Resources

Resources

One-step inequalities: word problems

Nick is doing some math exercises on a website called Khon Academy. In Khon Academy, you have to get at least

70\%

of the problems in an exercise right in order to gain proficiency.

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So far, Nick answered one question, and he didn't get it right. Suppose he answers

75\%

of the following

q

questions correctly and gains proficiency in the exercise.

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Write an inequality in terms of

q

that models the situation.

\frac{1}{4} c+3 d

c=6

d=7

The recommended maximum speed for the German Autobahn expressway is

v_{\text{max}}

kilometers per hour. Write an inequality that describes

v

, any speed slower than the recommended maximum speed.

Jenna worked out for more than

25

minutes total. She spent

10

minutes running, and

x

minutes each on

2

stations. How much time could she have spent on each station?

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Write an inequality to represent the situation. Use

x

to represent the minutes spent on each of the stations.

A charity organization had to sell

18

tickets to their fundraiser just to cover necessary production costs. They sold each ticket for

\$45

y

represent the net profit (in dollars) when they have sold

x

tickets. Which of the following information about the graph of the relationship is given?

Marie is leaving her cat, Shadow, with a pet-sitter while she's away on vacation. Marie packs up toys, and a note that says Shadow eats at most

12

ounces of food each day.

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Write an inequality to represent the number of ounces of food that Shadow might eat in a day,

f

Frannie thinks she has raised a prize-winning giant pumpkin. It weighs

324

pounds! In order to get the first place prize, the pumpkins of her competitors can weigh

324

pounds at most.

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Write an inequality to represent the weight, in pounds, of the other pumpkins if Frannie is to win the first place prize,

w

g(x)=8x-5

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Which of the following is equivalent to

g(g(x))

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1

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64x-10

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64x-45

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64x^{2}+25

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64x^{2}-80x+25

The light from the moon, in lux, on the night of the

t^{\text{th}}

2016

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L(t)=0.25-\sin\left(\frac{2\pi(t-2)}{28.5}\right)

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What is the period of the light from the moon? Give an exact answer.

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A chameleon is looking for prey. Let positive numbers represent the elevation of prey above the chameleon and negative numbers represent the elevation of prey below the chameleon.

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The chameleon spots a fly at

4\,\text{m}

and a grasshopper at

-6\,\text{m}

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What does an elevation of

0\,\text{m}

represent in this situation?

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1

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(A) The elevation of the fly

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(B) The elevation of the chameleon

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(C) The elevation of the grasshopper

149\,\text{km}

from the university and drives

95\,\text{km}

closer every hour. Valente is

170\,\text{km}

from the university and drives

110\,\text{km}

closer every hour. Let

t

represent the time, in hours, since Chiamaka and Valente started driving toward the university. Complete the inequality to represent the times when Valente is closer than Chiamaka to the university.

A solid line in the

xy

plane passes through the points

(0, -4)

(1, -1)

(2, 2)

. The line divides the plane into two halves. The top half is shaded. A solid line in the

xy

plane passes through the points

(0, -4)

(1, -1)

(2, 2)

. The line divides the plane into two halves. The top half is shaded. If the shaded region in the graph represents the solution set to an inequality, which of the following could be the inequality?

Mindy and Troy combined ate

9

pieces of wedding cake. Mindy ate

3

pieces of cake and Troy had

\frac{1}{4}

of the total cake. How many pieces of cake were there in all?

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\square

A standard deck of cards has

52

total cards divided evenly into

4

suits - there are

13

13

13

13

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Ayana and Emil are playing a game that involves drawing

2

cards from a standard deck without replacement to start the game. If neither of the cards are spades, then Ayana goes first. Otherwise, Emil goes first.

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Is this a fair way to decide who goes first? Why or why not?

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1

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(A) No, there is a higher probability that Ayana goes first.

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B No, there is a higher probability that Emil goes first.

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(C) Yes, they both have an equal probability of going first.

Mindy and Troy combined ate

9

pieces of wedding cake. Mindy ate

3

pieces of cake and Troy had

\frac{1}{4}

of the total cake. How many pieces of cake were there in all?

Use the disk method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the

x

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y=\frac{1}{\sqrt{1+x^{2}}}, \quad y=0, \quad x=-1, \quad x=1

20

\$

to spend on taxi fare. The ride costs

5

\$

2.50

per kilometer. Write an inequality to determine the distance in kilometers,

d

, you can ride for

20

\$

Calculate the target markup when the unit product cost is

\$ 32

, there is revenue of

\$ 6

per unit, and the target markup percentage is

40 \%

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\$ 18.80

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\$ 12.20

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\$ 12.80

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\$ 14.20

36S+64U \geq 288

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Choo Kheng must travel at least

kilometers in a submarine in order to reach her destination. The submarine travels at a constant speed on the water's surface, and travels at a constant speed underwater. In the given inequality,

S

represents the number of hours the submarine can travel on the water's surface and

U

represents the number of hours it can travel underwater in order to reach Choo Kheng's destination. If the submarine travels for

hours on the water's surface, what is the least number of hours the submarine must travel underwater in order for Choo Kheng to reach her destination?

The cost of a sleeping bag has been discounted by

\$22

, so that the sale price of the bag is

\$52

. Find the original price of the sleeping bag,

p

, by solving the equation

p-22=52

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The original price of the sleeping bag was

\$\square

A

has side lengths of

6 \mathrm{~cm}

3.5 \mathrm{~cm}

. The side lengths of rectangle

B

are proportional to the side lengths of rectangle

A

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What could be the side lengths of rectangle

B

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2

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3 \mathrm{~cm}

1.75 \mathrm{~cm}

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5 \mathrm{~cm}

2.5 \mathrm{~cm}

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7 \mathrm{~cm}

7 \mathrm{~cm}

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12 \mathrm{~cm}

5 \mathrm{~cm}

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5.25 \mathrm{~cm}

9 \mathrm{~cm}

The drama club is selling tickets to their play to raise money for the show's expenses. Each student ticket sells for

\$ 5.50

and each adult ticket sells for

\$ 9.50

. The drama club must make at least

\$ 660

from ticket sales to cover the show's costs. Write an inequality that could represent the possible values for the number of student tickets sold,

s

, and the number of adult tickets sold,

a

, that would satisfy the constraint.

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n

d

dimes. She has no less than

\$ 1

worth of coins altogether. Write this situation as an inequality.

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Penelope is working two summer jobs, making

\$ 7

per hour washing cars and

\$ 13

per hour landscaping. Penelope must earn a minimum of

\$ 95

this week. Write an inequality that would represent the possible values for the number of hours washing cars,

w

, and the number of hours landscaping,

l

, that Penelope can work in a given week.

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Lydia is working two summer jobs, tutoring and landscaping. She must work no less than

9

hours altogether between both jobs in a given week. Write an inequality that would represent the possible values for the number of hours tutoring,

t

, and the number of hours landscaping,

l

, that Lydia can work in a given week.

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Caroline and her children went into a restaurant and will buy drinks and tacos. Each drink costs

\$ 2.75

and each taco costs

\$ 3

. Caroline has a total of

\$ 50

to spend on drinks and tacos. Write an inequality that would represent the possible values for the number of drinks purchased,

d

, and the number of tacos purchased,

t

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p

n

nickels. He has no more than

\$ 1

worth of coins altogether. Write this situation as an inequality.

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An online furniture store sells chairs for

\$ 100

each and tables for

\$ 650

each. The store must sell a minimum of

\$ 3700

worth of chairs and tables each day. Write an inequality that could represent the possible values for the number of tables sold,

t

, and the number of chairs sold,

c

, that would satisfy the constraint.

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An online furniture store sells chairs for

\$ 150

each and tables for

\$ 350

each. The store must sell at least

\$ 6600

worth of chairs and tables each day. Write an inequality that could represent the possible values for the number of tables sold,

t

, and the number of chairs sold,

c

, that would satisfy the constraint.

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d

q

quarters. She has a minimum of

\$ 2

worth of coins altogether. Write this situation as an inequality.

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Alejandro and his children went into a grocery store and will buy apples and mangos. Each apple costs

\$ 2

and each mango costs

\$ 1

. Alejandro has a total of

\$ 20

to spend on apples and mangos. Write an inequality that would represent the possible values for the number of apples purchased,

a

, and the number of mangos purchased,

m

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Oliver and his children went into a grocery store and will buy bananas and peaches. Each banana costs

\$ 0.60

and each peach costs

\$ 1

. Oliver has a total of

\$ 5

to spend on bananas and peaches. Write an inequality that would represent the possible values for the number of bananas purchased,

b

, and the number of peaches purchased,

p

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Jeriel is working two summer jobs, washing cars and landscaping. He can work a maximum of

15

hours altogether between both jobs in a given week. Write an inequality that would represent the possible values for the number of hours washing cars,

w

, and the number of hours landscaping,

l

, that Jeriel can work in a given week.

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The drama club is selling tickets to their play to raise money for the show's expenses. Each student ticket sells for

\$ 5

and each adult ticket sells for

\$ 9.50

. The drama club must make a minimum of

\$ 570

from ticket sales to cover the show's costs. Write an inequality that could represent the possible values for the number of student tickets sold,

s

, and the number of adult tickets sold,

a

, that would satisfy the constraint.

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Audrey and her children went into a grocery store and will buy apples and peaches. She must buy a maximum of

8

apples and peaches altogether. Write an inequality that would represent the possible values for the number of apples purchased,

a

, and the number of peaches purchased,

p

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Landon and his children went into a movie theater and will buy candies and pretzels. Each candy costs

\$ 5

and each pretzel costs

\$ 4.25

. Landon has a total of

\$ 45

to spend on candies and pretzels. Write an inequality that would represent the possible values for the number of candies purchased,

c

, and the number of pretzels purchased,

p

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The cumulative costs of a tankless water heater and tank water heater are

\$4,090

each at the end of the

7^{\text{th}}

year of use. The cumulative costs of the tankless heater increase

\$180

per year, but for the tank heater, they increase by

11\%

per year. By the end of the

10^{\text{th}}

year, about how much more will the cumulative costs of the tank heater be than those of the tankless heater?

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1

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\$964

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\$1,504

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\$4,625

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\$5,723

In a board game, negative number cards represent moving backward and positive number cards represent moving forward. For example, Nuha drew a card with

4

and moved forward

4

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Raha draws a card with

0

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0

represent in the board game?

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1

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(A) Going back to the starting space

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(B) Not moving any spaces

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(C) Losing all points

A publishing company prints newspapers and magazines.

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N

represent the number of newspapers and

M

represent the number of magazines the company can print with its daily supply of ink cartridges.

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0.5N + 2.5M \leq 6000

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The company wants to print

8000

newspapers daily. How many magazines at most can the company print daily with the remaining number of ink cartridges?

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1

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(A) The company can print

800

magazines at most.

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(B) The company can print

2500

magazines at most.

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(C) The company can print

6000

magazines at most.

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(D) The company can print

8000

magazines at most.

Amy hikes down a slope to a lake that is

10.2\,\text{meters}

below the trail. Then Amy jumps into the lake, and swims

1.5\,\text{meters}

down. She wonders what her new height is relative to the trail.

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Which of the following equations matches the situation above?

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1

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-10.2+1.5= \,?

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10.2-1.5= \,?

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-10.2-1.5= \,?

A jar of jelly bean candies weighs

8

ounces. A container of caramel candies weighs

40

ounces. L\'ucia buys

10

pounds worth of jelly bean candy jars and caramel candy containers for her big party. Given that there are

16

ounces in a pound, which of the following equations correctly relates the number of jars of jelly bean candies,

j

, and the number of containers of caramel candies,

c

, that L\'ucia bought?

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1

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\frac{8}{16}j+\frac{40}{16}c=10

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8j+40c=10

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\frac{16}{8}j+\frac{16}{40}c=10

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16\cdot 8j+16\cdot 40c=10

Four players are playing a game together. They each gain or lose points based on their actions.

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5

points for letting a fish get away.

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10

points for collecting a mushroom.

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100

points for disturbing a bear.

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80

points for building a fire.

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Whose action has the greatest effect on their scores?

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1

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1.5 \cdot 1.09^{t}

models the housing costs, in thousands of dollars, for summer term

t

years since Chinedu applied to his college.

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1

09

represent in this expression?

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1

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(A) Chinedu applied to his school

9

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(B) The summer housing costs were

\$ 1,090

the year that Chinedu applied to his college.

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(C) The summer housing costs increase by

9 \%

Consider the following problem:

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The depth of the water in Harsha's bird bath is changing at a rate of

r(t)=0.25 t-0.1

millimeters per hour (where

t

is the time in hours). At time

t=0

, the depth of the water is

35

millimeters. What is the depth of the water at

t=3

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Which expression can we use to solve the problem?

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1

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35+\int_{2}^{3} r(t) d t

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35+\int_{3}^{3} r(t) d t

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35+\int_{3}^{4} r(t) d t

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35+\int_{0}^{3} r(t) d t

The amount of gasoline stored at a fuel station is decreasing at a rate of

r(t)

liters per hour (where

t

is the time in hours).

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\int_{20}^{21} r(t) d t=-450

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1

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20^{\text {th }}

hour, the amount of gasoline at the fuel station decreased by

450

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(B) During the first

20

hours, the amount of gasoline at the fuel station decreased by

450

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(C) During the first

21

hours, the amount of gasoline at the fuel station decreased by

450

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21^{\text {st }}

hour, the amount of gasoline at the fuel station decreased by

450

Consider the following problem:

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The depth of the water in Harsha's bird bath is changing at a rate of

r(t)=0.25 t-0.1

millimeters per hour (where

t

is the time in hours). At time

t=0

, the depth of the water is

35

millimeters. What is the depth of the water at

t=3

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Which expression can we use to solve the problem?

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1

\newline

35+\int_{3}^{3} r(t) d t

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35+\int_{3}^{4} r(t) d t

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35+\int_{0}^{3} r(t) d t

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35+\int_{2}^{3} r(t) d t

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h(x)=x^{5}-2 x^{4}+10 x^{2}-10 x \text {. }

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Below is Omar's attempt to write a formal justification for the fact that there exists a value

c

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-2<c<2 \text { such that } h^{\prime}(c)=6 \text {. }

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Is Omar's justification complete? If not, why?

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Omar's justification:

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\begin{array}{l} h(-2)=-4 \text { and } \\ h(2)=20 \text {, so } \\ \frac{h(2)-h(-2)}{2-(-2)}=6 . \end{array}

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So, according to the mean value theorem, there exists a value

c

somewhere in the interval

-2<c<2

h^{\prime}(c)=6

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1

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(A) Yes, Omar's justification is complete.

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(B) No, Omar didn't establish that the average rate of change of

h

[-2,2]

6

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(C) No, Omar didn't establish that

h

is differentiable.

f(x)=\sqrt{x+\sqrt{x^{3}}}

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Below is Lena's attempt to write a formal justification for the fact that the equation

f^{\prime}(x)=\frac{2}{3}

has a solution where

0<x<9

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Is Lena's justification complete? If not, why?

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Lena's justification:

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f(0)=0

f(9)=6

, so the average rate of change of

f

over the interval

0 \leq x \leq 9

\frac{2}{3}

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So, according to the mean value theorem,

f^{\prime}(x)=\frac{2}{3}

must have a solution somewhere in the interval

0<x<9

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1

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(A) Yes, Lena's justification is complete.

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(B) No, Lena didn't establish that the average rate of change of

f

[0,9]

\frac{2}{3}

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(C) No, Lena didn't establish that

f

is differentiable.

g(x)=5-e^{x}

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Below is Sean's attempt to write a formal justification for the fact that the equation

g(x)=0

has a solution where

1 \leq x \leq 4

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Is Sean's justification complete? If not, why?

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Sean's justification:

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g

is defined for all real numbers, and exponential functions are continuous at all points in their domains.

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So, according to the intermediate value theorem,

g(x)=0

must have a solution somewhere between

x=1

x=4

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1

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(A) Yes, Sean's justification is complete.

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(B) No, Sean didn't establish that

0

g(1)

g(4)

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(C) No, Sean didn't establish that

g