it is either more positive or less D x = 8; x = 12
–
negative than that number. When these numbers are
multiplied by a negative value, their roles are ANSWER:
reversed. That is, the number that was more positive D
is now more negative than the other number. Thus, it Solve each equation. Check your solutions.
is now less than that number and the inequality
symbol needs to be reversed. 54.
ANSWER:
SHORT RESPONSERogelio found a cookie
50.
recipe that requires cup of sugar and 2 cups of
flour. How many cups of sugar would he need if he
used 6 cups of flour? 55.
ANSWER:
ANSWER:
STATISTICS The mean score for Samantha s first
51. ’
six algebra quizzes was 88. If she scored a 95 on her 56.
next quiz, what will her mean score be for all 7 ANSWER:
quizzes?
A C 91
89
B D ASTRONOMYPluto travels in a path that is not
92 57.
90
circular. Pluto’s farthest distance from the Sun is
ANSWER: 4539 million miles, and its closest distance is 2756
A million miles. Write an equation that can be solved to
SAT/ACT The average of five numbers is 9. The find the minimum and maximum distances from the
52. Sun to Pluto.
average of 7 other numbers is 8. What is the average
of all 12 numbers? ANSWER:
F
G POPULATIONIn 2005, the population of Bay City
58.
was 19,611. For each of the next five years, the
H population decreased by an average of 715 people
per year.
J a. What was the population in 2010?
b. If the population continues to decline at the same
K rate as from 2005 to 2010, what would you expect
the population to be in 2025?
ANSWER:
ANSWER:
F a. 16,036
What is the complete solution of the equation b. 5311
53.
GEOMETRY The formula for the surface area of a
? 59.
A x = 8; x = 12 cylinder is .
B x = 8; x = 12 a. Use the Distributive Property to rewrite the
– formula by factoring out the greatest common factor
C x = 8; x = 12
– – of the two terms.
D x = 8; x = 12
– b. Find the surface area for a cylinder with radius 3
ANSWER: centimeters and height 10 centimeters using both
D formulas. Leave the answer in terms of .
Solve each equation. Check your solutions. c. Which formula do you prefer? Explain your
reasoning.
54.
ANSWER:
ANSWER: a.
b.
c. b
Sample answer: The formula in part is quicker.
55.
CONSTRUCTIONThe Sawyers are adding a
60.
family room to their house. The dimensions of the
ANSWER: room are 26 feet by 28 feet. Show how to use the
Distributive Property to mentally calculate the area
of the room.
ANSWER:
56.
ANSWER:
Solve each equation. Check your solutions.
61.
ASTRONOMYPluto travels in a path that is not
57.
circular. Pluto’s farthest distance from the Sun is
ANSWER:
4539 million miles, and its closest distance is 2756
million miles. Write an equation that can be solved to
find the minimum and maximum distances from the
Sun to Pluto.
62.
ANSWER:
ANSWER:
POPULATIONIn 2005, the population of Bay City
58.
was 19,611. For each of the next five years, the 63.
population decreased by an average of 715 people
ANSWER:
per year.
a. What was the population in 2010?
b. If the population continues to decline at the same
rate as from 2005 to 2010, what would you expect
the population to be in 2025? 64.
ANSWER:
ANSWER:
a. 16,036
b. 5311
GEOMETRY The formula for the surface area of a 65.
59.
cylinder is . ANSWER:
a. Use the Distributive Property to rewrite the
formula by factoring out the greatest common factor
of the two terms.
b. Find the surface area for a cylinder with radius 3 66.
centimeters and height 10 centimeters using both ANSWER:
formulas. Leave the answer in terms of .
c. Which formula do you prefer? Explain your
reasoning.
ANSWER:
a.
b.
c. b
Sample answer: The formula in part is quicker.
CONSTRUCTIONThe Sawyers are adding a
60.
family room to their house. The dimensions of the
room are 26 feet by 28 feet. Show how to use the
Distributive Property to mentally calculate the area
of the room.
ANSWER:
Solve each equation. Check your solutions.
61.
ANSWER:
62.
ANSWER:
63.
ANSWER:
64.
ANSWER:
65.
ANSWER:
66.
ANSWER:
Solve each inequality. Then graph the solution
set on a number line.
1.
ANSWER:
2.
ANSWER:
3.
ANSWER:
4.
ANSWER:
5.
ANSWER:
6.
ANSWER:
7.
ANSWER:
Solve each inequality. Then graph the solution
set on a number line.
1.
ANSWER:
8.
ANSWER:
2.
ANSWER:
YARD WORK
9. Tara is delivering bags of mulch.
Each bag weighs 48 pounds, and the push cart
weighs 65 pounds. If her flat-bed truck is capable of
3. hauling 2000 pounds, how many bags of mulch can
Tara safely take on each trip?
ANSWER:
ANSWER:
40 bags
Solve each inequality. Then graph the solution
set on a number line.
4.
10.
ANSWER:
ANSWER:
5.
11.
ANSWER:
ANSWER:
6.
12.
ANSWER:
ANSWER:
13.
7.
ANSWER:
ANSWER:
14.
8.
ANSWER:
ANSWER:
YARD WORK 15.
9. Tara is delivering bags of mulch.
Each bag weighs 48 pounds, and the push cart
weighs 65 pounds. If her flat-bed truck is capable of ANSWER:
hauling 2000 pounds, how many bags of mulch can
Tara safely take on each trip?
ANSWER:
40 bags 16.
Solve each inequality. Then graph the solution ANSWER:
set on a number line.
10.
ANSWER:
17.
ANSWER:
11.
ANSWER:
18.
ANSWER:
12.
ANSWER:
19.
ANSWER:
13.
ANSWER:
20.
ANSWER:
14.
ANSWER:
21.
ANSWER:
15.
ANSWER:
GYMNASTICS In a gymnastics competition, an
22.
athlete’s final score is calculated by taking 75% of
the average technical score and adding 25% of the
artistic score. All scores are out of 10, and one
16. gymnast has a 7.6 average technical score. What
artistic score does the gymnast need to have a final
ANSWER:
score of at least 8.0?
ANSWER:
9.2
17. Define a variable and write an inequality for
each problem. Then solve.
ANSWER: Twelve less than the product of three and a number
23.
is less than 21.
ANSWER:
3x – 12 < 21; x < 11
18.
24. The quotient of three times a number and 4 is at least
ANSWER: 16.
–
ANSWER:
19.
25. The difference of 5 times a number and 6 is greater
ANSWER: than the number.
ANSWER:
26. The quotient of the sum of 3 and a number and 6 is
20. less than 2.
–
ANSWER:
ANSWER:
HIKINGDanielle can hike 3 miles in an hour, but
27.
she has to take a one-hour break for lunch and a
one-hour break for dinner. If Danielle wants to hike
21. at least 18 miles, solve to determine
how many hours the hike should take.
ANSWER:
ANSWER:
at least 8 hours
Solve each inequality. Then graph the solution
GYMNASTICSIn a gymnastics competition, an set on a number line.
22.
athlete’s final score is calculated by taking 75% of
28.
the average technical score and adding 25% of the
artistic score. All scores are out of 10, and one ANSWER:
gymnast has a 7.6 average technical score. What
artistic score does the gymnast need to have a final
score of at least 8.0?
ANSWER:
9.2 29.
Define a variable and write an inequality for ANSWER:
each problem. Then solve.
23. Twelve less than the product of three and a number
is less than 21.
ANSWER:
3x – 12 < 21; x < 11
30.
24. The quotient of three times a number and 4 is at least
16. ANSWER:
–
ANSWER:
31.
25. The difference of 5 times a number and 6 is greater
than the number. ANSWER:
ANSWER:
1-5 Solving Inequalities
26. The quotient of the sum of 3 and a number and 6 is
less than 2. 32.
–
ANSWER:
ANSWER:
33.
HIKINGDanielle can hike 3 miles in an hour, but
27.
she has to take a one-hour break for lunch and a
one-hour break for dinner. If Danielle wants to hike ANSWER:
at least 18 miles, solve to determine
how many hours the hike should take.
ANSWER:
at least 8 hours
34.
Solve each inequality. Then graph the solution
ANSWER:
set on a number line.
28.
ANSWER:
35.
29.
ANSWER:
ANSWER:
36.
30.
ANSWER:
ANSWER:
MONEYJin is selling advertising space in Central
37.
City Magazine to local businesses. Jin earns 3%
31. commission for every advertisement he sells plus a
salary of $250 a week. If the average amount of
ANSWER: money that a business spends on an advertisement is
$500, how many advertisements must he sell each
week to make a salary of at least $700 that week?
a. Write an inequality to describe this situation.
b. Solve the inequality and interpret the solution.
32.
ANSWER:
ANSWER:
a.
b.
He must sell at least 30 advertisements.
eSolutions Manual - Powered by Cognero Page3
Define a variable and write an inequality for
33.
each problem. Then solve.
38. One third of the sum of 5 times a number and 3 is
ANSWER: less than one fourth the sum of six times that number
and 5.
ANSWER:
34.
39. The sum of one third a number and 4 is at most the
ANSWER: sum of twice that number and 12.
ANSWER:
SENSE-MAKINGThe sides of square ABCD are
40.
35. extended to form rectangle DEFG. If the perimeter
of the rectangle is at least twice the perimeter of the
ANSWER: square, what is the maximum length of a side of
square ABCD?
36.
ANSWER:
ANSWER:
9 in.
MONEYJin is selling advertising space in Central
37.
MARATHONSJamie wants to be able to run at
City Magazine to local businesses. Jin earns 3% 41.
commission for every advertisement he sells plus a least the standard marathon distance of 26.2 miles. A
salary of $250 a week. If the average amount of good rule for training is that runners generally have
money that a business spends on an advertisement is enough endurance to finish a race that is up to 3
$500, how many advertisements must he sell each times his or her average daily distance.
week to make a salary of at least $700 that week? a. If the length of her current daily run is 5 miles,
a. Write an inequality to describe this situation. write an inequality to find the amount by which she
b. Solve the inequality and interpret the solution. needs to increase her daily run to have enough
endurance to finish a marathon.
ANSWER: b. Solve the inequality and interpret the solution.
a.
b. ANSWER:
He must sell at least 30 advertisements. a.
Define a variable and write an inequality for b
. In order to have enough endurance to
each problem. Then solve. run a marathon, Jamie should increase the distance
38. One third of the sum of 5 times a number and 3 is of her average daily run by at least 3.73 miles.
less than one fourth the sum of six times that number
and 5. MODELINGThe costs for renting a car from Ace
42.
Car Rental and from Basic Car Rental are shown in
ANSWER: the table. For what mileage does Basic have the
better deal? Use the inequality
. Explain why this inequality
The sum of one third a number and 4 is at most the works.
39.
sum of twice that number and 12.
ANSWER:
SENSE-MAKINGThe sides of square ABCD are
40.
extended to form rectangle DEFG. If the perimeter ANSWER:
of the rectangle is at least twice the perimeter of the Basic has the better deal as long as you are traveling
square, what is the maximum length of a side of more than 80 miles. Yes, this is the correct inequality
square ABCD? to use. Sample explanation: It works because the
inequality finds the mileage at which Ace’s charge is
greater than Basic’s charge.
MULTIPLE REPRESENTATIONSIn this
43.
exercise, you will explore graphing inequalities on a
coordinate plane.
a. TABULAR
Organize the following into a table.
Substitute 5 points into the inequality .
State whether the resulting statement is true or
ANSWER: false.
9 in. b. GRAPHICAL
Graph . Also graph
MARATHONSJamie wants to be able to run at
41. the 5 points from the table. Label all points that
least the standard marathon distance of 26.2 miles. A resulted in a true statement with a T. Label all points
good rule for training is that runners generally have that resulted in a false statement with an F.
enough endurance to finish a race that is up to 3
c. VERBAL
times his or her average daily distance. Describe the pattern produced by the
a. If the length of her current daily run is 5 miles, points you have labeled. Make a conjecture about
write an inequality to find the amount by which she which points on the coordinate plane would result in
needs to increase her daily run to have enough true and false statements.
endurance to finish a marathon.
ANSWER:
b. Solve the inequality and interpret the solution. a. Sample answer:
ANSWER:
a.
b
. In order to have enough endurance to
run a marathon, Jamie should increase the distance
of her average daily run by at least 3.73 miles.
MODELINGThe costs for renting a car from Ace
42.
Car Rental and from Basic Car Rental are shown in
the table. For what mileage does Basic have the
better deal? Use the inequality
. Explain why this inequality
works.
b. Sample answer:
ANSWER:
Basic has the better deal as long as you are traveling
more than 80 miles. Yes, this is the correct inequality
to use. Sample explanation: It works because the
inequality finds the mileage at which Ace’s charge is
greater than Basic’s charge.
c. Sample answer: The points on or above the line
result in true statements, and the points below the
MULTIPLE REPRESENTATIONSIn this
43. line result in false statements. This is true for all
exercise, you will explore graphing inequalities on a points on the coordinate plane.
coordinate plane.
a. TABULAR
Organize the following into a table.
44. CHALLENGEIf and , then
Substitute 5 points into the inequality . . What is ?
State whether the resulting statement is true or
false.
ANSWER:
b. GRAPHICAL (a + b) < 4
Graph . Also graph
the 5 points from the table. Label all points that 45. ERROR ANALYSISMadlynn and Emilie were
resulted in a true statement with a T. Label all points comparing their homework. Is either of them
that resulted in a false statement with an F. correct? Explain your reasoning.
c. VERBAL
Describe the pattern produced by the
points you have labeled. Make a conjecture about
which points on the coordinate plane would result in
true and false statements.
ANSWER:
a. Sample answer:
ANSWER:
No; sample answer: Madlynn reversed the inequality
sign when she added 1 to each side. Emilie did not
reverse the inequality sign at all.
46. REASONINGDetermine whether the following
statement is sometimes, always, or never true.
Explain your reasoning.
The opposite of the absolute value of a negative
number is less than the opposite of that number.
ANSWER:
Sample answer: Always; the opposite of the absolute
value of a negative number will always be a negative
value, while the opposite of a negative number will
b. Sample answer: always be a positive value. A negative value will
always be less than a positive value.
47. CHALLENGEGiven with sides
and , determine
the values of x such that exists.
ANSWER:
Using the Triangle Inequality Theorem, we know
that the sum of the lengths of any 2 sides of a
triangle must be greater than the length of the
remaining side. This generates 3 inequalities to
c. Sample answer: The points on or above the line examine.
result in true statements, and the points below the
line result in false statements. This is true for all
points on the coordinate plane.
44. CHALLENGEIf and , then
. What is ?
In order for all 3 conditions to be true, x must be
ANSWER: greater than 0.2.
(a + b) < 4
48. OPEN ENDEDWrite an inequality for which the
45. ERROR ANALYSISMadlynn and Emilie were solution is all real numbers in the form
comparing their homework. Is either of them . Explain how you know this.
correct? Explain your reasoning.
ANSWER:
Sample answer: ; This has a
solution set of all real numbers because it simplifies
to or . This indicates that for
any real value of x the inequality is equivalent to
, that is the left side will always be 1 greater than the
ANSWER: right side.
No; sample answer: Madlynn reversed the inequality
sign when she added 1 to each side. Emilie did not
reverse the inequality sign at all. 49. WRITING IN MATHWhy does the inequality
symbol need to be reversed when multiplying or
dividing by a negative number?
46. REASONINGDetermine whether the following
statement is sometimes, always, or never true.
Explain your reasoning. ANSWER:
The opposite of the absolute value of a negative Sample answer: When one number is greater than
another number, it is either more positive or less
number is less than the opposite of that number. negative than that number. When these numbers are
multiplied by a negative value, their roles are
ANSWER: reversed. That is, the number that was more positive
Sample answer: Always; the opposite of the absolute is now more negative than the other number. Thus, it
value of a negative number will always be a negative is now less than that number and the inequality
value, while the opposite of a negative number will symbol needs to be reversed.
always be a positive value. A negative value will
always be less than a positive value.
50. SHORT RESPONSERogelio found a cookie
47. CHALLENGEGiven with sides recipe that requires cup of sugar and 2 cups of
and , determine flour. How many cups of sugar would he need if he
the values of x such that exists. used 6 cups of flour?
ANSWER:
Using the Triangle Inequality Theorem, we know ANSWER:
that the sum of the lengths of any 2 sides of a
triangle must be greater than the length of the
remaining side. This generates 3 inequalities to
examine. 51. STATISTICS The mean score for Samantha’s first
six algebra quizzes was 88. If she scored a 95 on her
next quiz, what will her mean score be for all 7
quizzes?
A C
89 91
B D
90 92
ANSWER:
A
In order for all 3 conditions to be true, x must be
greater than 0.2. 52. SAT/ACT The average of five numbers is 9. The
average of 7 other numbers is 8. What is the average
48. OPEN ENDEDWrite an inequality for which the of all 12 numbers?
solution is all real numbers in the form
. Explain how you know this. F
G
ANSWER:
Sample answer: ; This has a H
solution set of all real numbers because it simplifies
to or . This indicates that for J
any real value of x the inequality is equivalent to
, that is the left side will always be 1 greater than the K
right side.
49. WRITING IN MATHWhy does the inequality ANSWER:
symbol need to be reversed when multiplying or F
dividing by a negative number?
53. What is the complete solution of the equation
ANSWER: ?
Sample answer: When one number is greater than A x = 8; x = 12
another number, it is either more positive or less B x = 8; x = –12
negative than that number. When these numbers are
multiplied by a negative value, their roles are Cx = –8; x = –12
reversed. That is, the number that was more positive D x = –8; x = 12
is now more negative than the other number. Thus, it
is now less than that number and the inequality ANSWER:
symbol needs to be reversed. D
Solve each equation. Check your solutions.
50. SHORT RESPONSERogelio found a cookie
recipe that requires cup of sugar and 2 cups of 54.
flour. How many cups of sugar would he need if he ANSWER:
used 6 cups of flour?
ANSWER:
55.
ANSWER:
STATISTICS The mean score for Samantha s first
51. ’
six algebra quizzes was 88. If she scored a 95 on her
next quiz, what will her mean score be for all 7
quizzes?
56.
A C 91
89
B D 92 ANSWER:
90
ANSWER:
A
ASTRONOMYPluto travels in a path that is not
57.
SAT/ACT The average of five numbers is 9. The circular. Pluto’s farthest distance from the Sun is
52. 4539 million miles, and its closest distance is 2756
average of 7 other numbers is 8. What is the average million miles. Write an equation that can be solved to
of all 12 numbers? find the minimum and maximum distances from the
Sun to Pluto.
F
G ANSWER:
H
POPULATIONIn 2005, the population of Bay City
58.
J was 19,611. For each of the next five years, the
population decreased by an average of 715 people
per year.
K a. What was the population in 2010?
b. If the population continues to decline at the same
ANSWER: rate as from 2005 to 2010, what would you expect
F the population to be in 2025?
What is the complete solution of the equation ANSWER:
53.
? a. 16,036
A x = 8; x = 12 b. 5311
B x = 8; x = 12
– GEOMETRY The formula for the surface area of a
C x = 8; x = 12 59.
– – cylinder is .
D x = 8; x = 12
– a. Use the Distributive Property to rewrite the
formula by factoring out the greatest common factor
ANSWER:
D of the two terms.
b. Find the surface area for a cylinder with radius 3
Solve each equation. Check your solutions. centimeters and height 10 centimeters using both
54. formulas. Leave the answer in terms of .
c. Which formula do you prefer? Explain your
ANSWER: reasoning.
ANSWER:
a.
55. b.
c. b
Sample answer: The formula in part is quicker.
ANSWER:
CONSTRUCTIONThe Sawyers are adding a
60.
family room to their house. The dimensions of the
room are 26 feet by 28 feet. Show how to use the
56. Distributive Property to mentally calculate the area
of the room.
ANSWER:
ANSWER:
ASTRONOMYPluto travels in a path that is not
57.
circular. Pluto’s farthest distance from the Sun is Solve each equation. Check your solutions.
4539 million miles, and its closest distance is 2756
million miles. Write an equation that can be solved to 61.
find the minimum and maximum distances from the
ANSWER:
Sun to Pluto.
ANSWER:
62.
ANSWER:
POPULATIONIn 2005, the population of Bay City
58.
was 19,611. For each of the next five years, the
population decreased by an average of 715 people
per year.
a. What was the population in 2010? 63.
b. If the population continues to decline at the same
ANSWER:
rate as from 2005 to 2010, what would you expect
the population to be in 2025?
ANSWER:
a. 16,036 64.
b. 5311
ANSWER:
GEOMETRY The formula for the surface area of a
59.
cylinder is .
a. Use the Distributive Property to rewrite the 65.
formula by factoring out the greatest common factor
of the two terms. ANSWER:
b. Find the surface area for a cylinder with radius 3
centimeters and height 10 centimeters using both
formulas. Leave the answer in terms of .
66.
c. Which formula do you prefer? Explain your
reasoning. ANSWER:
ANSWER:
a.
b.
c. b
Sample answer: The formula in part is quicker.
CONSTRUCTIONThe Sawyers are adding a
60.
family room to their house. The dimensions of the
room are 26 feet by 28 feet. Show how to use the
Distributive Property to mentally calculate the area
of the room.
ANSWER:
Solve each equation. Check your solutions.
61.
ANSWER:
62.
ANSWER:
63.
ANSWER:
64.
ANSWER:
65.
ANSWER:
66.
ANSWER:
Solve each inequality. Then graph the solution
set on a number line.
1.
ANSWER:
2.
ANSWER:
3.
ANSWER:
4.
ANSWER:
5.
ANSWER:
6.
ANSWER:
7.
ANSWER:
Solve each inequality. Then graph the solution
set on a number line.
1.
ANSWER:
8.
ANSWER:
2.
ANSWER:
YARD WORK
9. Tara is delivering bags of mulch.
Each bag weighs 48 pounds, and the push cart
weighs 65 pounds. If her flat-bed truck is capable of
hauling 2000 pounds, how many bags of mulch can
3. Tara safely take on each trip?
ANSWER:
ANSWER:
40 bags
Solve each inequality. Then graph the solution
set on a number line.
4.
10.
ANSWER: ANSWER:
5.
11.
ANSWER:
ANSWER:
6.
12.
ANSWER:
ANSWER:
13.
7. ANSWER:
ANSWER:
14.
ANSWER:
8.
ANSWER:
15.
YARD WORK
9. Tara is delivering bags of mulch.
Each bag weighs 48 pounds, and the push cart ANSWER:
weighs 65 pounds. If her flat-bed truck is capable of
hauling 2000 pounds, how many bags of mulch can
Tara safely take on each trip?
16.
ANSWER:
40 bags
ANSWER:
Solve each inequality. Then graph the solution
set on a number line.
10.
ANSWER: 17.
ANSWER:
11.
18.
ANSWER:
ANSWER:
12.
ANSWER: 19.
ANSWER:
13.
ANSWER:
20.
ANSWER:
14.
ANSWER:
21.
ANSWER:
15.
ANSWER: GYMNASTICS In a gymnastics competition, an
22.
athlete’s final score is calculated by taking 75% of
the average technical score and adding 25% of the
artistic score. All scores are out of 10, and one
gymnast has a 7.6 average technical score. What
16. artistic score does the gymnast need to have a final
score of at least 8.0?
ANSWER:
ANSWER:
9.2
Define a variable and write an inequality for
17.
each problem. Then solve.
23. Twelve less than the product of three and a number
ANSWER: is less than 21.
ANSWER:
3x – 12 < 21; x < 11
18. The quotient of three times a number and 4 is at least
24.
–16.
ANSWER:
ANSWER:
19. The difference of 5 times a number and 6 is greater
25.
than the number.
ANSWER:
ANSWER:
26. The quotient of the sum of 3 and a number and 6 is
less than 2.
–
20.
ANSWER:
ANSWER:
HIKINGDanielle can hike 3 miles in an hour, but
27.
she has to take a one-hour break for lunch and a
one-hour break for dinner. If Danielle wants to hike
at least 18 miles, solve to determine
21. how many hours the hike should take.
ANSWER:
ANSWER:
at least 8 hours
Solve each inequality. Then graph the solution
set on a number line.
GYMNASTICSIn a gymnastics competition, an
22.
athlete s final score is calculated by taking 75% of 28.
’
the average technical score and adding 25% of the ANSWER:
artistic score. All scores are out of 10, and one
gymnast has a 7.6 average technical score. What
artistic score does the gymnast need to have a final
score of at least 8.0?
29.
ANSWER:
9.2
ANSWER:
Define a variable and write an inequality for
each problem. Then solve.
23. Twelve less than the product of three and a number
is less than 21.
ANSWER:
3x 12 < 21; x < 11 30.
–
The quotient of three times a number and 4 is at least ANSWER:
24.
–16.
ANSWER:
31.
The difference of 5 times a number and 6 is greater ANSWER:
25.
than the number.
ANSWER:
The quotient of the sum of 3 and a number and 6 is 32.
26.
less than 2. ANSWER:
–
ANSWER:
33.
HIKINGDanielle can hike 3 miles in an hour, but
27.
she has to take a one-hour break for lunch and a ANSWER:
one-hour break for dinner. If Danielle wants to hike
at least 18 miles, solve to determine
how many hours the hike should take.
ANSWER:
at least 8 hours 34.
Solve each inequality. Then graph the solution ANSWER:
set on a number line.
28.
ANSWER:
35.
ANSWER:
29.
ANSWER:
36.
ANSWER:
30.
ANSWER:
MONEYJin is selling advertising space in Central
37.
City Magazine to local businesses. Jin earns 3%
commission for every advertisement he sells plus a
31. salary of $250 a week. If the average amount of
money that a business spends on an advertisement is
ANSWER:
$500, how many advertisements must he sell each
week to make a salary of at least $700 that week?
a. Write an inequality to describe this situation.
b. Solve the inequality and interpret the solution.
32.
ANSWER:
ANSWER: a.
b.
He must sell at least 30 advertisements.
Define a variable and write an inequality for
each problem. Then solve.
33. One third of the sum of 5 times a number and 3 is
38.
less than one fourth the sum of six times that number
ANSWER: and 5.
ANSWER:
34. The sum of one third a number and 4 is at most the
39.
sum of twice that number and 12.
ANSWER:
ANSWER:
SENSE-MAKINGThe sides of square ABCD are
40.
extended to form rectangle DEFG. If the perimeter
35. of the rectangle is at least twice the perimeter of the
square, what is the maximum length of a side of
ANSWER: square ABCD?
36.
ANSWER:
1-5 Solving Inequalities ANSWER:
9 in.
MONEYJin is selling advertising space in Central MARATHONSJamie wants to be able to run at
37. 41.
City Magazine to local businesses. Jin earns 3% least the standard marathon distance of 26.2 miles. A
commission for every advertisement he sells plus a good rule for training is that runners generally have
salary of $250 a week. If the average amount of enough endurance to finish a race that is up to 3
money that a business spends on an advertisement is times his or her average daily distance.
$500, how many advertisements must he sell each a. If the length of her current daily run is 5 miles,
week to make a salary of at least $700 that week? write an inequality to find the amount by which she
a. Write an inequality to describe this situation. needs to increase her daily run to have enough
b. Solve the inequality and interpret the solution. endurance to finish a marathon.
b. Solve the inequality and interpret the solution.
ANSWER:
a. ANSWER:
b. a.
He must sell at least 30 advertisements.
b
. In order to have enough endurance to
Define a variable and write an inequality for run a marathon, Jamie should increase the distance
each problem. Then solve. of her average daily run by at least 3.73 miles.
38. One third of the sum of 5 times a number and 3 is
less than one fourth the sum of six times that number
MODELINGThe costs for renting a car from Ace
and 5. 42.
Car Rental and from Basic Car Rental are shown in
the table. For what mileage does Basic have the
ANSWER: better deal? Use the inequality
. Explain why this inequality
works.
The sum of one third a number and 4 is at most the
39.
sum of twice that number and 12.
ANSWER:
ANSWER:
SENSE-MAKINGThe sides of square ABCD are
40.
extended to form rectangle DEFG. If the perimeter Basic has the better deal as long as you are traveling
of the rectangle is at least twice the perimeter of the more than 80 miles. Yes, this is the correct inequality
square, what is the maximum length of a side of to use. Sample explanation: It works because the
square ABCD? inequality finds the mileage at which Ace s charge is
’
greater than Basic’s charge.
MULTIPLE REPRESENTATIONSIn this
43.
exercise, you will explore graphing inequalities on a
coordinate plane.
a. TABULAR
Organize the following into a table.
Substitute 5 points into the inequality .
State whether the resulting statement is true or
false.
ANSWER:
b. GRAPHICAL
9 in. Graph . Also graph
the 5 points from the table. Label all points that
MARATHONSJamie wants to be able to run at
41. resulted in a true statement with a T. Label all points
least the standard marathon distance of 26.2 miles. A that resulted in a false statement with an F.
good rule for training is that runners generally have
c. VERBAL
enough endurance to finish a race that is up to 3 Describe the pattern produced by the
eSolutions Manual - Powered by Cognero Page4
times his or her average daily distance. points you have labeled. Make a conjecture about
a. If the length of her current daily run is 5 miles, which points on the coordinate plane would result in
write an inequality to find the amount by which she true and false statements.
needs to increase her daily run to have enough
endurance to finish a marathon. ANSWER:
b. Solve the inequality and interpret the solution. a. Sample answer:
ANSWER:
a.
b
. In order to have enough endurance to
run a marathon, Jamie should increase the distance
of her average daily run by at least 3.73 miles.
MODELINGThe costs for renting a car from Ace
42.
Car Rental and from Basic Car Rental are shown in
the table. For what mileage does Basic have the
better deal? Use the inequality
. Explain why this inequality
works.
b. Sample answer:
ANSWER:
Basic has the better deal as long as you are traveling
more than 80 miles. Yes, this is the correct inequality
to use. Sample explanation: It works because the
inequality finds the mileage at which Ace s charge is
’
c. Sample answer: The points on or above the line
greater than Basic’s charge. result in true statements, and the points below the
line result in false statements. This is true for all
MULTIPLE REPRESENTATIONSIn this
43. points on the coordinate plane.
exercise, you will explore graphing inequalities on a
coordinate plane.
a. TABULAR 44. CHALLENGEIf and , then
Organize the following into a table.
Substitute 5 points into the inequality . . What is ?
State whether the resulting statement is true or
false. ANSWER:
(a + b) < 4
b. GRAPHICAL
Graph . Also graph
45. ERROR ANALYSISMadlynn and Emilie were
the 5 points from the table. Label all points that comparing their homework. Is either of them
resulted in a true statement with a T. Label all points correct? Explain your reasoning.
that resulted in a false statement with an F.
c. VERBAL
Describe the pattern produced by the
points you have labeled. Make a conjecture about
which points on the coordinate plane would result in
true and false statements.
ANSWER:
a. Sample answer: ANSWER:
No; sample answer: Madlynn reversed the inequality
sign when she added 1 to each side. Emilie did not
reverse the inequality sign at all.
46. REASONINGDetermine whether the following
statement is sometimes, always, or never true.
Explain your reasoning.
The opposite of the absolute value of a negative
number is less than the opposite of that number.
ANSWER:
Sample answer: Always; the opposite of the absolute
value of a negative number will always be a negative
value, while the opposite of a negative number will
always be a positive value. A negative value will
b. Sample answer: always be less than a positive value.
47. CHALLENGEGiven with sides
and , determine
the values of x such that exists.
ANSWER:
Using the Triangle Inequality Theorem, we know
that the sum of the lengths of any 2 sides of a
triangle must be greater than the length of the
remaining side. This generates 3 inequalities to
examine.
c. Sample answer: The points on or above the line
result in true statements, and the points below the
line result in false statements. This is true for all
points on the coordinate plane.
44. CHALLENGEIf and , then
. What is ?
In order for all 3 conditions to be true, x must be
greater than 0.2.
ANSWER:
(a + b) < 4
48. OPEN ENDEDWrite an inequality for which the
solution is all real numbers in the form
45. ERROR ANALYSISMadlynn and Emilie were . Explain how you know this.
comparing their homework. Is either of them
correct? Explain your reasoning.
ANSWER:
Sample answer: ; This has a
solution set of all real numbers because it simplifies
to or . This indicates that for
any real value of x the inequality is equivalent to
, that is the left side will always be 1 greater than the
right side.
ANSWER:
No; sample answer: Madlynn reversed the inequality
sign when she added 1 to each side. Emilie did not 49. WRITING IN MATHWhy does the inequality
reverse the inequality sign at all. symbol need to be reversed when multiplying or
dividing by a negative number?
46. REASONINGDetermine whether the following
statement is sometimes, always, or never true. ANSWER:
Explain your reasoning. Sample answer: When one number is greater than
The opposite of the absolute value of a negative another number, it is either more positive or less
negative than that number. When these numbers are
number is less than the opposite of that number. multiplied by a negative value, their roles are
reversed. That is, the number that was more positive
ANSWER: is now more negative than the other number. Thus, it
Sample answer: Always; the opposite of the absolute is now less than that number and the inequality
value of a negative number will always be a negative symbol needs to be reversed.
value, while the opposite of a negative number will
always be a positive value. A negative value will
always be less than a positive value. 50. SHORT RESPONSERogelio found a cookie
recipe that requires cup of sugar and 2 cups of
47. CHALLENGEGiven with sides
and , determine flour. How many cups of sugar would he need if he
the values of x such that exists. used 6 cups of flour?
ANSWER:
ANSWER:
Using the Triangle Inequality Theorem, we know
that the sum of the lengths of any 2 sides of a
triangle must be greater than the length of the
remaining side. This generates 3 inequalities to 51. STATISTICS The mean score for Samantha’s first
examine. six algebra quizzes was 88. If she scored a 95 on her
next quiz, what will her mean score be for all 7
quizzes?
A C
89 91
B D
90 92
ANSWER:
A
In order for all 3 conditions to be true, x must be 52. SAT/ACT The average of five numbers is 9. The
greater than 0.2. average of 7 other numbers is 8. What is the average
of all 12 numbers?
48. OPEN ENDEDWrite an inequality for which the
solution is all real numbers in the form F
. Explain how you know this. G
ANSWER:
Sample answer: ; This has a H
solution set of all real numbers because it simplifies J
to or . This indicates that for
any real value of x the inequality is equivalent to K
, that is the left side will always be 1 greater than the
right side.
ANSWER:
49. WRITING IN MATHWhy does the inequality F
symbol need to be reversed when multiplying or
dividing by a negative number? 53. What is the complete solution of the equation
?
ANSWER: A x = 8; x = 12
Sample answer: When one number is greater than B x = 8; x = –12
another number, it is either more positive or less
negative than that number. When these numbers are Cx = –8; x = –12
multiplied by a negative value, their roles are D x = –8; x = 12
reversed. That is, the number that was more positive
is now more negative than the other number. Thus, it ANSWER:
is now less than that number and the inequality D
symbol needs to be reversed. Solve each equation. Check your solutions.
50. SHORT RESPONSERogelio found a cookie 54.
recipe that requires cup of sugar and 2 cups of
ANSWER:
flour. How many cups of sugar would he need if he
used 6 cups of flour?
ANSWER: 55.
ANSWER:
51. STATISTICS The mean score for Samantha’s first
six algebra quizzes was 88. If she scored a 95 on her
next quiz, what will her mean score be for all 7
quizzes? 56.
A C ANSWER:
89 91
B D
90 92
ANSWER:
A 57. ASTRONOMYPluto travels in a path that is not
circular. Pluto’s farthest distance from the Sun is
52. SAT/ACT The average of five numbers is 9. The 4539 million miles, and its closest distance is 2756
average of 7 other numbers is 8. What is the average million miles. Write an equation that can be solved to
of all 12 numbers? find the minimum and maximum distances from the
Sun to Pluto.
F
ANSWER:
G
H 58. POPULATIONIn 2005, the population of Bay City
was 19,611. For each of the next five years, the
J population decreased by an average of 715 people
per year.
K a. What was the population in 2010?
b. If the population continues to decline at the same
rate as from 2005 to 2010, what would you expect
ANSWER: the population to be in 2025?
F
ANSWER:
53. What is the complete solution of the equation a. 16,036
? b. 5311
A x = 8; x = 12 59. GEOMETRY The formula for the surface area of a
B x = 8; x = –12 cylinder is .
C x = –8; x = –12 a. Use the Distributive Property to rewrite the
D x = –8; x = 12 formula by factoring out the greatest common factor
ANSWER: of the two terms.
D b. Find the surface area for a cylinder with radius 3
Solve each equation. Check your solutions. centimeters and height 10 centimeters using both
formulas. Leave the answer in terms of .
54.
c. Which formula do you prefer? Explain your
reasoning.
ANSWER:
ANSWER:
a.
b.
55.
c. b
Sample answer: The formula in part is quicker.
ANSWER:
CONSTRUCTIONThe Sawyers are adding a
60.
family room to their house. The dimensions of the
room are 26 feet by 28 feet. Show how to use the
Distributive Property to mentally calculate the area
56. of the room.
ANSWER:
ANSWER:
57. ASTRONOMYPluto travels in a path that is not Solve each equation. Check your solutions.
circular. Pluto’s farthest distance from the Sun is
4539 million miles, and its closest distance is 2756 61.
million miles. Write an equation that can be solved to
find the minimum and maximum distances from the ANSWER:
Sun to Pluto.
ANSWER:
62.
ANSWER:
58. POPULATIONIn 2005, the population of Bay City
was 19,611. For each of the next five years, the
population decreased by an average of 715 people
63.
per year.
a. What was the population in 2010?
b. If the population continues to decline at the same ANSWER:
rate as from 2005 to 2010, what would you expect
the population to be in 2025?
ANSWER:
64.
a. 16,036
b. 5311 ANSWER:
59. GEOMETRY The formula for the surface area of a
cylinder is .
65.
a. Use the Distributive Property to rewrite the
formula by factoring out the greatest common factor ANSWER:
of the two terms.
b. Find the surface area for a cylinder with radius 3
centimeters and height 10 centimeters using both
66.
formulas. Leave the answer in terms of .
ANSWER:
c. Which formula do you prefer? Explain your
reasoning.
ANSWER:
a.
b.
c. b
Sample answer: The formula in part is quicker.
CONSTRUCTIONThe Sawyers are adding a
60.
family room to their house. The dimensions of the
room are 26 feet by 28 feet. Show how to use the
Distributive Property to mentally calculate the area
of the room.
ANSWER:
Solve each equation. Check your solutions.
61.
ANSWER:
62.
ANSWER:
63.
ANSWER:
64.
ANSWER:
65.
ANSWER:
66.
ANSWER:
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