Jessica plots the data points relating the amount of money she needs to repay a loan and the number of months she has been making payments.

A 2-column table with 5 rows. The first column is labeled months with entries 6, 12, 18, 24, 30. The second column is labeled amount to repay (dollar sign) with entries 2,700; 2,110; 1,110; 870; 220.
A graph shows months labeled 5 to 60 on the horizontal axis and amount to repay (dollar sign) on the vertical axis. A line shows a downward trend.

She calculates two regression models. Which is true?

The linear model better represents the situation because the amount she owes is decreasing by about the same amount every 6 months.
The linear model better represents the situation because according to the exponential model, the repayment amount will never be 0.
The exponential model better represents the situation because the amount she owes decreases by about the same amount every 6 months.
The exponential model better represents the situation because according to the linear model, the repayment amount will eventually be negative.

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse's length equals the sum of the squares of the other two sides' lengths. This can be proven via the areas of squares with sides corresponding to the triangle's sides, demonstrating that a²+b²=c².

The Pythagorean theorem is a fundamental principle in geometry, named after the Greek mathematician Pythagoras. It states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be written as: a²+b²=c².

To prove this, we can begin with a right-angled triangle with sides a, b, and hypotenuse c. If we square the length of c (i.e., c²), this is equivalent to the area of a square with side length c. Similarly, a² and b² represent the areas of squares with side lengths a and b, respectively.

If we add together the areas of the two smaller squares (a²+b²), this equals the area of the larger square (c²). This proves the Pythagorean theorem. For instance, consider a right triangle with sides of lengths 3, 4, and 5. The squares would have areas 9 and 16, which add up to 25 - the area of the square on the hypotenuse.

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The Pythagorean Theorem states that a² + b² = c². A step-by-step proof involves constructing a square with the right-angled triangle and equating the areas. This logical deduction confirms the theorem's correctness.

The Pythagorean Theorem is one of the fundamental results in Geometry, which states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b): a² + b² = c². Here is a step-by-step proof:

Thus, we have proven that the theorem holds true using logical deductions.

Answer:

both are done

look at picture

1. 13/36

2. 3/16

Answer:

Step-by-step explanation:

Let x represent the amount that she needs to save each month for the next year.

Stephanie is saving money to buy a new computer. So far she has saved $200. This means that the total amount that she would have saved in y months is

200 + xy

Since there are 12 months in a year,

an inequality to show how much she needs to save each month for the next year so she has at least $1200 to spend on the computer is

200 + 12x ≥ 1200

12x ≥ 1200 - 200

12x ≥ 1000

x ≥ 1000/12

x ≥ 83.33

Answer:

7.41 m/s

Step-by-step explanation:

Force of a car inclined at an angle

F=mg Sin Θ where; m = 2956 kg , g = 9.8m/s² , Θ = 35⁰

F = 2956 x 9.8 x Sin 35 = 16615.82 N

Net force due to friction = 16615.82 - 3080 = 13535.82 N

To calculate acceleration; F= Ma

13535.82 = 2956 x a

a = 4.58 m/s²

From equation of motion

[tex]V^{2} = V_{o} ^{2} + 2aS[/tex]

V² =  0 + (2 x 4.58 x 6)

V = [tex]\sqrt{54.95}[/tex]

V = 7.41 m/s

Answer:

f'(x) = [tex]-\frac{9}{x^2}[/tex]

Step-by-step explanation:

i) f(x) = 9 / x

ii) f'(x)  = [tex]$\lim_{h\to 0} \frac{f(x+h) - f(x)}{h} $ \hspace{0.2cm}[/tex]

        [tex]= $\lim_{h\to 0} \frac{\frac{9}{x+h} - \frac{9}{x} }{h} = \hspace{0.1cm} $\lim_{h\to 0} \frac{9x - 9(x+h)}{hx(x+h)} $ \hspace{0.1cm} = \hspace{0.1cm}$\lim_{h\to 0} \frac{-9h}{hx(x+h)} = $\lim_{h\to 0} \frac{-h}{x(x+h)} = \frac{-9}{x^2}$[/tex]

[tex]\bf (\stackrel{x_1}{-1}~,~\stackrel{y_1}{-7})\qquad (\stackrel{x_2}{-9}~,~\stackrel{y_2}{-2}) ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{-2}-\stackrel{y1}{(-7)}}}{\underset{run} {\underset{x_2}{-9}-\underset{x_1}{(-1)}}}\implies \cfrac{-2+7}{-9+1}\implies -\cfrac{5}{8}[/tex]

Answer:

The answer to your question is m = [tex]\frac{5}{8}[/tex]

Step-by-step explanation:

Data

A (-1, -7)

B (-9, -2)

Formula

slope = m = [tex]\frac{y2 - y1}{x2 - x1}[/tex]

Use the slope formula to find the answer. Just substitute the values and simplify them.

Substitution

x1 = -1

x2 = -9

y1 = -7

y2 = -2

                 m = [tex]\frac{-2 + 7}{-9 + 1}[/tex]

Simplification

                 m =- [tex]\frac{5}{8}[/tex]

Result

                  m =- [tex]\frac{5}{8}[/tex]

Answer:

Oh cool! The answer is 90 since a and c or parallel, b cuts through them perpendicularly, forming a right angle.

Step-by-step explanation:

Answer:

Yes to both.

Step-by-step explanation:

It is true that

[tex]\dfrac{x}{y}=1 \iff x=y[/tex]

In fact, if you know that x/y=1, just multiply both sides by y to get x=y. On the other hand, if you know that x=y (and they are not zero), you can divide both sides by y to get x/y=1.

Since the two expressions are equivalent, you can always use Phil's or Don's expression, at will.

Answer:

Question 1: Are the footprints of the two sheds similar?

Question 2: Tell whether the footprint of the least expensive shed is an enlargement or a reduction,

Question 3: Find the scale factor from the most expensive shed to the least expensive shed

Explanation:

Question 1: Are the footprints of the two sheds similar?

The footprints of the two sheds will be similar if their measures are proportional.

The ratio of the measures of the footprint of the most expensive shed is:

The ratio of the measures of the footprint of the least expensive shed is:

Since, the two ratios are equal, you conclude that the corresponding dimensions are proportional and the two sheds are similar.

Question 2: Tell whether the footprint of the least expensive shed is an enlargement or a reduction.

A reduction is a similar transformation (the image and the preimage are similar)  that maps the original figure into a smaller one.

Since the dimensions of the foot print of the least expensive shed, 10 feet wide by 14 feet long, are smaller than the dimensions of the most expensive shed, 15 feet wide by 21 feet long the you conclude that the former is a reduction of the latter.

Question 3: Find the scale factor from the most expensive shed to the least expensive shed.

To find the scale factor from the most expensive shed to the least expensive shed, you divide the measures of the corresponding dimensions. You can do it either with the widths or with the lengths.

Using the widths, you get:

That means that the scale factor from the most expensive shed to the least expensive shed is 3/2.

Using the lenghts, you should obtain the same scale factor:

Answer:i dont knowStep-by-step explanation:

Answer:

152x miles

Step-by-step explanation:

Jose runs x miles per day each week for 5 days.

For 3 days of the week he runs at a speed of 8 minutes per mile. So we have

3*8*x = 24x

For 2 days he runs 7 minutes per miles. We have

2*7*x = 14x

in one week, he runs 24x + 14x = 38x

In 4 weeks, he runs

4*38x = 152x miles

Answer:

Acceleration (a) [tex]=5.29\ m\./s^2[/tex]

Step-by-step explanation:

Given that

final velocity (v) [tex]=45\ m/s[/tex]

Initial velocity (u) [tex]=0\ m/s[/tex]

Time (t) [tex]=8.5\ seconds[/tex]

Then

[tex]v=u+at\ ...................................... (1)[/tex]

Where [tex]a[/tex] is acceleration.

put the value in equation [tex](1)[/tex]

[tex]45=0+a\times8.5\\a\times8.5=45[/tex]

dividing [tex]8.5[/tex] both sides

[tex]\frac{a\times8.5}{8.5}=\frac{45}{8.5}\\\\a=\frac{45}{8.5}\\\\a=\frac{450}{85}\\\\a=5.29\ m/s^2[/tex]

Answer:

the selling price of the house is estimated as $137,000

Step-by-step explanation:

Given data:

clear amount is $128,000

closing cost is $780

sales commission rate is 6%

let S be the selling price of house

from the information given in the question we have  following relation

[tex]128,000 \leq S - 0.06S - 780[/tex]

[tex]128,000 \leq 0.94S - 780[/tex]

adding 780 on both side, we get

[tex]128,000 - 780 \leq 0.94S[/tex]

divide the both side by 0.94, we get

[tex]137,000 \leq S[/tex]

therefor the selling price of the house is estimated as $137,000

Answer:

The solution of the inequality required for the situation is d  ≥ $13.46.

Step-by-step explanation:

i) the minimum deposit is $10.

ii) the balance before the statement is $-3.46

iii) a deposit d is made so that the fee did not have to be made

iv) therefore d + (-3.46)  ≥  10

                    d - 3.46 ≥ 10

    therefore d  ≥ 10 + 3.46

 therefore d  ≥ $13.46

Chris would have to deposit at least $13.46 into her account in order to avoid being charged a fee. This amount is determined by setting up and solving the inequality -3.46 + d ≥ 10, which represents her account balance.

In this situation, Chris is trying to avoid a fee by maintaining a balance of at least $10 in her account. Prior to making a deposit, her account balance is -$3.46. Thus, we need to find out the minimal deposit, d, that she needs to make in order to bring her balance up to $10. To do this, we can set up the following inequality:

-3.46 + d ≥ 10

To solve the inequality for d, we simply add 3.46 to both sides of the inequality:

d ≥ 13.46

Therefore, Chris must make a deposit of at least $13.46 to avoid incurring the fee.

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Answer:

  $5.25

Step-by-step explanation:

The two purchases can be described by the equations ...

  3s +2p = 29.75

  4s +3p = 42.00

If you multiply the first equation by 3 and subtract 2 times the second equation, the cost of a shirt pops out.

  3(3s +2p) -2(4s +3p) = 3(29.75) -2(42.00)

  9s +6p -8s -6p = 89.25 -84.00 . . . . . . . eliminate parentheses

  s = 5.25 . . . . . . . . . . collect terms

One shirt costs $5.25.

Answer:

200-50-40=B

B =110 blue pieces

Step-by-step explanation:

Step-by-step explanation:

0

Answer:

hot dogs sold=110

sodas sold=36

Step-by-step explanation:

146-36=110

110+36=146

Answer:91 Sodas and 55 hot dogs were sold.

Step-by-step explanation:

Let x represent the number of Sodas that were sold at the basketball game.

Let y represent the number of hot dogs that were sold at the basketball game.

At the basketball game a vendor sold a combined total of 146 sodas and hotdogs. This means that

x + y = 146 - - - - - - - - - - - - - - -1

The number of Sodas was 36 more than the number of hotdogs sold. This means that

x = y + 36

Substituting x = y + 36 into equation 1, it becomes

y + 36 + y = 146

2y + 36 = 146

2y = 146 - 36 = 110

y = 110/2 = 55

x = y + 36 = 55 + 36

x = 91

Answer:

Step-by-step explanation:

Triangle GHI is a right angle triangle.

From the given right angle triangle,

GH represents the hypotenuse of the right angle triangle.

With m∠H as the reference angle,

HI represents the adjacent side of the right angle triangle.

GI represents the opposite side of the right angle triangle.

To determine m∠H, we would apply

the Sine trigonometric ratio.

Sin θ = adjacent side/hypotenuse. Therefore,

Sin m∠H = 7/10 = 0.7

m∠H = Sin^-1(0.7)

m∠H = 44.4° to the nearest tenth.

Answer: gayness

Step-by-step explanation:

Answer:

B. [-1, ∞).

Step-by-step explanation:

g(x) = 3|x − 1| − 1

When x = 1  g(x) = 3(0) - 1 = -1.

As all negative vales of x will give positive values of  |x - 1| then  g(x) = -1 is its minimum value. The graph will be shaped like a letter V with the vertex at (1,-1).

Therefore the range is  [-1, ∞).

The range of g(x)=3|x-1|-1 is [-1,∞) i.e. option B is correct.

The set of all the outputs of a function is known as the range of the function .

According to the given question

we have,

A function, g(x) = 3|x-1| - 1

Lets, find the value of g(x) for the different values of "x"

when,

x=0 ⇒ g(0) = -1

x=1 ⇒ g(1) = -1

x=2 ⇒g(2) = 2

Similarly, we can check for the negative values of x.

So, for all the negative values of x we will gives only positive values for g(x) and  only at x=0, g(x) = -1 , which is its minimum value .

⇒ The range of given function g(x) is {-1,∞).

Hence , option B is correct.

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Answer:

There were 6 people against the bill.

Step-by-step explanation:

Given:

In a recent poll,only 8% of the people surveyed were against a new bill making it mandatory to recycle.

Now, to find of the 75 people surveyed were against the bill.

Total number of people = 75.

Percent of the people surveyed were against the bill = 8%.

Now, to get the number of people who were against the bill:

8% of 75

[tex]=\frac{8}{100}\times 75[/tex]

[tex]=0.08\times 75[/tex]

[tex]=6.[/tex]

Therefore, there were 6 people against the bill.

Answer:

It takes 60 hours for Alex to complete the job alone.

Step-by-step explanation:

Given:

Number of hours Brandon alone can finish job =20 hours

Let the number of hours required by Alex alone to complete the job be 'x' hrs.

We need to find the number of hours required by Alex alone to complete the job.

Solution:

Now we can say that;

Rate at which Alex can complete the job alone = [tex]\frac1x[/tex] job/hour

Rate at which Brandon can complete the job alone = [tex]\frac{1}{20}[/tex] job /hour

Also Given:

Number of hours required for both to complete the job = 15

So rate of both complete the job  = [tex]\frac{1}{15}[/tex] job/hour

Now we can say that;

rate of both complete the job is equal to sum of Rate at which Alex can complete the job alone and Rate at which Brandon can complete the job alone.

framing in equation form we get;

[tex]\frac1x+\frac{1}{20}=\frac{1}{15}[/tex]

Now taking LCM for making the denominators same we get;

[tex]\frac{20}{20x}+\frac{x}{20x}=\frac{1}{15}[/tex]

Now denominators are common so we will solve the numerators.

[tex]\frac{20+x}{20x}=\frac{1}{15}[/tex]

Now Using cross multiplication we get;

[tex]15(20+x)=20x[/tex]

Applying Distributive property we get;

[tex]15\times20+15\times x =20x\\\\300+15x=20x[/tex]

Combining like terms we get;

[tex]300=20x-15x\\\\300=5x[/tex]

Now Dividing both side by 5 we get;

[tex]\frac{300}{5}=\frac{5x}{5}\\\\x=60\ hrs[/tex]

Hence It takes 60 hours for Alex to complete the job alone.

To find how long it will take Alex to complete the job by himself, we first establish the individual work rates of Alex and Brandon. We know the combined work rate (1/15 jobs per hour) and Brandon's work rate (1/20 jobs per hour). Alex's work rate is thus 1/15 - 1/20 = 1/60 jobs per hour, which means it will take him 60 hours to do the job alone.

This question requires knowledge of work rate calculations in Mathematics. The scenario involves two workers, Alex and Brandon, who can complete a cleaning task together in 15 hours. It's also given that Brandon can do the job alone in 20 hours, and we need to calculate how long it would take for Alex to complete the job by himself.

First, let's define their rates of work. If Brandon can complete the job in 20 hours, his work rate is 1 job per 20 hours, or 1/20 jobs per hour. Similarly, if Alex and Brandon together can complete the job in 15 hours, their combined work rate is 1 job per 15 hours, or 1/15 jobs per hour.

To find Alex's work rate, we subtract Brandon's work rate from the combined work rate. Thus, Alex's work rate is 1/15 - 1/20 = 1/60 jobs per hour. This means it would take Alex 60 hours to complete the job on his own.

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Answer:

The coordinates of the point that partitions the directed line segment PT in a 3:2 ratio will be (5, 11).

Step-by-step explanation:

Given the points

What are the coordinates of the point that partitions the directed line segment PT in a 3:2 ratio?

Let D be the Divided point of the directed line segment PT.

SECTION FORMULA

The point (x, y) which partitions the line segment of the points (x₁, y₁) and

(x₂, y₂) in a ratio [tex]m:n[/tex] will be:

                                            [tex]\left(\frac{m\:x_2+n\:x_1}{m+n},\:\frac{m\:y_2+n\:y_1}{m+n}\right)[/tex]

Here,

Substituting the values in the above formula

[tex]D\left(x\right)=\left(\frac{3\times \:7\:+\:2\times 2}{3+2},\:\frac{3\times 17\:+\:2\times 2}{3+2}\right)[/tex]

[tex]D\left(x\right)=\left(\frac{21\:+\:4}{5},\:\frac{51\:+\:4}{5}\right)[/tex]

[tex]D\left(x\right)=\left(\frac{25}{5},\:\frac{55}{5}\right)[/tex]

As

[tex]\frac{25}{5}=5,\:\frac{55}{5}=11[/tex]

So

[tex]$D(x)=(5, 11)[/tex]

Therefore, the coordinates of the point that partitions the directed line segment PT in a 3:2 ratio will be (5, 11).

Answer:

880 in³

Step-by-step explanation:

L = 20, W = 10, H = 8

2(h × W) + 2(h × L) + 2(W × L)

= 2(8*10) + 2(8*20) + 2(10*20)

= 2(80) + 2(160) + 2(200)

= 160 + 320 + 400

= 880 in³

(A) Probability of the black card is 1 /2.
(B) Probability of a red card is 26/51.
(C) Probability of selecting a black card and then selecting a red card is 13/51.

Probability can be defined as the ratio of favorable outcomes to the total number of events.

Here,
Total number of cards in the deck = 52
Total number of black cards = 26
Total number of red cards = 26
Probability of selecting a black card = 26 / 52 = 1 / 2
Now, the total number of card become after selecting 1 black card = 51
Probability of selecting a red card = 26 / 51
Now,
The probability of selecting a black card and then selecting a red card is,
= 1 / 2 * 26 / 51
= 13 / 51

Thus, (A) the Probability of a black card is 1 /2.
(B) Probability of a red card is 26/51.
(C) Probability of selecting a black card and then selecting a red card is 13/51.

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The probability of selecting a black card first and then a red card is [tex]\(\frac{13}{51}\).[/tex]

To find the probability of selecting a black card and then selecting a red card from a standard deck of 52 playing cards without replacement, we need to follow these steps:

1. Calculate the probability of selecting a black card first:

  - There are 26 black cards in a standard deck (13 spades and 13 clubs).

  - The probability of selecting a black card first is:

  [tex]\[ P(\text{Black card first}) = \frac{26}{52} = \frac{1}{2} \][/tex]

2. Calculate the probability of selecting a red card second, given that the first card was black:

  - After drawing the first black card, there are 51 cards left in the deck.

  - There are still 26 red cards in the deck (since the black card was not replaced).

  - The probability of selecting a red card given that the first card was black is:

   [tex]\[ P(\text{Red card second} \mid \text{Black card first}) = \frac{26}{51} \][/tex]

3. Calculate the combined probability of both events occurring:

  - The probability of both events happening (selecting a black card first and then a red card) is the product of the individual probabilities:

    [tex]\[ P(\text{Black card first and Red card second}) = P(\text{Black card first}) \times P(\text{Red card second} \mid \text{Black card first}) \][/tex]

    Substituting the values we found:

    [tex]\[ P(\text{Black card first and Red card second}) = \left(\frac{1}{2}\right) \times \left(\frac{26}{51}\right) = \frac{26}{102} = \frac{13}{51} \][/tex]

Therefore, the probability of selecting a black card first and then a red card is [tex]\(\frac{13}{51}\).[/tex]

Answer:

circle b

Step-by-step explanation:

Answer:

Step-by-step explanation:

amosc: marcos62280 ( ;

Answer: B) 4.963±0.019.

Step-by-step explanation:

Confidence interval for population mean ( when population standard deviation is not given) is given by :-

[tex]\overline{x}\pm t^*\dfrac{s}{\sqrt{n}}[/tex]

, where [tex]\overline{x}[/tex] = Sample  mean

n= Sample size

s= sample standard deviation

t* = critical t-value.

As per given:

n= 50

Degree of freedom = n-1 =49

[tex]\overline{x}= 4.963\ lb[/tex]

s= 0.067 lb

For df = 49 and significance level of 0.05 , the critical two-tailed t-value ( from t-distribution table) is 2.010.

Now , substitute all values in the formula , we get

[tex]4.963\pm (2.010)\dfrac{0.067}{\sqrt{50}}\\\\ 4.963\pm (2.010)(0.0094752)\\\\ 4.963\pm0.019045152\approx4.963\pm0.019[/tex]

Hence,  a 95% confidence interval for the mean weight (in pounds) of the mulch produced by this company is [tex]4.963\pm0.019[/tex].

Thus , the correct answer is B) 4.963±0.019.

Answer:

the correct answer is B) 4.963±0.019.

Step-by-step explanation:

Answer:

Work done is 1.02J

Step-by-step explanation:

Work (W) done in stretching a spring = 1/2Fe

F (force) = 6 lb = 6×4.4482N = 26.6892N

e (extension) = 11in - 8in = 3in = 3×0.0254m = 0.0762m

W = 1/2 × 26.6892 × 0.0762 = 1.02J

Answer: the weight of each bags are 1.85kg, 1.85kg and 3.05 kg

Step-by-step explanation:

Total weight of the three bags is

6 3/4 kg. Converting to decimal, it becomes 6.75 kg.

Two of them have the same weight. Let x represent the bags of equal weight. Their total weight would be 2x. The third bag is heavier than each of the bags of equal weight by 1 1/5 kg. Converting to decimal, it becomes 1.2kg. It means that the weight of the third bag would be

x + 1.2

Therefore, for the three bags,

2x + x + 1.2 = 6.75

3x = 6.75 - 1.2 = 5.55

x = 5.55/3 = 1.85

Weight of the third bag is

1.85 + 1.2 = 3.05kg

To solve for the weights of the bags, we set up an equation and find that the weight of each of the lighter bags is 1 17/20 kg, and the weight of the heavier bag is 3 1/20 kg.

The question asks us to calculate the weight of each bag of sweets when given the total weight and the additional weight of the heavier bag. Let's denote the weight of the two equal bags as 'x' kilograms each. According to the problem, the third bag weighs 'x + 1 1/5' kg, which is heavier by 1 1/5 kg than each of the other two bags. The total weight of the three bags is 6 3/4 kg.

Starting with this information, we can set up an equation to find the value of 'x': 2x + (x + 1 1/5 kg) = 6 3/4 kg. To solve for 'x', first simplify the right side of the equation by converting the mixed fraction to an improper fraction: 6 3/4 kg = 27/4 kg. Now, convert 1 1/5 kg to an improper fraction as well: 1 1/5 kg = 6/5 kg. The equation now is: 2x + x + 6/5 = 27/4 kg.

Combining like terms, we have: 3x = 27/4 kg - 6/5 kg. To combine these fractions, we need a common denominator, which would be 20 in this case. This changes the equation to: 3x = (27/4) * (5/5) - (6/5) * (4/4) = 135/20 kg - 24/20 kg = 111/20 kg. To find 'x', divide both sides by 3: x = (111/20 kg) / 3 = (111/20) * (1/3) kg = 37/20 kg = 1 17/20 kg. This is the weight of each of the lighter bags.

Subsequently, the weight of the heavier bag can be found by adding the additional weight: x + 6/5 kg = (37/20 kg) + (6/5 kg) = (37/20) + (24/20) = 61/20 kg = 3 1/20 kg.

Therefore, the weight of the two bags with equal weight is 1 17/20 kg each, and the weight of the heavier bag is 3 1/20 kg.

Answer:

Option c

Step-by-step explanation:

given that limit x tending to 4 of the function (3x-4) is 8

This implies for all values of x such that for epsilon >0 arbitrary small ,

[tex]||x-4||<\epsilon[/tex], we get

|f(x)-8|<3epsilon

this is equivalent to the option c.

Proof:

Consider

[tex]||x-4||<\epsilon\\3||x-4||<3\epsilon\\||3x-12||<3\epsilon\\||3x-4|-8| <3\epsilon[/tex]

Hence it follows that option C is right.

The correct statement to use when proving that limx→4(3x−4)=8 is option b: Given 0 < | x - 4 | < epsilon/3, then | (3x - 4) - 8 | < epsilon. This uses the formal definition of limit and abides the epsilon-delta parameters definition to provide the correct proof.

To verify the given limit, we need to utilize the formal definition of a limit. This formal definition gives us a systematic method to prove a limit based on epsilon-delta parameters. The purpose is to show that as x gets closer and closer to 4 (with| x - 4 | being smaller than an arbitrary positive delta), the expression (3x-4) increasingly approaches 8 (with |(3x - 4) - 8| getting within an epsilon range).

The correct choice is: b. Given 0 < | x - 4 | < epsilon/3, then | (3x - 4) - 8 | < epsilon. In this case, 'epsilon/3' in | x - 4 | < 'epsilon/3' is the delta in the epsilon-delta definition of limit. Essentially, it captures the notion that for every epsilon > 0, there exists a delta > 0 such that 0 < | x - 4 | < delta implies | (3x - 4) - 8 | < epsilon, thereby proving the limit.

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