An equipment rental business currently charges $32. 00 per power tool rental and averages 24 rentals a day. The company recently performed a study and found that for every $2. 00 increase in rental price, the average number of customers decreased by one rental.


The company determined that the function f(x) = (32 + 2x)(24 – x) can be used to represent the total revenue earned from the rentals, where x represents the number of rental price increases and f(x) represents the revenue earned.


Part A: Explain what the expressions (32 + 2x) and (24 – x) represent in the given scenario.


Part B: What is the revenue after 3 rental price increases? Show your work or explain how you found your answer.


Part C: What rental price would give the maximum revenue for the company?

The drink mix that is needed to make one cup of drink is 21/20

Ali uses 21/2 scoops of drink mix to make 10 cups off drinks

The amount of drink mix needed to make one cup can be calculated as follows

21/2= 10

x= 1

cross multiply both sides

10x= 21/2

Divide by the coefficient of x which is 10

x= 21/2 ÷ 10

x= 21/2 × 1/10

x= 21/20

Hence the drink mix needed to make one cup is 21/20

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The value of the side lengths b and h in the right-angle triangle is [tex]3\sqrt{2} , and[/tex] 8.

Trigonometric ratios are the ratios of a right triangle's sides. The sine (sin), cosine (cos), and tangent are three often used trigonometric ratios. (tan).

The given figure is a right-angle triangle.

To find the value of b and h we need to apply the trigonometric ratio.

In the first triangle,

[tex]cos45 = \frac{adjacent}{hypotenuse}[/tex]

[tex]cos45 = \frac{b}{6}[/tex]

[tex]\frac{1}{\sqrt{2} } = \frac{b}{6}[/tex]

[tex]b = \frac{1}{\sqrt{2} } * 6[/tex]

[tex]b = 3\sqrt{2}[/tex]

In the second triangle

[tex]cos60 = \frac{adjacent}{hypotenuse} \\cos60 = \frac{4}{h} \\\frac{1}{2} = \frac{4}{h} \\h= 2 *4\\h = 8[/tex]

Therefore the value of the b and h is [tex]3\sqrt{2}[/tex] , and 8 respectively.

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The new measurement based on the given scale factor of 4 is 36. The scale factor is the ratio of the new size of an object to its original size. In this case, the scale factor is 4, which means the new size is 4 times larger than the original size.

If the original measurement is 9, then the new measurement can be calculated by multiplying the original measurement by the scale factor.
New measurement = Original measurement x Scale factor
New measurement = 9 x 4
New measurement = 36
Therefore, the new measurement based on the given scale factor of 4 is 36.

To explain it further, imagine you have a drawing that is 9 inches wide. If you were to increase the scale factor to 4, the new drawing would be 4 times larger, which means it would be 36 inches wide. This concept is commonly used in architecture, engineering, and other fields where scaling drawings or models is necessary to represent them accurately. Understanding scale factors is important in order to make accurate and proportional changes to objects and designs.

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6.7 times 10 to the power of 8 in scientific notation is [tex]6.7\times 10^8.[/tex]

We must adhere to the norms of the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division and Addition and Subtraction, in that sequence to conduct the order of operations with scientific notation for 6.7 × 108.

Write the number 6.7.

Multiply it by 10 raised to the power of 8. This means you move the decimal point 8 places to the right.

[tex]6.7\times 10^8[/tex]

The final answer is 670,000,000 in standard form or [tex]6.7\times 10^8[/tex] in scientific notation.

Therefore, 6.7 times 10 to the power of 8 in scientific notation is [tex]6.7\times 10^8.[/tex]

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So, to the nearest hundredth, the value of f(26.5) is approximately 59.81.

To find the value of f(26.5) for the given exponential function f(x), we will first determine the base and initial value of the function using the given points f(4.5) = 22 and f(14) = 40.
An exponential function has the form f(x) = ab^x, where a is the initial value and b is the base.
1. Write the two equations using the given points:
22 = ab^(4.5)
40 = ab^(14)
2. Divide the second equation by the first to eliminate the initial value, a:
(40/22) = (ab^(14))/(ab^(4.5))
3. Simplify and solve for the base, b:
1.81818 = b^(9.5)
b ≈ 1.05459
4. Now, plug b back into one of the original equations to solve for the initial value, a:
22 = a(1.05459)^4.5
a ≈ 16.08364
5. With a and b found, we can now find the value of f(26.5):
f(26.5) = 16.08364 * (1.05459)^26.5
f(26.5) ≈ 59.81
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The value of exponential function f(26. 5), to the nearest hundredth is 144.42.

To find the value of f(26.5) for the exponential function, we can use the given data points and apply the exponential function formula.

Let's assume the exponential function is of the form: f(x) = a * b^x, where a and b are constants.

Using the given data points:

f(4.5) = 22

f(14) = 40

Substituting these values into the exponential function equation:

22 = a * b^4.5 ---(1)

40 = a * b^14 ---(2)

Dividing equation (2) by equation (1), we can eliminate the constant 'a':

40/22 = (a * b^14) / (a * b^4.5)

1.8182 = b^(14 - 4.5)

1.8182 = b^9.5

To find the value of b, we take the 9.5th root of 1.8182:

b ≈ 1.109

Now we can substitute the value of b into equation (1) to solve for 'a':

22 = a * (1.109)^4.5

a ≈ 4.95

Therefore, the exponential function can be approximated as: f(x) ≈ 4.95 * (1.109)^x

Now we can find the value of f(26.5):

f(26.5) ≈ 4.95 * (1.109)^26.5 ≈ 144.42

Hence, the value of f(26.5), to the nearest hundredth, is approximately 144.42.

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The balance of the CD after 6 years will be $678.35.

To calculate the balance of the CD after 6 years, we need to use the formula:

[tex]A = P(1 + r/n)^{(nt)[/tex]

Where:
A = the balance after 6 years
P = the initial deposit of $500
r = the annual interest rate of 2.25%
n = the number of times the interest is compounded per year (biweekly = 26 times per year)
t = the number of years (6)

Plugging in the values, we get:

A = [tex]500(1 + 0.0225/26)^{(26*6)[/tex]
A = 500(1.001727)¹⁵⁶
A = 500(1.3567)
A = $678.35

Therefore, the balance of the CD after 6 years will be $678.35.

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Answer:

about 12.67 cm²

Step-by-step explanation:

The area of a sector of a circle is given by:

A = (θ/360) x πr²

where A is the area of the sector, θ is the central angle of the sector (in degrees), and r is the radius of the circle.

In this case, we are given the central angle of the sector, which is 86.4 degrees, and the radius of the circle, which is 4.1 cm. Therefore, we can calculate the area of the sector formed by angle CBA as follows:

A = (86.4/360) x π(4.1)²

A ≈ 12.67 cm²

So, the area "about 12.67 cm²" is a possible area of the sector formed by angle CBA.

To determine if any of the other given areas are possible, we can calculate the central angle of each sector using the same formula as above, and then check if it matches the given angle of 86.4 degrees.

For the area "about 23.35 cm²":

23.35 = (θ/360) x π(4.1)²

θ ≈ 149.6 degrees

The central angle of this sector is approximately 149.6 degrees, which is not equal to the given angle of 86.4 degrees. Therefore, the area "about 23.35 cm²" is not a possible area of the sector formed by angle CBA.

For the area "about 3.09 cm²":

3.09 = (θ/360) x π(4.1)²

θ ≈ 19.16 degrees

The central angle of this sector is approximately 19.16 degrees, which is not equal to the given angle of 86.4 degrees. Therefore, the area "about 3.09 cm²" is not a possible area of the sector formed by angle CBA.

For the area "about 40.14 cm²":

40.14 = (θ/360) x π(4.1)²

θ ≈ 256.4 degrees

The central angle of this sector is approximately 256.4 degrees, which is not equal to the given angle of 86.4 degrees. Therefore, the area "about 40.14 cm²" is not a possible area of the sector formed by angle CBA.

Therefore, the only possible area of the sector formed by angle CBA is "about 12.67 cm²".

The arc measure of major arc BDC in degrees is 240 degrees.

To find the arc measure of major arc BDC in degrees, you'll need to provide more information about the given circle or angles within it. However, I can guide you on how to find the arc measure once you have the necessary information.

1. Determine the measure of the central angle corresponding to the major arc BDC. This can be done by subtracting the measure of the minor arc from 360 degrees.

2. Use the central angle measure to find the arc measure of major arc BDC. Since the arc measure is equal to the measure of the central angle in degrees, the arc measure of major arc BDC will be the same as the central angle measure you found in step 1.

Please provide more information or details about the given circle or angles to help you find the arc measure of major arc BDC.

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The value of the scale factor for the similar figures is 1/3

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

The similar figures

The corresponsing sides of the similar figures are

Original = 12

New = 4

Using the above as a guide, we have the following:

Scale factor = New /Original

substitute the known values in the above equation, so, we have the following representation

Scale factor = 4/12

Evaluate

Scale factor = 1/3

Hence, the scale factor for the similar figures is 1/3

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To find the extremes of 4x−4y subject to the condition x2 + 2y2 = 1, we can use the method of Lagrange multipliers.

First, we set up the Lagrange equation:

∇f(x,y) = λ∇g(x,y)

where f(x,y) = 4x-4y and g(x,y) = x2 + 2y2 - 1.

Taking partial derivatives, we have:

∂f/∂x = 4
∂f/∂y = -4
∂g/∂x = 2x
∂g/∂y = 4y

Setting these equal to their respective Lagrange multipliers, we have:

4 = 2λx
-4 = 4λy
x2 + 2y2 = 1

Solving for x and y in terms of λ, we get:

x = 2λ/4 = λ/2
y = -λ/4

Substituting these back into the constraint equation, we have:

(λ/2)2 + 2(-λ/4)2 = 1
λ2/4 + λ2/8 = 1
3λ2/8 = 1
λ2 = 8/3

Taking the positive and negative square roots of λ2, we have:

λ = ±2√2/3

Substituting these values back into x and y, we get:

For λ = 2√2/3:
x = (2√2/3)/2 = √2/3
y = -(2√2/3)/4 = -√2/6

For λ = -2√2/3:
x = (-2√2/3)/2 = -√2/3
y = -(-2√2/3)/4 = √2/6

Now we can find the extreme values of f(x,y) by plugging in these values of x and y:

f(√2/3, -√2/6) = 4(√2/3) - 4(-√2/6) = 4√2
f(-√2/3, √2/6) = 4(-√2/3) - 4(√2/6) = -4√2

Therefore, the maximum value of 4x-4y subject to the condition x2 + 2y2 = 1 is 4√2 and the minimum value is -4√2.

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None of the given equations have infinitely many solutions.

To identify which equation does not have infinitely many solutions among the given options.

1) 0 = 6x + 4 = 2(3x + 2)
2) 2x + 5 - 5x + 2 + 3x = 7
3) -3x + 13 + 4x – 5 = 7x + 8
4) 4x - 8 = 2(2x + 3)

Let's analyze each equation:

1) The equation can be simplified to 0 = 6x + 4, which is not true for all x, so it does not have infinitely many solutions.
2) Simplifying the equation, we get 0 = 7, which is false for any x, so it does not have infinitely many solutions.
3) Simplifying the equation, we get 1x + 8 = 7x + 8, which can be further simplified to -6x = 0, or x = 0. Since it has only one solution, it does not have infinitely many solutions.
4) Expanding the equation, we get 4x - 8 = 4x + 6. It is false for any x, so it does not have infinitely many solutions.

Therefore, none of the given equations have infinitely many solutions.

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To test whether shoes cost less on average at Payless than at Famous Footwear, we can conduct a two-sample t-test for the difference in means.

The null hypothesis is that there is no difference in the mean prices between the two stores, and the alternative hypothesis is that the mean price at Payless is less than the mean price at Famous Footwear.

We can calculate the t-statistic using the formula:

t = [tex](x1 - x2 - 0) / sqrt(s1^2/n1 + s2^2/n2)[/tex]

where x1 and x2 are the sample means, s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes.

Substituting the values given in the table, we get:

t =[tex](21.39 - 45.66 - 0) / sqrt((7.47^2/30) + (16.54^2/30))[/tex] = -10.78

The degrees of freedom for this test is (30-1) + (30-1) = 58.

Using a t-table or calculator, we find the p-value to be very small, much less than 0.05.

Therefore, we reject the null hypothesis and conclude that there is convincing evidence at the 0.05 significance level that shoes cost less on average at Payless than at Famous Footwear.

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Answer:

? equals 80

Step-by-step explanation:

The function given is y = 18000(0.76)^t, where y represents the value of the car and t represents the time in years.

This is an exponential decay function, meaning that the value of the car decreases over time. To determine if the value of the function will ever be 0, we would need to find if there exists a time t when y = 0. Let's analyze the function:

0 = 18000(0.76)^t

In an exponential decay function, the base (0.76 in this case) is between 0 and 1, so as time (t) increases, (0.76)^t will approach 0, but it will never actually reach 0. Thus, the value of the car will keep decreasing over time but will never be exactly 0.

In summary, the value of the function, which represents the car's value, will never be 0, but it will get infinitely close to 0 as time progresses. This is a characteristic of exponential decay functions, where the value never reaches 0 but approaches it as time goes on.

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The derivative of the function y = 2(2^x)-2 is 12 * 2^x ln(2) or 12ln(2)x(2^x-1).

To find the derivative of the function y = 2(2^x)-2, we will use the power rule and the chain rule of differentiation.

Apply the power rule to the function y = 2(2^x)-2. The power rule states that if f(x) = x^n, then f'(x) = nx^(n-1).

y' = [2(2^x)-2]'

= 2[(2^x)-2]'

= 2ln(2^x)'

Apply the chain rule to (2^x)'. The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x))h'(x). In this case, g(x) = 2^x, so g'(x) = ln(2)*2^x.

y' = 2ln(2^x)'

= 2ln(2^x)

= 2ln(2)x(2^x-1)

Therefore, the derivative of the function y = 2(2^x)-2 is 12 * 2^x ln(2) or 12ln(2)x(2^x-1).

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To estimate the proportion of admitted hospital patients who have type 2 diabetes at Orlando Health Hospitals within a certain margin of error and confidence level,

we can use the following formula to determine the necessary sample size:

[tex]n = [(Z^2 * p * q) / E^2] / [1 + ((Z^2 * p * q) / E^2N)][/tex]

where:

n = sample size

Z = z-score for the desired confidence level (in this case, 1.81 for a 93% confidence level)

p = estimated proportion of patients with type 2 diabetes (unknown)

q = 1 - p

E = margin of error (0.06)

N = population size (unknown)

Since we do not know the estimated proportion of patients with type 2 diabetes or the population size,

we can assume a conservative estimate of p = q = 0.5, which maximizes the sample size required.

Plugging in the values, we get:

[tex]n = [(1.81^2 * 0.5 * 0.5) / 0.06^2] / [1 + ((1.81^2 * 0.5 * 0.5) / 0.06^2N)][/tex]

Simplifying, we get:

[tex]n = 1242.95 / [1 + (2.4 / N)][/tex]

To satisfy the requirements of the problem, we need to round up to the nearest whole number, so we need a sample size of at least 1243 patients.

Note that if we had a better estimate for p or N, we could use those values in the formula to get a more precise sample size.

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The volume of small cone-shaped cups is 11.8 in³.

To find the volume of the small shake in a cone-shaped cup that is 7 inches tall and has a diameter of 3 inches, we can use the formula for the volume of a cone:

V = 1/3 πr²h

where V = volume

r = radius

h = height of the cone

Given, diameter of come is 3 inches

We know r = d/2

r = 3/2

= 1.5

Substituting the value in the formula

V = 1/3 × 3.14 × 7 × (1.5)²

= 11.78

Rounding to nearest tenth

V = 11.8

Hence, the volume of small cone-shaped cups is 11.8 in³.

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The minimum dimensions of the box will be 7 cm × 6 cm × 6 cm

A dimension is described as the measurement of something in physical space such as length, width, or height.

We know that the there will be maximum dimension when the height of the cylinder and the radius of the hemisphere are aligned together.

Maximum height = 4 cm + 3 cm = 7 cm

Maximum diameter = 2 × 3 cm = 6 cm

Therefore, we can see that the minimum dimensions of the box are :

7 cm × 6 cm × 6 cm.

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x^(1/2) = 0

This equation has no real solutions, since no real number raised to any power can equal 0.

Therefore, the sum of all values of x satisfying r*(x) = 0 or P(x) is undefined is just 2, the only value of x for which r*(x) - 0.

To find the values of x for which r*(x) - 0 or P(x) is undefined, we first need to determine what r*(x) and P(x) are.

r*(x) is the derivative of f(x), which we can find using the product rule:

r*(x) = (1/2)x^(-1/2)(x-4) + x^1/2(1)

Simplifying this expression, we get:

r*(x) = (x-2)/sqrt(x)

To find the values of x for which r*(x) - 0, we can set r*(x) equal to 0 and solve for x:

(x-2)/sqrt(x) = 0

x - 2 = 0

x = 2

So the only value of x for which r*(x) - 0 is x = 2.

Next, we need to find the values of x for which P(x) is undefined. P(x) is undefined when the denominator of the expression for f(x) is equal to 0, since division by 0 is undefined. The denominator of f(x) is x^(1/2), so we need to solve the equation x^(1/2) = 0:

x^(1/2) = 0

This equation has no real solutions, since no real number raised to any power can equal 0.

Therefore, the sum of all values of x satisfying r*(x) = 0 or P(x) is undefined is just 2, the only value of x for which r*(x) - 0.

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The absolute maximum value of the function on the interval (-5, 11) is 670, which occurs at x = -5, and the absolute minimum value is approximately 400.847, which occurs at x ≈ 1.154.

To find the absolute maximum and minimum values of the function f(x) = -2x^3 + 8x + 400 on the interval (-5, 11), we need to consider the critical points and the endpoints of the interval.

First, we find the derivative of the function:

f'(x) = -6x^2 + 8

Setting f'(x) = 0 to find the critical points, we get:

-6x^2 + 8 = 0

x^2 = 4/3

x = ±√(4/3)

Since only √(4/3) is within the interval (-5, 11), this is the only critical point we need to consider.

Next, we evaluate the function at the endpoints of the interval:

f(-5) = -2(-5)^3 + 8(-5) + 400 = 670

f(11) = -2(11)^3 + 8(11) + 400 = -1666

Finally, we evaluate the function at the critical point:

f(√(4/3)) = -2(√(4/3))^3 + 8(√(4/3)) + 400 ≈ 400.847

Therefore, the absolute maximum value of the function on the interval (-5, 11) is 670, which occurs at x = -5, and the absolute minimum value is approximately 400.847, which occurs at x ≈ 1.154.

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The total number of bags farmers need to buy to fertilize the lawn twice a year is 3.

The label on the fertilizer bag is 4lb bag can fertilize 5000 Sq. Ft.

4lb = 5000

1lb = 5000/4

1lb = 1250

1lb bag can fertilize 1250 Sq. Ft.

To fertilize 7400 sq. Ft. lawn twice a year

Total = 7400 + 7400

Total = 14800

No. of 1lb bag can need to fertilize 14800 sq. Ft. lawn = 14800/1250

No. of 1lb bag can need to fertilize 14800 sq. Ft. lawn = 11.84

As each bag is 4lb

No. of bags needed = 11.84/4

No. of bags needed = 2.96 ≈ 3

Total no. of bag needed is 3

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Step-by-step explanation:

there is no explanation about x so wierd

When we look at [tex]f(t)=1/7 t^3-4t-12[/tex], this is a cubic equation, and solving it analytically is not straightforward.

To find the value of r such that f(r) = 0, we need to solve the equation:

[tex]1/7 r^3 - 4r - 12 = 0[/tex]

This is a cubic equation, and solving it analytically is not straightforward.

Yet, it is possible to obtain the value of r that meets the equation using numerical schemes such as Newton-Raphson or bisection. Additionally, one can take advantage of calculation tools and graphical software to calculate an estimation of r.

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The management fee Mike must pay this year is $84.

To calculate the management fee Mathew must pay this year, we need to find 0.50 percent of $16,800.

First, let's convert the percentage to a decimal by dividing it by 100. So, 0.50 percent is equal to 0.50/100 = 0.005.

Now, multiply the total investment amount by the management fee rate:

Management fee = $16,800 x 0.005 = $84.

Therefore, Mathew must pay $84.00 as the management fee this year for his investment in the New Colony Pacific Region mutual fund (rounded to 2 decimal places).

In summary, management fees are a percentage of the total asset value invested in a mutual fund, which goes toward compensating the fund managers for their services. In this case, Mathew's management fee is 0.50 percent, which equals $84.00 for the year.

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When researchers record how much of an experimental medication is in a person's bloodstream every hour, they are measuring the medication's concentration over time. This information is important because it can help determine the medication's effectiveness and potential side effects.

The half-life of a medication is the time it takes for half of the drug to be eliminated from the body. In this case, the half-life of the experimental medication is about 6 hours.

Knowing the half-life of a medication is important because it can help predict how long it will take for the drug to be eliminated from the body and when the next dose should be administered. For example, if a medication has a half-life of 6 hours, it means that after 6 hours, half of the medication will be eliminated from the body.

After another 6 hours, half of the remaining medication will be eliminated, and so on.
By monitoring the concentration of the medication in a person's bloodstream every hour, researchers can determine how quickly the drug is being absorbed and eliminated from the body. z

This information can help optimize dosing and minimize potential side effects. Overall, understanding the pharmacokinetics of a medication is crucial for safe and effective use in clinical practice.

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The crate can hold 175 bricks.

First, we need to convert the volume of the crate from cubic meters to cubic centimeters because the volume of each brick is given in cubic centimeters.

1 m = 100 cm

Volume of crate = 2.8 m3 = 2.8 x (100 cm)3 = 2,800,000 cm3

Now we can find the number of bricks that can be packed into the crate by dividing the volume of the crate by the volume of each brick:

Number of bricks = Volume of crate / Volume of each brick

= 2,800,000 cm3 / 16,000 cm3

= 175 bricks

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Let x be the number of pinon nuts each animal has collected. To make the chipmunk have ten more pinon nuts than the squirrel, the squirrel would have to give the chipmunk 10 pinon nuts.

So, after the exchange, the squirrel would have x - 10 pinon nuts, and the chipmunk would have x + 10 pinon nuts.

Since they are each giving an equal amount, the total number of pinon nuts remains the same. Therefore, we can set up the equation:

x + (x - 10) = 2x - 10

Simplifying and solving for x, we get:

2x - 10 = 2x

-10 = 0

This is a contradiction, so there is no solution that satisfies the conditions of the problem.

Therefore, the problem is not well-defined and there is no answer.

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The solution of the system of equations is given by the ordered pair (-4, 5).

In order to graph the solution to the given system of equations on a coordinate plane, we would use an online graphing calculator to plot the given system of equations and then take note of the point of intersection;

x - y = -9    ......equation 1.

3x + 4y = 8  ......equation 2.

Based on the graph shown in the image attached above, we can logically deduce that the solution to this system of equations is the point of intersection of the lines on the graph representing each of them, which lies in Quadrant II, and it is given by the ordered pairs (-4, 5).

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Absolute measures of dispersion give the actual spread or variability in the original units of measurement, while relative measures of dispersion express the dispersion relative to the mean or some other characteristic of the data.

Measures of dispersion are used to describe the spread or variability of a set of data. There are two common types of measures of dispersion: absolute measures and relative measures.

Absolute measures of dispersion, such as the range, interquartile range (IQR), and standard deviation, give an actual value or measurement of the spread in the original units of measurement.

For example, the range is simply the difference between the maximum and minimum values in a data set, while the standard deviation is a measure of how far each value is from the mean.

Relative measures of dispersion, such as the coefficient of variation (CV), express the dispersion relative to the mean or some other characteristic of the data. These measures are useful when comparing the variability of different sets of data that have different units of measurement or different means

For example, the CV is the ratio of the standard deviation to the mean, expressed as a percentage, and it can be used to compare the variability of different data sets that have different means.

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Answer:

x = (-2 + 3) * (4-(-8))

Step-by-step explanation: