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Find a value using two-variable equations

Susan's cell phone plan includes unlimited texting. The amount she pays for texting each month can be described by the function

y = 20

y

is the number of dollars spent on texting as a function of

x

, the number of texts.

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What is true about the function?

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(A)It is linear because the rate of change is zero.

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(B)It is linear because the rate of change is increasing.

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(C)It is nonlinear because the rate of change is zero.

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(D)It is nonlinear because the rate of change is increasing.

The equation shows how the cost of buying pencils is related to the number of pencils purchased.

c= 0.5p+2

p

represents the number of pencils, and the variable

c

represents the total cost. How much does it cost to buy

10

This equation shows how the amount of time that a receptionist named Trevor spends on the phone is related to the number of phone calls he routes to employees.

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t = 2p

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p

represents the number phone calls he routes, and the variable

t

represents the number of minutes he is on the phone. If he routes

4

phone calls, how much time will Trevor have spent on the phone in total?

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This equation shows how the money Lanberry Middle School spends on workbooks is related to the total number of workbooks they buy.

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d = 20t

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t

represents the number of workbooks bought by the school, and the variable

d

represents the total cost of those workbooks. How much would it cost Lanberry Middle School to buy a total of

1

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\$

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This equation shows how the cost of Ann's retirement party depends on the number of guests.

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c = 2g

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g

represents the number of guests, and the variable

c

represents the cost in dollars. If there are

5

guests, how much will Ann's retirement party cost?

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\$

\_\_\_\_

This equation shows how the total number of necklaces Kendra owns is related to the amount of money she spends on additional necklaces.

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n = d

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d

represents the amount of money she spends on additional necklaces, and the variable

n

represents the total number of necklaces she owns. With

\$10

to spend on new necklaces, how many total necklaces can Kendra own?

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_____ necklaces

This equation shows how the total number of books Clara has read depends on the number of months she has been part of a book club.

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b = m + 8

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m

represents the number of months she has been a member of the book club, and the variable

b

represents the number of books that she has read. After belonging to the book club for

9

months, how many books will Clara have read in all?

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This equation shows how the cost of Tanner's birthday party depends on the number of guests.

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c = g + 12

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g

represents the number of guests, and the variable

c

represents the cost in dollars. If there are

7

guests, how much will Tanner's birthday party cost?

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\$

Several years ago, Annie bought a computer for her small business. For tax purposes, Annie declares a constant rate of depreciation, or loss of value, for the computer. The graph shows the linear relationship between the value of the computer and the number of years Annie has owned it.

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Complete the equation to model the relationship between the value of the computer,

V

, in dollars and the number of years Annie has owned it,

t

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V = ____ t + ____

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Complete the sentences.

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The original purchase price of the computer is represented by the Choices: initial value, rate of change, x-intercept, which is Choices:

200

\$

1,600

\$

1,800

\$

2,000

\$

. The change in the computer's value over time is represented by the Choices: initial value, rate of change, x-intercept, which is Choices:

t

1

\$

t

3

\$

t

5

\$

200

\$

The average monthly temperature changes from month to month. Suppose that, for a given city, we can model the average temperature in a month with the following function.

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f(t)=60-27 \sin\left(\frac{\pi}{6}t\right)

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In this equation,

f(t)

is the average temperature in a month in degrees Fahrenheit, and

t

is the month of the year (January

=1

=2,\dots

). Find the following. If necessary, round to the nearest hundredth.

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Time for one full cycle of

f

\square

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Number of cycles of

f

\square

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Minimum average temperature in a month:

\square

After a certain medicine is ingested, the number of harmful bacteria remaining in the body declines rapidly. The relationship between the elapsed time

t

, in minutes, since the medicine is ingested, and the number of harmful bacteria remaining in the body,

H_{\text{minute}}(t)

, is modeled by the following function:

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H_{\text{minute}}(t)=500,000,000*(0.2)^t

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Complete the following sentence about the rate of change in the number of harmful bacteria remaining in the body in seconds.

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Round your answer to two decimal places.

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Every second, the number of harmful bacteria remaining in the body decays by a factor of

\square

Frank catches a cold. The next day he goes to school and

3

of his classmates catch a cold. Each day thereafter, the number of people catching a cold triples. Let

C

be the number of people who catch a cold on day

t

. Which of the following best explains the relationship between

t

C

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1

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(A) The relationship is exponential because

C

6

t

t=1

t=2

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(B) The relationship is linear because

C

increases by a factor of

3

t

1

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(C) The relationship is exponential because

C

increases by a factor of

3

t

1

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(D) The relationship is linear because

C

2

t

t=0

t=1

Fang and Meng have a large rose garden. When the roses first bloomed, they initially picked

4

fresh roses. Each day thereafter, as more roses bloomed, they quadrupled the number of roses they picked. Let

R

be the number of roses picked on day

t

. Which of the following best explains the relationship between

t

R

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1

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(A) The relationship is exponential because the values of

R

always increase as the values of

t

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(B) The relationship is linear because the values of

R

always increase as the values of

t

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(C) The relationship is linear because the values of

R

are always even integers.

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(D) The relationship is exponential because

R

increases by a factor of

4

t

1

Prachi conducted an experiment to determine the relationship between the number of hours,

t

, spent studying the night before the test and the number of questions,

q

, answered correctly on the test. She used the given equation to model her results.

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q=-3(t-2)^{2}+45

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What is the best interpretation of the number

2

in Prachi's model?

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1

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(A) In this experiment, a student who did not study would answer

2

questions correctly.

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(B) In this experiment, a student answered

2

more questions correctly for every hour of study.

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(C) In this experiment, a student who studied for

2

hours would answer the most questions correctly.

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(D) In this experiment, the first

2

hours of studying did not influence the number of questions the student answered correctly.

In the month of June, the temperature in Johannesburg, South Africa, varies over the day in a periodic way that can be modeled approximately by a trigonometric function.

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The lowest temperature is usually around

3^{\circ} \mathrm{C}

, and the highest temperature is around

18^{\circ} \mathrm{C}

. The temperature is typically halfway between the daily high and daily low at both

10

10

p.m., and the highest temperatures are in the afternoon.

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Find the formula of the trigonometric function that models the temperature

T

in Johannesburg

t

hours after midnight. Define the function using radians.

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T(t)=\square

Eric owns and operates the Hot Ham food truck. The expression

3.25 b+2 h

gives the cost, in dollars, of

b

h

hot dogs. What is the cost, in dollars, of

4

6

\tan A = \frac{3}{4}

\sin A \cos A = \frac{12}{25}

We can compare the intensities of earthquakes using the Richter scale

M=\log\frac{I}{S}

21/11/2022

, West Java recorded an earthquake with magnitude

M=5.6

. This was the deadliest earthquake to hit Indonesia since the

28/9/2018

Sulawesi earthquake. The Sulawesi earthquake had a magnitude of

M=7.5

. How many times more intense was the first earthquake than the second earthquake?

xy = ax + by

y=mx+c

Y = \frac{x}{y}

X = x

The list below shows the temperature Lillian recorded at

11

00

p.m. for four days last January.

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11^{\circ} \mathrm{F}

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-30^{\circ} \mathrm{F}

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-19^{\circ} \mathrm{F}

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-4^{\circ} \mathrm{F}

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On which of the days did Lillian record the lowest (coldest) temperature?

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The list below shows the temperature Casho recorded at

11

00

p.m. for four days last January.

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-28^{\circ} \mathrm{F}

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5^{\circ} \mathrm{F}

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-1^{\circ} \mathrm{F}

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10^{\circ} \mathrm{F}

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On which of the days did Casho record the lowest (coldest) temperature?

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The list below shows the temperature Jaya recorded at

11

00

p.m. for four days last January.

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-22^{\circ} \mathrm{F}

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-19^{\circ} \mathrm{F}

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3^{\circ} \mathrm{F}

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-12^{\circ} \mathrm{F}

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On which of the days did Jaya record the highest (warmest) temperature?

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The list below shows the temperature Elizabeth recorded at

11

00

p.m. for four days last January.

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-14^{\circ} \mathrm{F}

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-3^{\circ} \mathrm{F}

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-26^{\circ} \mathrm{F}

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-7^{\circ} \mathrm{F}

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On which of the days did Elizabeth record the highest (warmest) temperature?

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Without dividing, determine if

59

736

is divisible by

2

and explain how you know.

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59,736 \square

2

Without dividing, determine if

22

128

is divisible by

3

and explain how you know.

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22,128 \square

3

The given equation shows the number of possible distinct passwords,

p

L

, where each character is selected from

n

permitted characters.

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p=n^{L}

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How does the number,

p

, of possible distinct passwords change if the length is increased by

3

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1

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p

is multiplied by

n^{3}

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p

is multiplied by

3 n

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p

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p

is multiplied by

3

The number of years since Keith graduated from middle school can be represented by the equation

y=a-14

y

is the number of years and

a

is his age. Is the relationship between the number of years since Keith graduated and his age proportional or nonproportional?

Wei is standing in wavy water and notices the depth of the waves varies in a periodic way that can be modeled by a trigonometric function. He starts a stopwatch to time the waves. After

1.1

seconds, and then again every

3

seconds, the water just touches his knees. Between peaks, the water recedes to his ankles. Wei's ankles are

12\,\text{cm}

off the ocean floor, and his knees are

55\,\text{cm}

off the ocean floor. Find the formula of the trigonometric function that models the depth

D

t

seconds after Wei starts the stopwatch. Define the function using radians.

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D(t)=\square

g(x)=-3 x+5

x

g(x)=-1

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f(x)=5 x-3

x

f(x)=-8

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g(x)=2 x+1

x

g(x)=-9

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h(x)=2 x-1

x

h(x)=5

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f(x)=-x-3

x

f(x)=0

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g(x)=-x+3

x

g(x)=1

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f(x)=-3 x-4

x

f(x)=8

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h(x)=-x-4

x

h(x)=-8

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h(x)=-x-3

x

h(x)=7

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f(x)=-5 x-3

x

f(x)=-3

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In an arithmetic sequence, the first term,

a_{1}

9

, and the third term,

a_{3}

25

. Which number represents the common difference of the arithmetic sequence?

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d=5

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d=6

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d=7

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d=8

In an arithmetic sequence, the first term,

a_{1}

7

, and the fifth term,

a_{5}

31

. Which number represents the common difference of the arithmetic sequence?

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d=3

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d=4

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d=5

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d=6

Determine the value of

y

x

1

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y=\frac{40}{x+3}

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A pendulum is swinging next to a wall. The distance from the bob of the swinging pendulum to the wall varies in a periodic way that can be modeled by a trigonometric function.

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The function has period

0.8

seconds, amplitude

6\,\text{cm}

H=15\,\text{cm}

t=0.5

seconds, the bob is at its midline, moving towards the wall.

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Find the formula of the trigonometric function that models the distance

H

from the pendulum's bob to the wall after

t

seconds. Define the function using radians.

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H(t)=\square

Pluto's distance from the sun varies in a periodic way that can be modeled approximately by a trigonometric function.

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Pluto's maximum distance from the sun (aphelion) is

7.4

billion kilometers. Its minimum distance from the sun (perihelion) is

4.4

billion kilometers. Pluto last reached its perihelion in the year

1989

, and will next reach its perihelion in

2237

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Find the formula of the trigonometric function that models Pluto's distance

D

from the sun (in billion

\text{km}

t

2000

. Define the function using radians.

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D(t)=\square

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How far will Pluto be from the sun in

2022

? Round your answer, if necessary, to two decimal places.

\newline

\square

\text{km}

y=3x+2

Find the value of

y

x=4

Genesis's math teacher finds that there's roughly a linear relationship between the amount of time students spend on their homework and their weekly quiz scores. This relationship can be represented by the equation

y=6.3 x+66

y

represents the expected quiz score and

x

represents hours spent on homework that week. What is the meaning of the

x

y=90

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A student's expected quiz score if they spent

1

hour on their homework.

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A student's expected quiz score if they spent

90

hours on their homework.

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The change in expected quiz score for every additional one hour students spend on their homework.

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The number of hours a student should spend on their homework to expect a score of

90

There's a roughly linear relationship between the number of times a species of cricket will chirp in one minute and the temperature outside. For a certain type of cricket, this relationship can be expressed using the formula

T=0.3 c+37

T

represents the temperature in degrees Fahrenheit and

c

represents the number of times the cricket chirps in one minute. What is the meaning of the

T

c=87

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The number of times the cricket could be expected to chirp in one minute if it's

87^{\circ} \mathrm{F}

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The change in temperature in degrees Fahrenheit for each additional cricket chirp in one minute.

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The expected temperature in degrees Fahrenheit if the cricket has chirped o times per minute.

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The expected temperature in degrees Fahrenheit if the cricket has chirped

87

times per minute.

Khloe is going to the amusement park, where she has to pay a set price of admission and another price for tickets to go on each of the rides. The total amount of money Khloe will spend is given by the equation

y=22+1.5 x

y

represents the total amount of money, in dollars and cents, and

x

represents the number of rides Khloe goes on. What could the number

1

5

represent in the equation?

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The change in the total amount of money she will spend for every additional ride she goes on.

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The total amount of money Khloe will spend if she goes on

1

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The total amount of money Khloe will spend if she goes on o rides.

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The total amount of money Khloe will spend if she goes on

100

\mathrm{x}

in simplest form.

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5=\frac{1}{3}(8 x-12)

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Which of the following are true statements about the expression

-7+7

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2

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(A) The expression describes the number that is

7

-7

on the number line.

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(B) The expression describes the number that is

7

to the right of

-7

on the number line.

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(C) The expression equals

-14

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(D) The expression equals

0

Which of the following are true statements about the expression

-8+8

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2

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(A) The expression describes the number that is

8

-8

on the number line.

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(B) The expression describes the number that is

8

to the right of

-8

on the number line.

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(C) The expression equals

-16

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(D) The expression equals

0