The ratio of the length to breadth
of a room is 7:5. Find the length
of the room if the breadth is
10m

The length of side X in simple radical form with a rational denominator is 10/√3.

In Mathematics and Geometry, a 30-60-90 triangle is also referred to as a special right-angled triangle and it can be defined as a type of right-angled triangle whose angles are in the ratio 1:2:3 and the side lengths are in the ratio 1:√3:2.

This ultimately implies that, the length of the hypotenuse of a 30-60-90 triangle is double (twice) the length of the shorter leg (adjacent side), and the length of the longer leg (opposite side) of a 30-60-90 triangle is √3 times the length of the shorter leg (adjacent side):

Adjacent side = 5/√3

Hypotenuse, x = 2 × 5/√3

Hypotenuse, x = 10/√3.

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Missing information:

The question is incomplete and the complete question is shown in the attached picture.

r = r2

​1 − r=(−2 i^ −2 j^ +0 k^ )−(4 i^ −4 j^ +0 k^ )

⇒ r =−6 i^ +2 j^ +0 k^

∴∣ r ∣= (−6)2 +(2) 2 +0 2

= 36+4

​ = 40

​ =210

​

A quantity or phenomenon with independent qualities for both magnitude and direction is called a vector. The term can also refer to a quantity's mathematical or geometrical representation. Velocity, momentum, force, electromagnetic fields, and weight are a few examples of vectors in nature. The above option D is appropriate.

Computer graphics known as vector graphics allow for the direct creation of visual pictures using geometric structures such as points, lines, curves, and polygons that are defined on a Cartesian plane.

Any pathogen that conveys and spreads an infectious agent into other living things is referred to be a vector. These vectors could be bacteria or parasites.

The above option D is correct.

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To find the area of a parallelogram, you need to multiply the base by the height. In this case, the given dimensions are 60ft (base) and 67ft (height), and you need to subtract 52ft from the height.

New height = 67ft - 52ft = 15ft

Area of the parallelogram = Base * Height = 60ft * 15ft = 900 square feet.

Therefore, the area of the parallelogram is 900 square feet.

~~~Harsha~~~

Answer: D

Step-by-step explanation:

We are 95% confident that the true mean time for the execution of an order is between 23.36 and 25.24 minutes.

confidence interval = mean ± 1.96 * standard deviation / ✓(sample size)

In this case, the mean is 24.3 minutes, the standard deviation is 3.2 minutes, and the sample size is 45. Plugging these values into the formula, we get the following confidence interval:

confidence interval = 24.3 ± 1.96 * 3.2 / ✓(45)

= 24.3 ± 0.94

= (23.36, 25.24)

This means that we are 95% confident that the true mean time for the execution of an order is between 23.36 and 25.24 minutes.

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Joel Friedlander is a stockbroker from New York Securities and you are curious about the time that elapses between the placement of a

sales order and its execution. Joel sampled 45 orders and found that the mean time for execution it was 24.3 minutes, with a standard deviation of 3.2 minutes. Help Joel with the construction of a 95% confidence interval for the mean time for the execution of an order

Answer:

 (b)  f(x) = -1/x⁹ and g(x) = -8x +4

  (d)  f(x) = x⁹ and g(x) = -1/(-8x +4) . . . . . alternate solution

Step-by-step explanation:

You want to decompose h(x) = -1/(-8x +4)⁹ into f(x) and g(x) such that h(x) = f(g(x)).

The composition h(x) = f(g(x)) means that the function g(x) will replace x in the definition of f(x).

It is often convenient to look at the order of operations when asked to decompose a function like this. Here, the parenthetical expression (-8x+4) is raised to the 9th power and its opposite reciprocal is found. This suggests that f(x) can be a function that finds the opposite reciprocal of a 9th power,  matching choice B. Thus, a reasonable choice is ...

  (b)  f(x) = -1/x⁹ and g(x) = -8x +4

We note that the reciprocal of a 9th power is also the 9th power of a reciprocal. A negative sign is preserved by the odd power. This means that another reasonable choice for the decomposition is ...

  (d)  f(x) = x⁹ and g(x) = -1/(-8x +4)

__

Additional comment

We list choice B first because that one is probably the one you're supposed to claim as the answer. However, this question has two correct decompositions among those listed. You may want to discuss this with your teacher.

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Markdowns are usually applied before any additional discounts or promotions. In this case, the markdown reduced the regular price by 33%, resulting in the final price of $26.80.

Jules Girard bought 1 gallon of paint at a 33% markdown from its regular price of $39.99. To calculate the final price after the markdown, we can apply the percentage decrease to the regular price.

The markdown of 33% means that the price is reduced by 33% of the regular price. To calculate the amount of the markdown, we can multiply the regular price by the percentage decrease:

Markdown amount = 33% of $39.99 = 0.33 * $39.99

Next, we can subtract the markdown amount from the regular price to find the final price:

Final price = Regular price - Markdown amount = $39.99 - (0.33 * $39.99)

Calculating the expression within parentheses:

Final price = $39.99 - $13.19

Final price = $26.80

Therefore, Jules Girard bought 1 gallon of paint at a 33% markdown, resulting in a final price of $26.80.

Markdowns are usually applied before any additional discounts or promotions. In this case, the markdown reduced the regular price by 33%, resulting in the final price of $26.80.

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The 36th derivative of f(x) = cos(2x) is [tex]2^{17}[/tex]* cos(2x).

To find the 36th derivative of the function f(x) = cos(2x), we can apply the chain rule repeatedly. The chain rule states that if we have a composite function y = f(g(x)), then its derivative is given by dy/dx = f'(g(x)) * g'(x).

Let's start by finding the first few derivatives of f(x) = cos(2x):

f'(x) = -2sin(2x)

f''(x) = -4cos(2x)

f'''(x) = 8sin(2x)

f''''(x) = 16cos(2x)

We observe a pattern where the derivatives of cos(2x) alternate between sin(2x) and cos(2x), with the signs changing accordingly.

Based on this pattern, we can see that the 36th derivative will be:

f^(36)(x) =[tex](-1)^{17} * 2^{17} *[/tex] cos(2x)

Simplifying this expression, we have:

f^(36)(x) = [tex]2^{17} * cos(2x)[/tex]

Therefore, the 36th derivative of f(x) = cos(2x) is[tex]2^{17[/tex] * cos(2x).

It's important to note that in this case, the number 36 is even, and since the derivatives of cos(2x) follow a repeating pattern every 4 derivatives, the sign (-1) raised to the power of 17 accounts for the change in sign in the 36th derivative.

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The measure of the inscribed angle G in the circle is 85 degree.

An inscribed angle is simply an angle with its vertex on the circle and whose sides are chords.

The relationhip between an an inscribed angle and intercepted arc is expressed as:

Inscribed angle = 1/2 × intercepted arc.

Since the circle equals 360 degrees, we can determine the intercepted arc DF.

Hence:

Arc DF = 360 - ( 70 + 120 )

Arc DF = 360 - 190

Arc DF = 170°

Now, plug the value into the above formula:

Inscribed angle = 1/2 × intercepted arc.

Inscribed angle G = 1/2 × 170°

Inscribed angle G = 85°

Therefore, angle G has a measure of 85 degree.

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Answer:
The factors correlate with the x-intercepts. For example, with the factor (x - 3), there is an x-intercept at (3, 0). Likewise, with the factor (x + 2), there is an x-intercept at (-2, 0). This suggests that the opposite sign of the factors are the x-intercepts in the graph.

Answer:

The answer is down below

Step-by-step explanation:

v-u=◇velocity

[tex] = 9 - 5[/tex]

◇velocity =4ms‐¹

acceleration =◇velocity/time

[tex]a = \frac{4}{5} [/tex]

[tex]a = 0.8m {s}^{ - 2} [/tex]

Change in velocity:

[tex] \sf \:final \: velocity - initial \: velocity = change \: in \: velocity[/tex]

[tex] \therefore \tt \delta \: v = 9 - 5 \\ \tt = 4m {s}^{ - 1} [/tex]

To find acceleration:

[tex] \rm \: a = \frac{ \triangle \: v}{\triangle \: t} [/tex]

[tex] \rm \: a = \frac{ 4}{5 - 0} = \frac{4}{5} = 0.8m {s}^{ - 1} [/tex]

a. This situation would be a binomial experiment if the following conditions are met:

The trials are independent: The selection of one part does not affect the selection of the other.

Each trial has two possible outcomes: In this case, defective or non-defective.

The probability of success (finding a defective part) remains constant for each trial: In this case, the probability is 3% or 0.03.

The number of trials is fixed: We are selecting two parts, so the number of trials is predetermined.

b. Here is a tree diagram representing the two-trial experiment:

          D                ND

        /   \            /   \

       D     ND         D     ND

The branches represent the two trials, with D representing a defective part and ND representing a non-defective part.

c. To find the number of experimental outcomes resulting in exactly one defect, we can observe that there are two outcomes that meet this criterion: D-ND and ND-D. Both represent one defective part and one non-defective part.

d. The probabilities associated with finding no defects, exactly one defect, and two defects can be calculated as follows:

Probability of finding no defects: (1 - probability of finding a defective part) * (1 - probability of finding a defective part) = (1 - 0.03) * (1 - 0.03) = 0.97 * 0.97 = 0.9409.

Probability of finding exactly one defect: (probability of finding a defective part) * (probability of finding a non-defective part) + (probability of finding a non-defective part) * (probability of finding a defective part) = 0.03 * 0.97 + 0.97 * 0.03 = 0.0582.

Probability of finding two defects: (probability of finding a defective part) * (probability of finding a defective part) = 0.03 * 0.03 = 0.0009.

Therefore, the probabilities associated with finding no defects, exactly one defect, and two defects are approximately 0.9409, 0.0582, and 0.0009, respectively.

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Answer:

Please attach a photo to this so I can better answer your question

Answer:

$100

Step-by-step explanation:

Since 10 vases cost $1000, you can find the cost vase by dividing $1000 by 10.

$1000 ÷ 10 = $100

Each vase costs $100.

Using Pythagorean theorem, the length of the ladder is 10ft

In mathematical terms, if y and z are the lengths of the two shorter sides (also known as the legs) of a right triangle, and x is the length of the hypotenuse, the Pythagorean theorem can be expressed as:

x² = y² + z²

In the questions given, the only one we can use Pythagorean theorem to solve is the one with ladder since it's forms a right-angle triangle.

To calculate the length of the ladder, we can write the formula as;

x² = 8² + 6²

x² = 64 + 36

x² = 100

x = √100

x = 10

The length of the ladder is 10 feet

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Answer:

To solve the equation 12^(x + 1) = 79 for x, we need to isolate the exponent.

Step 1: Subtract 1 from both sides of the equation:

12^(x + 1) - 1 = 79 - 1

12^(x + 1) - 1 = 78

Step 2: Rewrite 12^(x + 1) as (12^x)(12^1) using the exponent property:

(12^x)(12) - 1 = 78

Step 3: Simplify the left side of the equation:

12(12^x) - 1 = 78

Step 4: Add 1 to both sides of the equation:

12(12^x) = 78 + 1

12(12^x) = 79

Step 5: Divide both sides of the equation by 12:

(12(12^x))/12 = 79/12

12^x = 79/12

Step 6: Take the logarithm (base 12) of both sides of the equation:

log12(12^x) = log12(79/12)

x = log12(79/12)

Therefore, the solution for x is x = log12(79/12)

Hope this helps!

The Billy's net pay is K646

The given information are; Billy's gross pay is K850.00, he is taxed 12% on his taxable income, and contributes 7% to Nusfund. His loan deduction is 5%.

To find out his net pay, the following steps must be taken:Firstly, calculate Billy's tax amount, Nusfund contribution, and loan deduction by using their respective percentages and the gross pay.

 tax amount= 12/100 × 850= K102 Refund contribution= 7/100 × 850= K59.50Loan deduction= 5/100 × 850= K42.50 Secondly, the sum of all the deductions made from the gross pay is calculated to obtain the total deductions.

Total deductions= K102 + K59.50 + K42.50= K204The final step is to subtract the total deductions from the gross pay to find Billy's net pay. Net pay= K850 − K204= K646.

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Step-by-step explanation:

To find the zeros of the polynomial p(x) = (2x^2 - 9x + 7)(x - 2), we set p(x) equal to zero and solve for x:

(2x^2 - 9x + 7)(x - 2) = 0

This equation will be satisfied if either of the factors on the left side equals zero. So we set each factor equal to zero and solve for x separately.

Setting 2x^2 - 9x + 7 = 0:

Using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a)

For this equation, a = 2, b = -9, and c = 7.

Plugging these values into the quadratic formula, we have:

x = (-(-9) ± √((-9)^2 - 4 * 2 * 7)) / (2 * 2)

x = (9 ± √(81 - 56)) / 4

x = (9 ± √25) / 4

x = (9 ± 5) / 4

So we have two solutions:

x1 = (9 + 5) / 4 = 14 / 4 = 3.5

x2 = (9 - 5) / 4 = 4 / 4 = 1

Setting x - 2 = 0:

x = 2

Therefore, the zeros of the polynomial p(x) = (2x^2 - 9x + 7)(x - 2) are x = 3.5, 1, and 2.

The length of BD of the given triangle is: 15

The perimeter of a triangle is simply defined as the length of the boundary of that triangle.

Now, we are given that:

AB = BC = a

Perimeter of triangle ABC = 50

Perimeter of triangle ABD = 40

Thus:

2a + 2b = 50

a + b = 25     ------(1)

Similarly:

a + b + h = 40

a + b = 40 - h   -----(2)

Put 40 - h for a + b in eq 1 to get:

40 - h = 25

h = 40 - 25

h = 15

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These are a legitimate probability distribution:

To have a legitimate probability distribution, the probabilities for all possible values of X must satisfy two conditions:

Each probability value must be between 0 and 1, inclusive.

The sum of all probability values must equal 1.

Check if the given probabilities satisfy these conditions:

P(X=-3) = 0.14 (valid)

P(X=1) = 0.1 (valid)

P(X=3) = 0.135 (valid)

P(X=5) = 0.306 (valid)

Now, calculate the remaining probability to check if it satisfies the second condition:

P(X=6) = 1 - (P(X=-3) + P(X=1) + P(X=3) + P(X=5))

P(X=6) = 1 - (0.14 + 0.1 + 0.135 + 0.306)

P(X=6) = 1 - 0.681

P(X=6) = 0.319

Since the calculated probability is valid (between 0 and 1), There is a legitimate probability distribution for the discrete random variable X:

P(X=-3) = 0.14

P(X=1) = 0.1

P(X=3) = 0.135

P(X=5) = 0.306

P(X=6) = 0.319

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Answer:

S = π(12)(20) + π(12^2) - (π(3)(5) + π(3^2))

= π(240 + 144 - (15 + 9))

= 360π = about 1,130.97 cm^2

Answer:

Step-by-step explanation: 5√2

The correct answer is B. 2, as the scale factor used to create the dilated polygon is 2.

To determine the scale factor used to create the image, we can compare the corresponding side lengths of the original polygon ABCD and the dilated polygon A'B'C'D'.

Let's calculate the lengths of the corresponding sides:

Side AB: The length of side AB is 3 - 1 = 2 units in the original polygon. In the dilated polygon, the length of side A'B' is 6 - 2 = 4 units.

Side BC: The length of side BC is -2 - (-1) = -1 units in the original polygon. In the dilated polygon, the length of side B'C' is -4 - (-2) = -2 units.

Side CD: The length of side CD is 1 - 3 = -2 units in the original polygon. In the dilated polygon, the length of side C'D' is 2 - 6 = -4 units.

Side DA: The length of side DA is -1 - (-2) = 1 unit in the original polygon. In the dilated polygon, the length of side D'A' is -2 - (-4) = 2 units.

Now, let's compare the corresponding side lengths:

AB: A'B' = 4 units / 2 units = 2

BC: B'C' = -2 units / -1 units = 2

CD: C'D' = -4 units / -2 units = 2

DA: D'A' = 2 units / 1 unit = 2

The scale factor used to create the image is the ratio of the corresponding side lengths.

In this case, all the corresponding side lengths have a ratio of 2.

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Chandler ended up 8 feet below sea level. The negative sign indicates that his final position is below sea level. This means that he has descended further into the lake compared to the starting point on the cliff.

Chandler started on the cliff at 32 feet above sea level. When he jumped for 40 feet, we need to determine the final position in relation to sea level.

Since Chandler jumped down, the distance below sea level will be calculated as a negative value. To find how far below sea level Chandler ended up, we subtract the jump distance (40 feet) from the starting height (32 feet above sea level):

32 feet - 40 feet = -8 feet

It's important to note that negative values are used here to represent the direction and magnitude of Chandler's descent relative to sea level

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Answer:

-8 feet

Step-by-step explanation:

Your overall grade, if the percentage scored in the categories are given, is 86. 35.

To calculate your overall grade, you'll need to multiply each individual grade by its respective weight and then add all these products together.

Tests:

= 85 % * 15%

= 12. 75

Quizzes:

= 90% * 10%

= 9. 0

Projects:

= 92% * 35%

= 32 .2

Participation:

= 80% * 35%

= 28. 0

Final Exam:

= 88% * 5%

= 4. 4

The overall grade is:

= 12.75 + 9.0 + 32.2 + 28.0 + 4.4

= 86. 35

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The scores are :
Tests: 85%

Quizzes: 90%

Projects: 92%

Participation: 80%

Final Exam: 88%

The total amount of interest Ron paid over the course of the loan is approximately $12,220.63.

To calculate the total amount of interest paid on Ron's student loan, we need to use the formula for compound interest. The formula for compound interest is:

A = P(1 + r/n)^(nt) - P

Where:

A is the total amount including principal and interest.

P is the principal amount (loan amount).

r is the annual interest rate (as a decimal).

n is the number of times the interest is compounded per year.

t is the number of years.

In this case, Ron's loan amount is $35,000, the annual interest rate is 3.8% (or 0.038 as a decimal), the loan term is 8 years, and the interest is compounded monthly (n = 12).

Plugging in these values into the formula:

A = $35,000(1 + 0.038/12)^(12*8) - $35,000

Calculating the exponent:

A = $35,000(1 + 0.00316667)^(96) - $35,000

Using a calculator to evaluate the expression within parentheses:

A = $35,000(1.00316667)^(96) - $35,000

Calculating the exponent:

A = $35,000(1.34898777) - $35,000

Subtracting the principal amount:

A = $47,220.63 - $35,000

A = $12,220.63

Therefore, the total amount of interest Ron paid over the course of the loan is approximately $12,220.63.

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The calculated sum of B1,B2,B3 and B4 is 6

From the question, we have the following parameters that can be used in our computation:

BN=(N-1)(N-2)(N-3)

Express properly

So, we have

B(N) = (N-1)(N-2)(N-3)

using the above as a guide, we have the following equations

B(1) = (1-1)(1-2)(1-3) = 0

B(2) = (2-1)(2-2)(2-3) = 0

B(3) = (3-1)(3-2)(3-3) = 0

B(4) = (4-1)(4-2)(4-3) = 6

When the above functions are added, we have

Sum = 0 + 0 + 0 + 6

Evaluate the sum

Sum = 6

Hence, the sum of B1,B2,B3 and B4 is 6

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Answer:0.00007593302

Step-by-step explanation:

The preparation of the branch's income statement for Santosh & Co. Chennai is as follows:

Branch of Santosh & Co. Chennai

For the year ended December 31, 2018

Sales revenue            $40,300

Cost of goods sold         4,200

Gross profit                 $36,100

Expenses:

Office expenses $36,000

Salaries                    3,600

Total expenses         $39,600

Loss                             $3,500

An income statement is a financial statement prepared at the end of an accounting period to determine the profit or loss generated by a business or branch.

The profit or loss is the difference between the total revenue and the total expenses for the accounting period.

Credit sales at branch 40,000

Office expenses by Head office 36,000

Cash remittance to branch for petty cash 9,000

Goods sent to Branch 7,200

Salaries paid by head office 3,600

Debtors 31.12.2018 30,000

Petty cash on 31.12.2018 16,200

Cash sales at branch 300

Stock of 31.12.2018 3,000

Sales revenue   $40,300 ($40,000 + $300)

Cost of goods sold $4,200 ($7,200 - $3,000)

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so the idea being, we have a system of equations of two variables and 4 equations, each one rendering a line, for this case these aren't equations per se, they're INEquations, so pretty much the function will be the same for an equation but we'll use > or < instead of =, but fairly the function is basically the same, the behaviour differs a bit.

we have a line passing through (-6,0) and (0,8), side one

we have a line passing through the x-axis and -6, namely (-6,0) and the y-axis and -4, namely (0,-4), side two

we have a line passing through (0,-4) and (6,4), side three

now, side four is simply the line connecting one and three.

the intersection of all four lines looks like the one in the picture below, so what are those lines with their shading producing that quadrilateral?

well, we have two points for all four, and that's all we need to get the equation of a line, once we get the equation, with its shading like that in the picture, we'll make it an inequality.

[tex](\stackrel{x_1}{-6}~,~\stackrel{y_1}{0})\qquad (\stackrel{x_2}{0}~,~\stackrel{y_2}{8}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{8}-\stackrel{y1}{0}}}{\underset{\textit{\large run}} {\underset{x_2}{0}-\underset{x_1}{(-6)}}} \implies \cfrac{8 -0}{0 +6} \implies \cfrac{ 8 }{ 6 } \implies \cfrac{4}{3}[/tex]

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{0}=\stackrel{m}{ \cfrac{4}{3}}(x-\stackrel{x_1}{(-6)}) \implies y -0 = \cfrac{4}{3} ( x +6) \\\\\\ y=\cfrac{4}{3}x+8\hspace{5em}\stackrel{\textit{side one} }{\boxed{y < \cfrac{4}{3}x+8}}[/tex]

[tex]\rule{34em}{0.25pt}\\\\ (\stackrel{x_1}{-6}~,~\stackrel{y_1}{0})\qquad (\stackrel{x_2}{0}~,~\stackrel{y_2}{-4}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{-4}-\stackrel{y1}{0}}}{\underset{\textit{\large run}} {\underset{x_2}{0}-\underset{x_1}{(-6)}}} \implies \cfrac{-4 -0}{0 +6} \implies \cfrac{ -4 }{ 6 } \implies - \cfrac{2}{3}[/tex]

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{0}=\stackrel{m}{- \cfrac{2}{3}}(x-\stackrel{x_1}{(-6)}) \implies y -0 = - \cfrac{2}{3} ( x +6) \\\\\\ y=-\cfrac{2}{3}x-4\hspace{5em}\stackrel{\textit{side two} }{\boxed{y > -\cfrac{2}{3}x-4}} \\\\[-0.35em] \rule{34em}{0.25pt}[/tex]

[tex]\stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{4}-\stackrel{y1}{(-4)}}}{\underset{\textit{\large run}} {\underset{x_2}{6}-\underset{x_1}{0}}} \implies \cfrac{4 +4}{6 -0} \implies \cfrac{ 8 }{ 6 } \implies \cfrac{4}{3}[/tex]

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{(-4)}=\stackrel{m}{ \cfrac{4}{3}}(x-\stackrel{x_1}{0}) \implies y +4 = \cfrac{4}{3} ( x -0) \\\\\\ y=\cfrac{4}{3}x-4\hspace{5em}\stackrel{ \textit{side three} }{\boxed{y > \cfrac{4}{3}x-4}} \\\\[-0.35em] \rule{34em}{0.25pt}[/tex]

[tex](\stackrel{x_1}{6}~,~\stackrel{y_1}{4})\qquad (\stackrel{x_2}{0}~,~\stackrel{y_2}{8}) ~\hfill~ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{8}-\stackrel{y1}{4}}}{\underset{\textit{\large run}} {\underset{x_2}{0}-\underset{x_1}{6}}} \implies \cfrac{ 4 }{ -6 } \implies - \cfrac{2}{3}[/tex]

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{4}=\stackrel{m}{- \cfrac{2}{3}}(x-\stackrel{x_1}{6}) \\\\\\ y=-\cfrac{2}{3}x+8\hspace{5em}\stackrel{ \textit{side four} }{\boxed{y < -\cfrac{2}{3}x+8}}[/tex]

now, we can make that quadrilateral a trapezoid by simply moving one point for "side four", say we change the point (0 , 8) and in essence slide it down over the line to  (-3 , 4).  Notice, all we did was slide it down the line of side one, that means the equation for side one never changed and thus its inequality is the same function.

now, with the new points for side for of (-3,4) and (6,4), let's rewrite its inequality

[tex](\stackrel{x_1}{-3}~,~\stackrel{y_1}{4})\qquad (\stackrel{x_2}{6}~,~\stackrel{y_2}{4}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{4}-\stackrel{y1}{4}}}{\underset{\textit{\large run}} {\underset{x_2}{6}-\underset{x_1}{(-3)}}} \implies \cfrac{4 -4}{6 +3} \implies \cfrac{ 0 }{ 9 } \implies 0[/tex]

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{4}=\stackrel{m}{ 0}(x-\stackrel{x_1}{(-3)}) \implies y -4 = 0 ( x +3) \\\\\\ y=4\hspace{5em}\stackrel{ \textit{side four changed} }{\boxed{y < 4}}[/tex]