Download The solution set is - Waynesville R

5-2 Solving Inequalities by Multiplication and Division
The Jefferson Band Boosters raised more than $5500 from sales of their $15 band DVD. Define
a variable, and write an inequality to represent the number of DVDs they sold. Solve the inequality and interpret your
solution.
Let d = the number of DVDs sold.
This means that the band sold at least 367 DVDs.
Solve each inequality. Graph the solution on a number line.
The solution set is {n|n < 60}.
The solution set is {r|r
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Page 1
The solution set is {r|r
5-2 Solving Inequalities by Multiplication and Division
The solution set is {c|
42}.
The solution set is {h|h < 10}.
9t > 108
The solution set is {t|t > 12}.
84 < 7v
The solution set is {v|v > 12}.
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The solution set is {t|t > 12}.
5-2 Solving Inequalities by Multiplication and Division
84 < 7v
The solution set is {v|v > 12}.
6x
The solution set is
.
5z
The solution set is {z|
8}.
Define a variable, write an inequality, and solve each problem. Then interpret your solution.
Mario purchases a prepaid phone plan for $50 at $0.13 per minute. How many minutes can
Mario talk on this plan?
Let m = the number of minutes that Mario can talk.
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So, Mario can talk up to 384 minutes.
Page 3
The solution set is {z|
8}.
5-2 Solving Inequalities by Multiplication and Division
Define a variable, write an inequality, and solve each problem. Then interpret your solution.
Mario purchases a prepaid phone plan for $50 at $0.13 per minute. How many minutes can
Mario talk on this plan?
Let m = the number of minutes that Mario can talk.
So, Mario can talk up to 384 minutes.
Rodrigo needs at least $560 to pay for his spring break expenses, and he is saving $25
from each of his weekly paychecks. How long will it be before he can pay for his trip?
Let p = the number of pay periods for which Rodrigo will need to save.
So, Rodrigo will need to save for 23 weeks.
Solve each inequality. Graph the solution on a number line.
The solution set is {m|
68}.
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The solution set is {a|a < 40}.
Page 4
The solution set is {m|
68}.
5-2 Solving Inequalities by Multiplication and Division
The solution set is {a|a < 40}.
The solution set is {c|c > 121}.
The solution set is {d|
68}.
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The solution set is {x|
20}.
Page 5
The solution set is {d|
68}.
5-2 Solving Inequalities by Multiplication and Division
The solution set is {x|
20}.
The solution set is {f |f < 432}.
The solution set is {h|h >21}.
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Page 6
The solution set is {h|h >21}.
5-2 Solving Inequalities by Multiplication and Division
The solution set is {j |
16}.
The solution set is {n|
108}.
6p
The solution set is {p |
16}.
4r < 64
The solution set is {r|r < 16}.
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The solution set is {p |
16}.
5-2 Solving Inequalities by Multiplication and Division
4r < 64
The solution set is {r|r < 16}.
32 > 2y
The solution set is {y|y > 16}.
26 < 26t
The solution set is {t|t > 1}.
6v > 72
The solution set is {v|v < 12}.
3z
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Page 8
The solution set is {v|v < 12}.
5-2 Solving Inequalities by Multiplication and Division
3z
The solution set is {z|
4b
11}.
3
The solution set is
.
2d < 5
The solution set is
.
7f > 5
The solution set is
.
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To remain on the cheerleading squad, Lakita must attend at least
Page 9
of the study table sessions
The solution set is
.
5-2 Solving Inequalities by Multiplication and Division
7f > 5
The solution set is
.
To remain on the cheerleading squad, Lakita must attend at least
of the study table sessions
offered. She attends 15 sessions. If Lakita met the requirements, what is the maximum number of study table
sessions?
Let s represent the number of sessions.
At most, 25 study table sessions are offered.
How many bracelets can Caitlin buy for herself and her friends if she wants to spend no more than
$22?
Let b represent the number of bracelets.
Caitlin can buy no more than 4 bracelets.
The National Honor Society at Pleasantville High School wants to raise at least $500 for a local charity.
Each student earns $0.50 for every quarter of a mile walked in a walk-a-thon. How many miles will the students
need to walk?
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Let m represent the number of quarter miles.
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5-2 Solving Inequalities by Multiplication and Division
Caitlin can buy no more than 4 bracelets.
The National Honor Society at Pleasantville High School wants to raise at least $500 for a local charity.
Each student earns $0.50 for every quarter of a mile walked in a walk-a-thon. How many miles will the students
need to walk?
Let m represent the number of quarter miles.
The students needs to walk at least 1000 quarter miles or at least 250 miles.
The American history classes are planning a trip to a local museum. Admission is $8 per person.
Determine how many people can go for $260.
Let p represent the number of people.
No more than 32 people can go to the museum.
If gasoline costs $3.15 per gallon, how many gallons of gasoline, to the nearest tenth, can Jan buy for
$24?
Let g represent the number of gallons of gasoline.
Jan can buy no more than 7.6 gallons of gasoline.
Match each inequality to the graph of its solution.
a.
b.
c.
d.
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5-2 Solving Inequalities by Multiplication and Division
Jan can buy no more than 7.6 gallons of gasoline.
Match each inequality to the graph of its solution.
a.
b.
c.
d.
The point is at 13.5, so the correct choice is b.
25j
The point is at 0.32, so the correct choice is a.
3.6p < 4.5
The point is at 1.25, so the correct choice is d.
2.3 < 5t
eSolutions
by so
Cognero
The Manual
point is- Powered
at 0.46,
the correct
choice is c.
Page 12
Fewer than 42 employees at a factory stated that they preferred fudge over fruit candy. This is about two
5-2 Solving Inequalities by Multiplication and Division
The point is at 1.25, so the correct choice is d.
2.3 < 5t
The point is at 0.46, so the correct choice is c.
Fewer than 42 employees at a factory stated that they preferred fudge over fruit candy. This is about two
thirds of the employees. How many employees are there?
Let e represent the number of employees.
So, there are fewer than 63 employees.
A certain travel agency employs more than 275 people at all of its branches. Approximately three fifths
of all the people are employed at the west branch. How many people work at the west branch?
Let p represent the number of people.
So, there are more than 165 people employed at the west branch.
MULTIPLE The equation for the volume of a pyramid is
Draw a pyramid with a square base b cm long and a height of h cm.
3
Suppose the pyramid has a volume of 72 cm . Write an equation to find the height.
Create a table showing the value of h when b = 1, 3, 6, 9, and 12.
Write an inequality for the possible lengths of b such that b < h. Write an inequality for the
possible lengths of h such that b > h.
a.
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b.
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5-2 Solving Inequalities by Multiplication and Division
So, there are more than 165 people employed at the west branch.
MULTIPLE The equation for the volume of a pyramid is
Draw a pyramid with a square base b cm long and a height of h cm.
3
Suppose the pyramid has a volume of 72 cm . Write an equation to find the height.
Create a table showing the value of h when b = 1, 3, 6, 9, and 12.
Write an inequality for the possible lengths of b such that b < h. Write an inequality for the
possible lengths of h such that b > h.
a.
b.
c.
b
h
1
216
3
24
6
6
9
12
d. b < h when 0 < b < 6; b > h when 6 < h.
Taro and Jamie are solving 6d
84. Is either of them correct? Explain your reasoning.
Taro is correct. Jamie reversed the inequality sign when dividing by 6. The inequality sign should only be reversed
when dividing or multiplying by a negative number.
CHALLENGE Solve each inequality for x. Assume that a > 0.
a.
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b.
c.
Page 14
Taro is correct.
Jamie by
reversed
the inequality
when dividing by 6. The inequality sign should only be reversed
5-2 Solving
Inequalities
Multiplication
andsign
Division
when dividing or multiplying by a negative number.
CHALLENGE Solve each inequality for x. Assume that a > 0.
a.
b.
c.
When dividing by a negative number, reverse the inequality symbol.
b.
c.
2
Determine whether x > 1 and x > 1 are equivalent. Explain.
2
Not equivalent; the inequalities do not have the same solution sets. If x = 2, then x = 4 and 4 > 1. The solution set
2
of x > 1 contains the subsets of real numbers given by x
x < 1.
Explain whether the statement If a > b, then
is sometimes, always, or never true.
Sometimes. Consider three cases.
Case I: a > 0 and b
Choose a = 3 and b
.
Case II: a < 0 and b
Choose a = 2 and b = 3. Then 2 > 3, but
.
Case III: a > 0 and b
Choose a = 2 and b = 3. Then 2 > 3, and
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The statement is only true when a > 0 and b < 0.
Page 15
Determine whether x > 1 and x > 1 are equivalent. Explain.
2
Not equivalent;
the inequalities
do not have
theDivision
same solution sets. If x = 2, then x = 4 and 4 > 1. The solution set
5-2 Solving
Inequalities
by Multiplication
and
2
of x > 1 contains the subsets of real numbers given by x
x < 1.
Explain whether the statement If a > b, then
is sometimes, always, or never true.
Sometimes. Consider three cases.
Case I: a > 0 and b
Choose a = 3 and b
.
Case II: a < 0 and b
Choose a = 2 and b = 3. Then 2 > 3, but
.
Case III: a > 0 and b
Choose a = 2 and b = 3. Then 2 > 3, and
The statement is only true when a > 0 and b < 0.
Create a real world situation to represent the inequality
Student answers may vary. Sample answer: The temperature never exceeds
.
.
WRITING IN MATH How are solving linear inequalities and linear equations similar? different?
Sample answer: The same processes are used when solving linear inequalities and equations that involve addition,
subtraction, multiplication, or division by a positive number. However, when a linear inequality is multiplied or divided
by a negative number, the inequality symbol must change directions so that the inequality remains true.
For example:
Inequality:
Equation:
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Juan s long-distance phone company charges 9 for each minute. Which inequality can be used to find how long he
can talk to a friend if he does not want to spend more than $2.50 on the call?
5-2 Solving
Multiplication
andThe
Division
Student Inequalities
answers may by
vary.
Sample answer:
temperature never exceeds
.
WRITING IN MATH How are solving linear inequalities and linear equations similar? different?
Sample answer: The same processes are used when solving linear inequalities and equations that involve addition,
subtraction, multiplication, or division by a positive number. However, when a linear inequality is multiplied or divided
by a negative number, the inequality symbol must change directions so that the inequality remains true.
For example:
Inequality:
Equation:
Juan s long-distance phone company charges 9 for each minute. Which inequality can be used to find how long he
can talk to a friend if he does not want to spend more than $2.50 on the call?
0.09 2.50m
0.09 2.50m
0.09m 2.50
0.09m 2.50
The rate 0.09 is the rate to be multiplied by each minute. So choices A and B can be eliminated. If he does not want
to spend more than $2.50, then choice D is correct.
Find the value of x.
Let y represent the height of the smaller triangle.
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0.09m
2.50
The rateInequalities
0.09 is the rate
be multiplied by
each
minute. So choices A and B can be eliminated. If he does not want
5-2 Solving
bytoMultiplication
and
Division
to spend more than $2.50, then choice D is correct.
Find the value of x.
Let y represent the height of the smaller triangle.
If y = 8, then
So, x is 10 inches.
What is the greatest rate of decrease of this function?
5
3
2
1
The greatest rate of decrease is when the slope is the steepest and negative. This means that the line will slope down
from left to right. So, the steepest line is between (
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5-2 Solving Inequalities by Multiplication and Division
So, x is 10 inches.
What is the greatest rate of decrease of this function?
5
3
2
1
The greatest rate of decrease is when the slope is the steepest and negative. This means that the line will slope down
from left to right. So, the steepest line is between (
So, the correct choice is F.
What is the value of x if 4x
2
3 = 2x?
2
So, the correct choice is C.
Solve each inequality. Check your solution, and then graph it on a number line.
8 + 4a < 6a
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5-2 Solving Inequalities by Multiplication and Division
So, the correct choice is C.
Solve each inequality. Check your solution, and then graph it on a number line.
8 + 4a < 6a
{a|a > 4}
2y + 11
7
24y
2b > 12b
Find the inverse of each function.
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5-2 Solving Inequalities by Multiplication and Division
Find the inverse of each function.
Pam is having blinds installed at her home. The cost c of installation for any number of blinds b
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can be described by c = 25 + 6.5b. Graph the equation and determine how much it would cost if Pam has 8 blinds
installed.
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5-2 Solving Inequalities by Multiplication and Division
Pam is having blinds installed at her home. The cost c of installation for any number of blinds b
can be described by c = 25 + 6.5b. Graph the equation and determine how much it would cost if Pam has 8 blinds
installed.
Evaluate c = 25 + 6.5b for b = 8.
So, it would cost Pam $77 to have 8 blinds installed.
A boater radioed for a helicopter to pick up a sick crew member. At that time, the boat and helicopter
were at the positions shown. How long will it take to reach the boat?
Let t represent the time.
It should take the helicopter 2 hours to reach the boat.
Solve each open sentence.
|x + 3| = 10
Case1:
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5-2 Solving Inequalities by Multiplication and Division
It should take the helicopter 2 hours to reach the boat.
Solve each open sentence.
|x + 3| = 10
Case1:
Case2:
{ 13, 7}
|2x
8| = 6
Case1:
Case2:
{1, 7}
|3x + 1| = 2
means the distance between 3x and 1 is 2. Since distance cannot be negative, the solution is the
empty set
.
Solve each equation.
4y + 11 = 19
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|3x + 1| = 2
means theby
distance
between 3x
and
1 is 2. Since distance cannot be negative, the solution is the
5-2 Solving Inequalities
Multiplication
and
Division
empty set .
Solve each equation.
4y + 11 = 19
2x
7 = 9 + 4x
+ 2x = 4x
(6w
8
3) = 3w + 12
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5-2 Solving Inequalities by Multiplication and Division
(6w
3) = 3w + 12
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