PLZ HELP! A train averages a speed of 90 miles per hour across the plains and 37.5 miles per hour through the mountains. If a trip of 300 miles took 3 hour and 48 minutes, how much of it was through the mountains?

Answers

Answer 1

The portion of the trip through the mountains was 112.5 miles.

To find this, first, let's denote the time spent in the plains as [tex]\( t_1 \)[/tex] and the time spent in the mountains as [tex]\( t_2 \)[/tex]. Since the total trip took 3 hours and 48 minutes (which is[tex]\( \frac{3}{4} \) hours[/tex]), we have:

[tex]\[ t_1 + t_2 = 3.75 \][/tex]

The distance traveled in the plains is [tex]\( d_1 = 90t_1 \)[/tex] and in the mountains is[tex]\( d_2 = 37.5t_2 \)[/tex]. The total distance traveled is 300 miles, so:

[tex]\[ d_1 + d_2 = 300 \][/tex]

Substituting the expressions for [tex]\( d_1 \)[/tex] and [tex]\( d_2 \):[/tex]

[tex]\[ 90t_1 + 37.5t_2 = 300 \][/tex]

We can rearrange the first equation to express [tex]\( t_1 \)[/tex] in terms of [tex]\( t_2 \)[/tex]:

[tex]\[ t_1 = 3.75 - t_2 \][/tex]

Substituting this into the second equation:

[tex]\[ 90(3.75 - t_2) + 37.5t_2 = 300 \][/tex]

[tex]\[ 337.5 - 90t_2 + 37.5t_2 = 300 \][/tex]

[tex]\[ 337.5 - 52.5t_2 = 300 \][/tex]

[tex]\[ 52.5t_2 = 37.5 \][/tex]

[tex]\[ t_2 = \frac{37.5}{52.5} \][/tex]

[tex]\[ t_2 = \frac{5}{7} \] hours[/tex]

Now, to find the distance traveled in the mountains:

[tex]\[ d_2 = 37.5 \times \frac{5}{7} \][/tex]

[tex]\[ d_2 = 112.5 \] miles[/tex]

So, the portion of the trip through the mountains was 112.5 miles.

Complete question

A train averages a speed of 90 miles per hour across the plains and 37.5 miles per hour through the mountains. If a trip of 300 miles took 3 hour and 48 minutes, how much of it was through the mountains?


Related Questions

there are some roses,lilies, and orchids in the vase. The number of roses is twice the number of lilies and the number of orchids is 5 more than the number of roses. if the total is 45, find the number of each type of flower

Answers

Answer:

The number of roses is 16

The number of lilies is 8

The number of orchids is 21

Step-by-step explanation:

Let

x-----> the number of roses

y-----> the number of lilies

z-----> the number of orchids

we know that

x=2y ----> y=x/2 ----> equation A

z=x+5 ---> equation B

x+y+z=45 ----> equation C

substitute equation A and equation B in equation C and solve for x

x+(x/2)+(x+5)=45

(5/2)x=45-5

(5/2)x=40

x=40*2/5

x=16 roses

Find the value of y

y=x/2 ----> y=16/2=8 lilies

Find the value of z

z=x+5 -----> z=16+5=21 orchids

therefore

The number of roses is 16

The number of lilies is 8

The number of orchids is 21

Which of the following give the correct graph and phase shift for y = 3 cos ( θ +20) – 4?​

Answers

Answer:

See picture attached below

Step-by-step explanation:

We can easily find the answer to your question, if we plot the equation with a graphing calculator or any plotting tool.

The equation is

y = 3 cos (θ + 20) - 4

See attached picture below

phase_shift = 20

max_amplitude = -1

min_amplitude = -7

Period = 2π

Raquel has created a program for a robot. The robot travels 10 meters and stops. The program is designed for wheels that are 2.5 cm in diameter. When Raquel replaces the wheels with ones having a 2.75 cm diameter, how should she adjust the number of rotations so that the robot travels the same distance?

Answers

Answer:

Raquel should adjust the number of rotations  to 115.81 rotations

Step-by-step explanation:

step 1

Find the circumference of the wheels that are 2.5 cm in diameter

The circumference is equal to

[tex]C=2\pi r[/tex]

we have

[tex]r=2.5/2=1.25\ cm[/tex]

assume

[tex]\pi =3.14[/tex]

substitute

[tex]C=2(3.14) (1.25)[/tex]

[tex]C=7.85\ cm[/tex]

step 2

Find the number of rotations

Divide 10 meters by the circumference

[tex]10\ m=1,000\ cm[/tex]

[tex]1,000/7.85=127.39\ rotations[/tex]

step 3

For a diameter of 2.75 cm find how should she adjust the number of rotations so that the robot travels the same distance

Find the circumference of the wheels

The circumference is equal to

[tex]C=2\pi r[/tex]

we have

[tex]r=2.75/2=1.375\ cm[/tex]

assume

[tex]\pi =3.14[/tex]

substitute

[tex]C=2(3.14) (1.375)[/tex]

[tex]C=8.635\ cm[/tex]

step 4

Find the number of rotations

Divide 10 meters by the circumference

[tex]10\ m=1,000\ cm[/tex]

[tex]1,000/8.635=115.81\ rotations[/tex]

Probability of getting HH

Answers

Answer:

The experimental probability is 3% greater than the theoretical probability.

Step-by-step explanation:

We are given the results of flipping two coins with their outcomes and number of times they were tossed.

We are to compare the experimental and theoretical probability of getting HH.

The theoretical Outcomes are: HH HT TH TT

So theoretical probability of getting HH = [tex]\frac{1}{4} \times 100[/tex] = 25%

Total number of outcomes = [tex]28+22+34+16[/tex] = 100

So experimental probability of getting HH = [tex]\frac{28}{100} \times 100[/tex] = 28%

Therefore, the experimental probability is 3% greater than the theoretical probability.

The surface area of a sphere is 16 square units. The radius of the sphere measures

Answers

The answer to this question is 2.

What value is equivalent to 8 · 9 − 2 · 5?

Answers

Answer:

6.4

Step-by-step explanation:

8.9 - 2.5 = 6.4

8 - 2 = 6

9 - 5 = 4

there you have it your answer 6.4

The center of a sphere is
a line segment from the center point to the surface of the sphere.
a fixed point equidistant from all points on the surface of the sphere.
a three-dimensional circle in which all points are equidistant from a fixed point.
the same as the base of the sphere.

Answers

Answer:

A fixed point equidistant from all points on the surface of the sphere

Step-by-step explanation:

we know that

The sphere is the set of all points in the space equidistant from a fixed point called the center of the sphere

therefore

The center of a sphere is a fixed point equidistant from all points on the surface of the sphere

In Geometry, the center of a sphere is: B. a fixed point equidistant from all points on the surface of the sphere.

What is a sphere?

A sphere can be defined as a round, three-dimensional solid geometric figure that has all its surface points (every point on its surface) at equal distances (equidistant) from the center.

In this context, we can infer and logically conclude that the center of a sphere simply refers to a fixed point that is equidistant or at equal distances from all points on the surface of the sphere.

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Write an inequality to describe the relationship between -1 1/2 and -1/4

Answers

Answer: -1 1/2 < -1/4

Explanation: Since the number is negative, -1 1/2 is further to the left on the number line, meaning it has less value.

Final answer:

The relationship between -1 1/2 and -1/4 can be expressed as an inequality. In this case, -1 1/2 is less than -1/4, so the inequality is -1 1/2 < -1/4.

Explanation:

The relationship between -1 1/2 and -1/4 may be written as an inequality. Remember, an inequality indicates that one number is larger or smaller than another. In this case, it shows which negative number, either -1 1/2 or -1/4, is larger. On the number line, a negative number situated to the right is larger than a number situated to the left. So, in terms of value, -1/4 is larger than -1 1/2 as it's less negative. This can be written as:

-1 1/2 < -1/4

.

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Write the equation of a line perpendicular to the given line and passing through the given point. y=3x+3(-1,-1) step by step

Answers

ANSWER

[tex]y = - \frac{1}{3} x -\frac{4}{3} [/tex]

EXPLANATION

The given line is

[tex]y = 3x + 3[/tex]

The given point is

[tex](-1,-1)[/tex]

The slope of the given line is

[tex]m = 3[/tex]

We found this by comparing

[tex]y = 3x + 3[/tex]

to

[tex]y = mx + b[/tex]

If two lines are perpendicular, then one is the negative reciprocal of the other.

Hence the slope of the required line is

[tex] - \frac{1}{3} [/tex]

Using the point-slope formula or otherwise, we can find the required equation.

[tex]y-y_1=m(x-x_1)[/tex]

We substitute the slope and point to get:

[tex]y + 1 = - \frac{1}{3} (x + 1)[/tex]

[tex]y = - \frac{1}{3} x -\frac{4}{3} [/tex]

WILL GIVE BRAINLIEST!!!


Solve.

5(b + 6) = 18

Answers

Answer:

b = -12/5

Step-by-step explanation:

5(b+6) = 18

5b + 30 = 18

5b = -12

b = -12/5

Answer:

b = -2.4

Step-by-step explanation:

5 ( b + 6 ) = 18

→ Expand brackets

5 b + 30 = 18

→ - 30 from both sides to isolate 5 b

5 b = - 12

→ ÷ Divide both sides 5 to isolate b

b = -2.4

Which angle in the translated trapezoid is congruent to angle S?

A.
angle Q apostrophe

B.
angle T apostrophe

C.
angle R apostrophe

D.
angle S apostrophe



Answers

Answer:

Option D. angle S apostrophe

Step-by-step explanation:

we know that

The transformation of the figure is a translation

The rule of the translation is

(x,y)-----> (x-3,y-7)

That means ----> left 3 units and down 7 units

Remember that a translation does not modify the internal angles of the figure as neither the length of their sides

so

∠S=∠S'

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Identify the roots of the quadratic function.
A) x = 0 and x = 4



B) y = 0 and y = 4

C) x = 0 and x = -4

D) y = 0 and y = -4

Answers

Answer:

Step-by-step explanation:

The roots of a function are the x-intercepts. By definition, the y-coordinate of points lying on the x-axis is zero. Therefore, to find the roots of a quadratic function, we set f (x) = 0, and solve the equation, ax2 + bx + c = 0.

Factor the expression

81-36xy

Answers

Answer: 9(9-4xy)

Step-by-step explanation:

You can factor out the 9, as 9*9 = 81 and 9*4 = 36.

So dividing both terms, you get 9(9-4xy)

If p(x) = x2 – 1 and q(x) = 5(x-1), which expression is equivalent to (p – q)(x)?

A. 5(x – 1) – x2 – 1
B. (5x – 1) – (x2 – 1)
C. (x2 – 1) – 5(x – 1)
D. (x2 – 1) – 5x – 1

please help!!!

Answers

Answer:

[tex]\large\boxed{C.\ (x^2-1)-5(x-1)}[/tex]

Step-by-step explanation:

[tex](f-g)(x)=f(x)-g(x)\\\\p(x)=x^2-1,\ q(x)=5(x-1)\\\\(p-q)(x)=(x^2-1)-5(x-1)[/tex]



Find the value of x.

Answers

Answer: [tex]x=76\°[/tex]

Step-by-step explanation:

By definition the angle formed by the intersection of two chords inside a circle is:

[tex]Angle\ formed\ by\ two\ Chords=\frac{sum\ of\ intercepted\ Arcs}{2}[/tex]

You can observe in the figure that  the angle formed by the intersection of the two chords measures 94°, and the intercepteed arcs are: x° and 112°.

Therefore, you need to substitute these values into the formula and solve for "x to find its value. Then:

[tex]94\°=\frac{x+112\°}{2}\\\\(94\°)(2)=x+112\°\\\\188\°-112\°=x\\\\x=76\°[/tex]

What is the definition of present value?A. the current value of a future sum of moneyB. the interest paid on a current sum of moneyC. the future value of a current sum of moneyD. the interest paid on a future sum of money

Answers

Answer:

The definition of present value is the current value of a future sum of money.

Choice A

Step-by-step explanation:

Present value (PV) is the current value of a future streams of cash flows or sum of money at a given expected rate of return by the investor. Future payment streams are discounted at the rate of return. The present value increases with the decrease in the rate of return or the discount rate and vice versa.

Option A is correct,  the definition of present value is the current value of a future sum of money.

Present value refers to the concept of determining the value of a future sum of money in terms of its current worth.

It takes into account factors such as the time value of money and discounting to calculate the value of future cash flows in today's terms.

By discounting future cash flows, the present value represents the amount of money that would need to be invested or received today to achieve the same value as the future sum of money.

Hence, the definition of present value of the current value of a future sum of money.

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Ashton surveyed some of the employees at his company about their cell phone habits. From the data, he concluded that most employees at his company use cell phones primarily for business. For which sample could this generalization be valid?

Answers

Answer:

It can't be A. since if you only look at managers, you are missing all the sales executives.

It may be C. this option is more random but doesn't guarantee that you will represent both groups of employee's. Also, each time you would conduct the survey, you will receive the exact same results since it is the same people. 

It isn't D. for the exact same reason as A. but you're missing managers now. 

Therefore the answer is B. Some managers and some sales executives selected at random. This way you get a sample from both categories, and within those groups, it is randomly selected.

 I hope this helps!

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Step-by-step explanation:

The generalization could be valid for a sample that accurately represents the entire employee population of Ashton's company.

This sample should be large enough to be statistically significant and should be selected randomly to avoid bias.

To calculate the sample size needed for a valid generalization, Ashton could use a confidence level and margin of error. Let's say he wants a 95% confidence level with a margin of error of 5%.

First, he needs to find the population size (total number of employees at the company). Let's assume there are 500 employees.

Next, he can use the formula for sample size calculation:

[tex]\[n = \frac{{Z^2 \cdot p \cdot (1-p)}}{{E^2}}\][/tex]

Where:

- (n) = sample size

-(Z) = Z-score corresponding to the desired confidence level (for 95% confidence level, Z = 1.96)

- (p) = estimated proportion of employees using cell phones primarily for business (from the survey data)

- (E) = margin of error (0.05 for 5%)

Let's say from the survey, Ashton found that 70% of employees use cell phones primarily for business.

Plugging in the values:

[tex]\[n = \frac{{1.96^2 \cdot 0.70 \cdot (1-0.70)}}{{0.05^2}}\][/tex]

[tex]\[n = \frac{{3.8416 \cdot 0.70 \cdot 0.30}}{{0.0025}}\][/tex]

[tex]\[n = \frac{{0.719856}}{{0.0025}}\][/tex]

[tex]\[n ≈ 287.94\][/tex]

So, Ashton would need a sample size of approximately 288 employees to make a valid generalization about the entire company.

To ensure the generalization is valid, Ashton needs to collect data from a sample that accurately represents the entire employee population. This sample should be large enough to be statistically significant and should be selected randomly to avoid bias.

Using statistical methods, Ashton can calculate the minimum sample size needed for a valid generalization. By setting a confidence level and margin of error, he can determine the sample size required to achieve a certain level of accuracy.

In this case, Ashton chose a 95% confidence level with a margin of error of 5%. He used a formula that takes into account the population size, estimated proportion of employees using cell phones primarily for business, and the margin of error.

After plugging in the values, he calculated that he would need a sample size of approximately 288 employees to make a valid generalization about the entire company.

So, for the conclusion that most employees at his company use cell phones primarily for business to be valid, Ashton should survey at least 288 randomly selected employees.

Complete question:

Ashton surveyed some of the employees at his company about their cell phone habits. From the data, he concluded that most employees at his company use cell phones primarily for business. For which sample could this generalization be valid?

ABC is reflected across the x-axis and then translated 4 units up to create A’B’C’. What are the coordinates of the vertices of A’B’C’?

Worth 25 points

Answers

The first option is the correct choice

The ratio of counselors to campers at a camp is 1 : 9. The ratio of campers who can swim to campers who cannot swim is 7 : 2. There are 13 counselors. How many campers can swim?

Answers

Answer:

91 campers can swim

Step-by-step explanation:

step 1

Find the number of campers

we know that

The ratio of counselors to campers at a camp is 1 : 9

so

by proportion

Find the number of campers if there are 13 counselors

Let

x-----> the number of campers

1/9=13/x

x=9*13=117 campers

step 2

How many campers can swim?

we know that

The ratio of campers who can swim to campers who cannot swim is 7 : 2

so

The ratio of total campers  to campers who can swim is 9 : 7

by proportion

Find how many campers can swim for a total of 117 campers

Let

x----> the number of campers that can swim

9/7=117/x

x=117*7/9

x=91 campers can swim

Does anyone know this?

Answers

Answer:

There are two inputs for which the output is 5.

The vertex of its graph is at (0,-2)

Step-by-step explanation:

ANSWER

2nd

4th

5th

EXPLANATION

The given function is

[tex]y = |x| - 2[/tex]

The value of this function cannot be negative: False.

y = |1| - 2=-1

Its graph has a V-shape. This is true because it is the graph of y=|x| shifted downwards by 2 units.

There is only one input for which this function is zero. False

both -2 and 2 will make this function zero.

There are two inputs for which this function is 5. True, x=7 and x=-7

The vertex is (0,-2). This is true because it is the graph of y=|x| shifted downwards by 2 units.

If it is 250 miles from New York to Boston and 120 miles from New York to Hartford, what percentage of the distance from New York to Boston is the distance from New York to Hartford?

Answers

Final answer:

The distance from New York to Hartford represents 48% of the distance from New York to Boston.

Explanation:

To find the percentage of the distance from New York to Boston that is the distance from New York to Hartford, we need to calculate the ratio of the two distances. First, divide the distance from New York to Hartford (120 miles) by the distance from New York to Boston (250 miles):

 

120 / 250 = 0.48

 

Convert the ratio to a percentage by multiplying by 100:

 

0.48 * 100 = 48%

 

Therefore, the distance from New York to Hartford represents 48% of the distance from New York to Boston.

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What is the equation for the line of reflection

Answers

Answer:

B

Step-by-step explanation:

The image is a reflection of the original in the y- axis

The equation of the y- axis is x = 0 → B

Answer:

The correct option is B.

Step-by-step explanation:

The coordinates of polygon ABCDE are A(-2,2), B(-1,2), C(1,1), D(0,-1) and E(-2,1).

The coordinates of polygon A'B'C'D'E' are A'(2,2), B'(1,2), C'(-1,1), D'(0,-1) and E'(2,1).

The relation between preimage and image is

[tex]P(x,y)\rightarrow P'(-x,y)[/tex]

From the given figure it is clear that the figure ABCDE reflected across the y-axis.

If a figure reflected across the y-axis, then

[tex](x,y)\rightarrow (-x,y)[/tex]

The equation of y-axis is x=0, so the given figure reflected across x=0.

Therefore the correct option is B.

What is the surface area of the regular pyramid below?

Answers

Answer:

648 sq units

Step-by-step explanation:

Area of the base= 12×12= 144 sq. units

Perimeter of the base=4×12= 48

Total surface area= 1/2×48×21 + 144

=648 sq units

ANSWER

648 square units.

EXPLANATION

The surface area of the regular pyramid is the area of the base plus the area of the 4 triangular faces.

We use the formula;

[tex]S.A = {l}^{2} + 4 \times \frac{1}{2} bh[/tex]

where l=12 units is the length of the square base and h=21 units is the vertical height of the triangular faces.

We substitute the values to get;

[tex]S.A = 12^{2} + 4 \times \frac{1}{2} \times 12 \times 21[/tex]

[tex]S.A = 144+ 504[/tex]

[tex]S.A =648 {units}^{2} [/tex]

Marita is cutting rolls of ribbon that are 3 feet long into 1/2- foot pieces. She needs fifteen 1/2- foot pieces for a project. She has 3 rolls of ribbon. Does she have enough to cut 15 pieces? Explain.

Answers

Answer: Yes

Step-by-step explanation:

3 * 2 = 6. So you can get 6 1/2 foot pieces out of 1 roll. She has 3 rolls. 6 * 3 = 18. 18 is more than 15.

Yes, She have enough ribbon to cut 15 pieces.

What is fraction?

The fraction is numerical representation of the numbers in the form of numerator and denominator.

We have,

Total rolls of ribbon = 3

Length of one roll of ribbon = 3 feet,

So,

Total Length of 3 roll of ribbon = 3 * 3 = 9 feet

And,

Marita cutting rolls into [tex]\frac{1}{2}[/tex] foot pieces.

So,

1 roll cut into pieces [tex]= \frac{3}{\frac{1}{2} } = \frac{3*2}{1}[/tex] = 6 pieces,

So,

3 roll cut into pieces = 6 * 3 = 18 pieces,

i.e.

18 pieces of  [tex]\frac{1}{2}[/tex] foot ,

And she needs [tex]15\frac{1}{2}[/tex] pieces of ribbon,

And total pieces of ribbon are 18.

Hence, we can say that Yes, She have enough ribbon to cut 15 pieces.

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caculaye the average rate of change of f(x)=x^2-1/x-4 for 2<=x<=6​

Answers

Answer:

4.75

Step-by-step explanation:

Given

f(x)=  (x^2-1)/(x-4)

The average rate of change for the interval a≤x≤b is given by:

Rate of change=  (f(b)-f(a))/(b-a)

In our question,

a=2

and

b=6

So,

f(2)=  ((2)^2-1)/(2-4)

=(4-1)/(-2)

= -3/2

And

f(6)=  ((6)^2-1)/(6-4)

=(36-1)/2

=  35/2

Rate of change=  ( 35/2-(-3/2))/(6-2)

=(35/2+3/2)/(6-2)

=  ((35+3)/2)/4

=(38/2)/4

=19/4

=4.75

The average rate of change is 4.75 ..

Answer:

Average rate of change =4.75.

Step-by-step explanation:

Given function is [tex]f\left(x\right)=\frac{x^2-1}{x-4}[/tex].

Now we need to find the average rate of change of f(x) for [tex]2\le x\le6[/tex].

So plug these values into average rate of change  (ARC) formula.

[tex]ARC=\frac{f\left(b\right)-f\left(a\right)}{b-a}[/tex]

[tex]ARC=\frac{f\left(6\right)-f\left(2\right)}{6-2}[/tex]

[tex]ARC=\frac{\frac{6^2-1}{6-4}-\frac{2^2-1}{2-4}}{4}[/tex]

[tex]ARC=\frac{\frac{36-1}{6-4}-\frac{4-1}{2-4}}{4}[/tex]

[tex]ARC=\frac{17.5-\left(-1.5\right)}{4}[/tex]

[tex]ARC=\frac{19}{4}[/tex]

[tex]ARC=4.75[/tex]

So the final answer is average rate of change =4.75.

Which statement describes the rate of change of the following function?
f(x) = 9x - 4
A.
The function has a constant rate of change, decreasing for all x at a rate of 4.
B.
The function has a constant rate of change, increasing for all x at a rate of 9.
C.
The function has a varying rate of change when x > 9.
D.
The function has a varying rate of change when x < 4.

Answers

Answer:

answerbis 2.25

Step-by-step explanation:

f(x)=9/4

f(x)=2.25

it would be a because of the -4

Options:
x = 2
y = 3
y = 2x
x = 4

Answers

Answer:

x = 2

Step-by-step explanation:

A vertical line has the equation x = a.  This line passes through points with x-coordinate of 2, so the equation is x = 2.

I have a box of 3 cupcakes. Each one is either chocolate or vanilla. What could o have in the box? What are all the ways to answer the question?

Answers

Answer:

one chocolate and two vanillatwo vanilla and one chocolatethree chocolatesthree vanillas

The native bird population in a city is decreasing at a rate of 10% per year due to industrialization of the area by humans. The population of native birds was 14,000 before the decrease began.
Complete the recursively-defined function to describe this situation.

f(0) = ____

f(n) = f(n - 1) · 0.9 , for n ≥ 1

After 3 years, _____ birds will remain.

Answers

Answer:

The recursively-defined function to describe this situation is:

f(0) = 14000 f(n) = f(n - 1) · 0.9 , for n ≥ 1 After 3 years, 10206 birds will remain.

Step-by-step explanation:

To solve the number of birds that there will be after three years, you must recognize that the function f(0) means the number of birds at time zero, in which the birds are complete (that is, the 14,000 birds), by their part, following the equation provided below we have the following calculation:

f(n) = f(n-1) * 0.9

So:

f(1) = f(0) * 0.9 = 14000 * 0.9 = 12600 f(2) = f(1) * 0.9 = 12600 * 0.9 = 11340 f(3) = f(2) * 0.9 = 11340 * 0.9 = 10206

Therefore, the final number of birds after three years in decline is 10206.

For which k are the roots of k(x2+1)=x2+3x–3 real and distinct?

Answers

Answer:

The solution for k is the interval (-3.5,1.5)

Step-by-step explanation:

we have

[tex]k(x^{2}+1)=x^{2}+3x-3[/tex]

[tex]kx^{2}+k=x^{2}+3x-3[/tex]

[tex]x^{2}-kx^{2}+3x-3-k=0[/tex]

[tex]}[1-k]x^{2}+3x-(3+k)=0[/tex]

we know that

If the discriminant is greater than zero . then the quadratic equation has two real and distinct solutions

The discriminant is equal to

[tex]D=b^{2}-4ac[/tex]

In this problem we have

a=(1-k)

b=3

c=-(3+k)

substitute

[tex]D=3^{2}-4(1-k)(-3-k)\\ \\D=9-4(-3-k+3k+k^{2})\\ \\D=9+12+4k-12k-4k^{2}\\ \\D=21-8k-4k^{2}[/tex]

so

[tex]21-8k-4k^{2} > 0[/tex]

solve the quadratic equation by graphing

The solution for k is the interval (-3.5,1.5)

see the attached figure

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