plssssssssssssssssssssss help me i will mark you the brainlyest Graph.





y= –34x–1

Answers

Answer 1

Answer:

the x-axis is 0,1

the y -axis is -1, - 35

sorry if  im rong

Step-by-step explanation:

Answer 2

Answer:

look at explanation

Step-by-step explanation:

the first thing to do when graphing is to look at your y coordinate, the last number without a variable behind it. this would be (-1).

mark -1 on your graph, which should be one slot to the left of zero.

the slope (the number with the variable behind it) can be read as rise/run. turn -34 into a fraction by simply putting it over 1. this will give you -34/1.

your rise in this case would be -34, which means from -1, go down 34 slots and put a nother dot. your run(denominator in fraction) is where you move horizontally. since the run is 1, move over 1 spot from where your last dot is and that is your final plot. connect the first dot you made tot he third dot you made and you have officially graphed it.


Related Questions

Please help!! Given the recursive formula shown, what are the first 4 terms of the sequence?

Answers

Answer:

C

Step-by-step explanation:

the fibrin definition tells us that the first term of the sequence a1 = 6

so, B is out.

and for every following term we always add 7 to every previous term to create the next one.

so, when I add 7 to 6, what is my second term in the sequence ? 7+6 = 13

not 42, not -7

therefore, only C is correct.

and 13+7 = 20

and 20+7 = 27

it all fits

Describe domain and range of the graph.

Answers

Domain: the set of possible values of the independent variable or variables of a function.
Range of graph: The range is the set of possible output values, which are shown on the y-axis. Keep in mind that if the graph continues beyond the portion of the graph we can see, the domain and range may be greater than the visible values.

A line passes through (2, −1) and (4, 5).
Which answer is the equation of the line?
A. −3x + 5y = 13
B. −3x + y = −7
C. −3x + y = 17
D. −3x + 5y = −13
Which answer is an equation in point-slope form for the given point and slope?
Point: (1, 9); Slope: 5
A. y − 1 = 5 (x + 9)
B. y − 9 = 5 (x − 1)
C. y + 9 = 5 (x−1)
D. y − 9 = 5 (x+1)

Answers

Answer:

−3x + y = −7       y - 9 = 5 (x - 1)

Step-by-step explanation:

y2 - y1 / x2 - x1

5 - (-1) / 4 - 2

6/2

= 3

slope intercept: −3x + y = −7

y - 9 = 5 (x - 1)

please help me with geometry

Answers

Answer:

A. If the side lengths are the same, then a triangle is not scalene.

Step-by-step explanation:

A triangle can be defined as a two-dimensional shape that comprises three (3) sides, three (3) vertices and three (3) angles.

Simply stated, any polygon with three (3) lengths of sides is a triangle.

In Geometry, a triangle is considered to be the most important shape.

Generally, there are three (3) main types of triangle based on the length of their sides and these include;

I. Equilateral triangle: it has all of its three (3) sides and interior angles equal.

II. Isosceles triangle: it has two (2) of its sides equal in length and two (2) equal angles.

III. Scalene triangle: it has all of its three (3) sides and interior angles different in length and size respectively.

the mean salary if of 5 employees is $35900. the median is $37000. the mode is $382000. If the median payed employee gets a $3100 raise, then…
New median:
New mode:

Answers

Answer:

Step-by-step explanation:

New median:40100

New mode:385100

If a cube has an edge of length e, then the lateral surface area is:

Answers

Answer:

The total lateral surface of this cube is 4*e^2

Step-by-step explanation:

A cube is a figure with all the sides of the same length, so each face of a cube is a square.

Remember that the area of a square of sidelength L is:

A = L^2

Now, when we want to find the lateral surface of a figure, we ignore the bases of the figure.

So, if a cube has 6 faces, if we ignore the two bases, we are left with 4 square faces.

And if the edge length is e, then each one of these four faces has an area:

A = e^2

So the total lateral surface is 4 times that:

S = 4*e^2

The total lateral surface of this cube is 4*e^2

Answer:

4e2

Step-by-step explanation:

I got it correct on founders edtell


What is the solution of the equation x2 -14x + 67 = 0?
A. X= 7+3in12
B.X = 7+31V-2
C.X= 7+ 2/8
D.X = 7+ 217

Answers

[tex]\sf\huge\underline\color{pink}{༄Answer:}[/tex]

What is the solution of the equation x^2 - 14x + 67 = 0?

[tex]\tt{First \: option. \: x = 7 \underline{+}3i\sqrt{2}}[/tex]

[tex]\color{pink}{==========================}[/tex]

#CarryOnLearning

[tex]\sf\color{pink}{༄⁂✰Bae \: Yoonah}[/tex]

A hospital director is told that 54% of the emergency room visitors are insured. The director wants to test the claim that the percentage of insured patients is under the expected percentage. A sample of 120 patients found that 60 were insured. Find the value of the test statistic. Round your answer to two decimal places.

Answers

Answer:

Z -0.879173965

Step-by-step explanation:

Z -0.879173965

ρ  0.5

π 0.54

n 120

The value of the test statistic is the z-score z = -0.88

What is a z-score?

The relationship between a value and the mean of a set of values is expressed numerically by a Z-score. The Z-score is computed using the standard deviations from the mean. A Z-score of zero indicates that the data point's score and the mean score are identical.

The Z-score is calculated using the formula:

z = (x - μ)/σ

where z: standard score

x: observed value

μ: mean of the sample

σ: standard deviation of the sample

Given data ,

Let the test statistic value be represented as z

Now , the probability of emergency room visitors are insured is q = 0.54

The total number of patients n = 120

The number of patients that were insured = 60

So , the percentage of people that were insured p = 60/120 = 0.5

Now , test statistic value z = ( p - q ) / [ √ ( q ( 1 - q )/n² ]

The value of z score is

z = [ 0.5 - 0.54 ] / √ 0.54 ( 1 - 0.54 ) / 120²

On simplifying the equation , we get

The value of z score is z = -0.88

Hence , the test statistic is z = -0.88

To learn more about z score click :

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Following are the published weights (in pounds) of all of the team members of Football Team A from a previous year.
177; 204; 211; 211; 232; 205; 185; 185; 178; 210; 206; 212; 184; 174;
185; 242; 188; 212; 215; 247; 241; 223; 220; 260; 245; 259; 278; 270;
280; 295; 275; 285; 290; 272; 273; 280; 285; 286; 200; 215; 185; 230;
250; 241; 190; 260; 250; 302; 265; 290; 276; 228; 265
Organize the data from smallest to largest value.

Part (a)
Find the median.
Part (b)
Find the first quartile. (Round your answer to one decimal place.)
Part (c)
Find the third quartile. (Round your answer to one decimal place.)
Part (d)
Construct a box plot of the data.

Answers

Answer:

im not sure but i do have a tip when you go to goo gle i want you to simplify your question (not what you asked on here) then copy your question as it is simple then search it on here because its slightly  for others to understand

Step-by-step explanation:

.

The graph below shows two polynomial functions, f(x) and g(x): Graph of f of x equals x squared minus 2 x plus 1. Graph of g of x equals x cubed plus 1 Which of the following statements is true about the graph above? (4 points) g(x) is an even degree polynomial with a positive leading coefficient. g(x) is an odd degree polynomial with a negative leading coefficient. f(x) is an even degree polynomial with a positive leading coefficient. f(x) is an odd degree polynomial with a negative leading coefficient.

Answers

9514 1404 393

Answer:

  (c)  f(x) is an even degree polynomial with a positive leading coefficient.

Step-by-step explanation:

The leading terms of the two functions are ...

  f(x): x² (even degree, positive coefficient: 1)

  g(x): x³ (odd degree, positive coefficient: 1)

Then it is true that ...

  f(x) is an even degree polynomial with a positive leading coefficient

In an experiment, the initial temperature of a solution is -5 °C. The solution is heated up at 3 °C per minute for 19 minutes and then it is cooled at 4 °C per minute for 6 minutes. Calculate the final temperature, in °C, of the solution.​

Answers

Answer:

28°C

Step-by-step explanation:

First you do 3*19=57°C

-5+57= 52°C

then you do 4*6=24 °C

as its being cooled you takeaway

52-24=28°C

Cual es el capital que prestado al 10% bimestral durante 6 meses y 10 días produce un interés de 1140

Answers

Respuesta:

3650

Explicación paso a paso:

Dado que :

Principal, P = capital prestado

Tasa anual, r = 10% * 6 = 60%

Interés = 1140

Periodo = 6 meses y 10 días = (6 * 30) +10 = 190 días

Conversión a años:

Periodo = 190/365

Usando la relación:

Interés = principal * tasa * tiempo

1140 = P * 60% * (190/365)

1140 = 0.3123287P

P = 1140 / 0,3123287

P = 3650

use de moivre's theorem to write
[2(cos 12 degrees + i sin 12 degrees)]^5
in standard form

Answers

DeMoivre's theorem says

(2 (cos(12°) + i sin(12°)))⁵ = 2⁵ (cos(5×12°) + i sin(5×12°))

… = 32 (cos(60°) + i sin(60°))

… = 32 (1/2 + √3/2 i )

… = 16 + 16√3 i

A person earns ​$23,600 one year and gets a​ 5% raise in salary. What is the new​ salary?

Answers

Answer:

24780

Step-by-step explanation:

might not be right but 24780 because 23600*5% is 1180 and add that to the original 23600 to get 24780

5% more is 105% in total, including the the 100% we start with, so:

23,600 * 1.05 = 24780

A lawyer commutes daily from his suburban home to his midtown office. The average time for a one-way trip is 24 minutes, with a standard deviation of 3.8 minutes. Assume the distribution of trip times to be normally distributed.
(a) What is the probability that a trip will take at least ½ hour?
(b) If the office opens at 9:00 A.M. and he leaves his house at 8:45 A.M. daily, what percentage of the time is he late for work?
(c) If he leaves the house at 8:35 A.M. and coffee is served at the office from 8:50 A.M. until 9:00 A.M., what is the probability that he misses coffee?
​(d) Find the length of time above which we find the slowest 10​% of trips.
(e) Find the probability that 2 of the next 3 trips will take at least one half
1/2 hour.

Answers

Answer:

Step-by-step explanation:

a) Probability-Above 30 min = 5.72% = .0572

b) Probability-Above 15 min =  99.11% = .9911

c) *Probability-Between  1 - 59.49% = .4051

d) 19.136 minutes  z = -1.28

a) The probability that trip will take at least 1/2 hour will be 0.0606.

b) The percentage of time the lawyer is late for work will be 99.18%.

c) The probability that lawyer misses coffee will be 0.3659.

d) The length of time above which we find the slowest 10​% of trips will be 0.5438.

e) The probability that exactly 2 out of 3 trips will take at least one half

1/2 hour is 0.0103.

What do you mean by normal distribution ?

A probability distribution known as a "normal distribution" shows that data are more likely to occur when they are close to the mean than when they are far from the mean.

Let assume the time taken for a one way trip be x .

x ⇒ N( μ , σ ²)

x  ⇒ N( 24 , 3.8 ²)

a)

The probability that trip will take at least 1/2 hour or 30 minutes will be :

P ( x ≥ 30)

= P [ (x - μ) / σ ≥ (30 - μ) / σ ]

We know that , (x - μ) / σ = z.

= P [ z ≥ (30 - 24) / 3.8)]

= P [ z ≥ 1.578 ]

= 1 - P [  z ≤ 1.578 ]

Now , using the standard normal table :

P ( x ≥ 30)

= 1 - 0.9394

= 0.0606

b)

The percentage of the time the lawyer is late for work will be :

P ( x  ≥ 15)

= P [ z ≥ -2.368 ]

= P [  z ≤ 2.368]

= 0.9918

or

99.18%

c)

The probability that lawyer misses coffee :

P ( 15 < x < 25 ) = P ( x < 25 ) - P ( x < 15)

= P [  z < 0.263] - P ( z < -2.368)

or

= 0.3659

d)

The length of time above which we find the slowest 10​% of trips :

P( x ≥ X ) ≤ 0.10

=  0.5438

e)

Let's assume that y represents the number of trips that takes at least half hour.

y ⇒ B ( n , p)

y ⇒ B ( 3 , 0.0606)

So , the probability that exactly 2 out of 3 trips will take at least one half

1/2 hour is :

P ( Y = 2 )

= 3C2 × (0.0606)² × ( 1 - 0.0606)

= 0.0103

Therefore , the answers are :

a) The probability that trip will take at least 1/2 hour will be 0.0606.

b) The percentage of time the lawyer is late for work will be 99.18%.

c) The probability that lawyer misses coffee will be 0.3659.

d) The length of time above which we find the slowest 10​% of trips will be 0.5438.

e) The probability that exactly 2 out of 3 trips will take at least one half

1/2 hour is 0.0103.

Learn more about normal distribution here :

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Find the length of the segment indicated. Round your answer to the nearest tenth if necessary.

Answers

Answer:

x=5

Step-by-step explanation:

im so bad at explaining things but i hope this helped

An exterior angle of a regular polygon cannot have the measure of
Select one:

a. 120

b. 40

c. 50

d. 90

e. 30


Answers

Option c 50 cannot be the measure of exterior angle. therefore only the angle which can evenly divides 360 can be the measure of exterior angle.
Yea it most definitely letter c

d(1)=2
d(n)=d(n−1)⋅(−2)^n

​What is the third term in the sequence?

Answers

Answer:

- 16

Step-by-step explanation:

The way this is given, you have to find the second term before you can fine the third.

a2 = d(n - 1) * (-2)^2

a2 = 2 * (-2)^2

a2 = 2 * 4

a2 = 8

a3 = d(3 - 1) * (-2)^3

a3 = 2 * (-2)^3

a3 = 2 * - 8

a3 = - 16

Give the domain and range of G={(6.0),(-9,-3),(1,-3)}

Answers

Answer:

Step-by-step explanation:

D={ 6  , -9  , 1 }

R={  0  ,-3 }

For an avid bird watcher, the probability of spotting a California Condor while birdwatching in the Grand Canyon area is 0.3. The probability of being able to take a clear picture of the bird suppose one is able to spot it is 0.8. What is the probability that the bird watcher is able to take a clear picture of a California Condor

Answers

Answer:

the probability of taking a clear picture of a California candor is .24

The number formed by adding 1 to the greatest 8 – digit number is?

Answers

Answer:

the answer would be 100,000,000

Greatest 8 Digit Number: 99,999,999

Adding 1 = 99999999+1

= 100,000,000

Must click thanks and mark brainliest

Please help with Question 2b

Answers

Answer:

MUST BE IN HLA, NOT FROM C TO ASSEMBLY.

PROGRAM 6: Same

Write an HLA Assembly language program that implements a function which correctly identifies when all four parameters are the same and returns a boolean value in AL (1 when all four values are equal; 0 otherwise). This function should have the following signature:

procedure theSam

Tìm thể tích của khối bao bởi mặt z=5+(x−4)^2+2y và mặt x=3,y=4 và mặt phẳng tọa độ.

Answers

Step-by-step explanation:

ccxiddidificifificici i i ivi i i i i i i i iivvii iix9difi


I need help completing this problem ASAP

Answers

Answer:

It is E. radicand

Step-by-step explanation:

Answer:

Radicand

Step-by-step explanation:

Radicand is the number found inside the radicals

Example:

[tex]\sqrt[4]{5} ; 2\sqrt[4]{5}[/tex]  are like terms

5 -> radicand and 4 is the index

Please help!! Can’t figure this out for the life of me.

Select the correct answer from each drop-down menu.

If _______, then AABC and ADEF are congruent by the ASA criterion.


If _______, then AABC and ADEF are congruent by the SAS criterion.


AABC and ADEF are congruent if ______


Answers

Answer:

Angle b is congruent to angle E

CA=FD

Step-by-step explanation:

If _______, then triangle ABC and triangle DEF are congruent by the ASA criterion.  ASA is angle side angle .  We know angle C= angle F and side CB = side FE We need to know angle B = angle E

If _______, then triangle ABC and triangle DEF are congruent by the SAS criterion.  SAS is side angle side,  we know side CB = side FE  and then angle C= angle F then we need side CA = side FD

If ∠ABC = ∠DEF, then ΔABC and ΔDEF are congruent by the ASA criterion.

If AC = DF, then ΔABC and ΔDEF are congruent by the SAS criterion.

What are congruent figures?

Two figures are said to be congruent of they have the same shape and all the corresponding sides and angles are congruent.

The HL (hypotenuse leg) congruence theorem states that if the hypotenuse and one leg of a triangle is congruent to another triangle, then both triangles are congruent.

In triangle ABC and DEF;

BC = EF and ∠ACB ≅ ∠DFE

Hence:

If ∠ABC = ∠DEF, then ΔABC and ΔDEF are congruent by the ASA criterion.

If AC = DF, then ΔABC and ΔDEF are congruent by the SAS criterion.

Find out more on congruent figures at: https://brainly.com/question/1675117

Hoang spends $10 on movie tickets, $50 on rent, and $3 on snacks. How much money did Hoang spend on variable expenses?

Answers

Answer:

63

Step-by-step explanation:

50+10=60 60+3=63 dollars

he spends 63$ on variable expenses

A 500-kg concrete block 95 cm longcm wide and cm high. It can exert three different pressures on a horizontal surface depending on which faceit rests on the highest pressure is

Answers

Pressure is force per unit area. If the block is sitting at rest on the surface, it applies a pressure of mg/A, where mg is the block's weight and A is the area of the face that makes contact with the surface.

The smaller the value of A, the larger the pressure. So the highest pressure the block can exert is achieved when it's resting on the face with the smallest dimensions.

If the block is a cube with side length 95 cm, then the pressure exerted by each face is the same.

give an example of a piecewise function

Answers

Answer:

f(x) = 6 when -5 < x ≤ -1

Step-by-step explanation:

It is estimated that 75% of all young adults between the ages of 18-35 do not have a landline in their homes and only use a cell phone at home.
(a) On average, how many young adults do not own a landline in a random sample of 100?
(b) What is the standard deviation of probability of young adults who do not own a landline in a simple random sample of 100?
(c) What is the proportion of young adults who do not own a landline?
(d) What is the probability that no one in a simple random sample of 100 young adults owns a landline?
(e) What is the probability that everyone in a simple random sample of 100 young adults owns a landline?
(f) What is the distribution of the number of young adults in a sample of 100 who do not own a landline?
(g) What is the probability that exactly half the young adults in a simple random sample of 100 do not own a landline?

Answers

Answer:

a) 75

b) 4.33

c) 0.75

d) [tex]3.2 \times 10^{-13}[/tex] probability that no one in a simple random sample of 100 young adults owns a landline

e) [tex]6.2 \times 10^{-61}[/tex] probability that everyone in a simple random sample of 100 young adults owns a landline.

f) Binomial, with [tex]n = 100, p = 0.75[/tex]

g) [tex]4.5 \times 10^{-8}[/tex] probability that exactly half the young adults in a simple random sample of 100 do not own a landline.

Step-by-step explanation:

For each young adult, there are only two possible outcomes. Either they do not own a landline, or they do. The probability of an young adult not having a landline is independent of any other adult, which means that the binomial probability distribution is used to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

And p is the probability of X happening.

The expected value of the binomial distribution is:

[tex]E(X) = np[/tex]

The standard deviation of the binomial distribution is:

[tex]\sqrt{V(X)} = \sqrt{np(1-p)}[/tex]

75% of all young adults between the ages of 18-35 do not have a landline in their homes and only use a cell phone at home.

This means that [tex]p = 0.75[/tex]

(a) On average, how many young adults do not own a landline in a random sample of 100?

Sample of 100, so [tex]n = 100[/tex]

[tex]E(X) = np = 100(0.75) = 75[/tex]

(b) What is the standard deviation of probability of young adults who do not own a landline in a simple random sample of 100?

[tex]\sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{100(0.75)(0.25)} = 4.33[/tex]

(c) What is the proportion of young adults who do not own a landline?

The estimation, of 75% = 0.75.

(d) What is the probability that no one in a simple random sample of 100 young adults owns a landline?

This is P(X = 100), that is, all do not own. So

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]P(X = 100) = C_{100,100}.(0.75)^{100}.(0.25)^{0} = 3.2 \times 10^{-13}[/tex]

[tex]3.2 \times 10^{-13}[/tex] probability that no one in a simple random sample of 100 young adults owns a landline.

(e) What is the probability that everyone in a simple random sample of 100 young adults owns a landline?

This is P(X = 0), that is, all own. So

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]P(X = 0) = C_{100,0}.(0.75)^{0}.(0.25)^{100} = 6.2 \times 10^{-61}[/tex]

[tex]6.2 \times 10^{-61}[/tex] probability that everyone in a simple random sample of 100 young adults owns a landline.

(f) What is the distribution of the number of young adults in a sample of 100 who do not own a landline?

Binomial, with [tex]n = 100, p = 0.75[/tex]

(g) What is the probability that exactly half the young adults in a simple random sample of 100 do not own a landline?

This is P(X = 50). So

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]P(X = 50) = C_{100,50}.(0.75)^{50}.(0.25)^{50} = 4.5 \times 10^{-8}[/tex]

[tex]4.5 \times 10^{-8}[/tex] probability that exactly half the young adults in a simple random sample of 100 do not own a landline.

help with 4b thank you. ​

Answers

First let's compute dx/dt

[tex]x = t - \frac{1}{t}\\\\x = t - t^{-1}\\\\\frac{dx}{dt} = \frac{d}{dt}\left(t - t^{-1}\right)\\\\\frac{dx}{dt} = 1-(-1)t^{-2}\\\\\frac{dx}{dt} = 1+\frac{1}{t^{2}}\\\\\frac{dx}{dt} = \frac{t^2}{t^{2}}+\frac{1}{t^{2}}\\\\\frac{dx}{dt} = \frac{t^2+1}{t^{2}}\\\\[/tex]

Now compute dy/dt

[tex]y = 2t + \frac{1}{t}\\\\y = 2t + t^{-1}\\\\\frac{dy}{dt} = \frac{d}{dt}\left(2t + t^{-1}\right)\\\\\frac{dy}{dt} = 2 - t^{-2}\\\\\frac{dy}{dt} = 2 - \frac{1}{t^2}\\\\\frac{dy}{dt} = \frac{2t^2}{t^2}-\frac{1}{t^2}\\\\\frac{dy}{dt} = \frac{2t^2-1}{t^2}\\\\[/tex]

From here, apply the chain rule to say

[tex]\frac{dy}{dx} = \frac{dy*dt}{dx*dt}\\\\\frac{dy}{dx} = \frac{dy}{dt} \times \frac{dt}{dx}\\\\\frac{dy}{dx} = \frac{dy}{dt} \div \frac{dx}{dt}\\\\\frac{dy}{dx} = \frac{2t^2-1}{t^2} \div \frac{t^2+1}{t^{2}}\\\\\frac{dy}{dx} = \frac{2t^2-1}{t^2} \times \frac{t^{2}}{t^2+1}\\\\\frac{dy}{dx} = \frac{2t^2-1}{t^2+1}\\\\[/tex]

We could use polynomial long division, or we could add 2 and subtract 2 from the numerator and do a bit of algebra like so

[tex]\frac{dy}{dx} = \frac{2t^2-1}{t^2+1}\\\\\frac{dy}{dx} = \frac{2t^2-1+2-2}{t^2+1}\\\\\frac{dy}{dx} = \frac{(2t^2+2)-1-2}{t^2+1}\\\\\frac{dy}{dx} = \frac{2(t^2+1)-3}{t^2+1}\\\\\frac{dy}{dx} = \frac{2(t^2+1)}{t^2+1}-\frac{3}{t^2+1}\\\\\frac{dy}{dx} = 2-\frac{3}{t^2+1}\\\\[/tex]

This concludes the first part of 4b

=======================================================

Now onto the second part.

Since t is nonzero, this means either t > 0 or t < 0.

If t > 0, then,

[tex]t > 0\\\\t^2 > 0\\\\t^2+1 > 1\\\\\frac{1}{t^2+1} < 1 \ \text{ ... inequality sign flip}\\\\\frac{3}{t^2+1} < 3\\\\-\frac{3}{t^2+1} > -3 \ \text{ ... inequality sign flip}\\\\-\frac{3}{t^2+1}+2 > -3 + 2\\\\2-\frac{3}{t^2+1} > -1\\\\-1 < 2-\frac{3}{t^2+1}\\\\-1 < \frac{dy}{dx}\\\\[/tex]

note the inequality signs flipping when we apply the reciprocal to both sides, and when we multiply both sides by a negative value.

You should find that the same conclusion happens when we consider t < 0. Why? Because t < 0 becomes t^2 > 0 after we square both sides. The steps are the same as shown above.

So both t > 0 and t < 0 lead to [tex]-1 < \frac{dy}{dx}[/tex]

We can say that -1 is the lower bound of dy/dx. It never reaches -1 itself because t = 0 is not allowed.

We could say that

[tex]\displaystyle \lim_{t\to0}\left(2-\frac{3}{t^2+1}\right)=-1\\\\[/tex]

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To establish the upper bound, we consider what happens when t approaches either infinity.

If t approaches positive infinity, then,

[tex]\displaystyle L = \lim_{t\to\infty}\left(2-\frac{3}{t^2+1}\right)\\\\\\\displaystyle L = \lim_{t\to\infty}\left(\frac{2t^2-1}{t^2+1}\right)\\\\\\\displaystyle L = \lim_{t\to\infty}\left(\frac{2-\frac{1}{t^2}}{1+\frac{1}{t^2}}\right)\\\\\\\displaystyle L = \frac{2-0}{1+0}\\\\\\\displaystyle L = 2\\\\[/tex]

As t approaches infinity, the dy/dx value approaches L = 2 from below.

The same applies when t approaches negative infinity.

So we see that [tex]\frac{dy}{dx} < 2[/tex]

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Since [tex]-1 < \frac{dy}{dx} \text{ and } \frac{dy}{dx} < 2[/tex], those two inequalities combine into the compound inequality [tex]-1 < \frac{dy}{dx} < 2[/tex]

So dy/dx is bounded between -1 and 2, exclusive of either endpoint.

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