y = y₁ = m(x − x₁)

Show your work in the space provided below.

1. Find the equation of the line that has a slope of -4 and passes through the point (-3, 7).
Show your work below:

2. Find the equation of the line that passes through the points (4, -3) and (8,9).
Show your work below:

The ratio of vegetable oil to water is  [tex]\frac{5}{8}[/tex] : 2 and the value of the ratio is 0.3125.

What is ratio?

By dividing two amounts of the same unit, it is possible to determine how much of one quantity is in the other. This is referred to as ratio in mathematics.

We are given that a particular recipe uses [tex]\frac{5}{8}[/tex] cup of vegetable oil and 2 cups of water.

So, from this, we get the ratio of vegetable oil to water as [tex]\frac{5}{8}[/tex] : 2.

Now, the value of the obtained ratio is

⇒ 0.625 : 2

⇒ 0.3125

Hence, the required solution has been obtained.

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The coordinates at which the tangent line to f(x) = (x-6/x)^8 is horizontal are (√6, f(√6)) and (-√6, f(-√6)), where f(x) is the given function.

To find the coordinates at which the tangent line to the function f(x) = (x-6/x)^8 is horizontal, we need to find the critical points of the function where the derivative is zero or undefined.

First, let's find the derivative of the function:

f(x) = (x-6/x)^8

f'(x) = 8(x-6/x)^7 * (1 - (-6/x^2))

Simplifying the second term, we get:

f'(x) = 8(x-6/x)^7 * (x^2+6)/x^2

Now we need to set the derivative equal to zero and solve for x:

8(x-6/x)^7 * (x^2+6)/x^2 = 0

(x^2+6) cannot be zero, so we can ignore that factor.

8(x-6/x)^7 = 0

(x-6/x) = 0

x^2 - 6 = 0

x = ±√6

So we have two critical points at x = √6 and x = -√6.

Now we need to determine whether these critical points correspond to a maximum, minimum, or inflection point. To do this, we can use the second derivative test.

Taking the derivative of the first derivative, we get:

f''(x) = 8(x-6/x)^6 * (56/x^3 + 7)

Evaluating the second derivative at x = √6, we get:

f''(√6) = 8(√6-6/√6)^6 * (56/√6^3 + 7)

f''(√6) > 0, so the function has a local minimum at x = √6.

Evaluating the second derivative at x = -√6, we get:

f''(-√6) = 8(-√6-6/-√6)^6 * (56/-√6^3 + 7)

f''(-√6) < 0, so the function has a local maximum at x = -√6.

Therefore, the coordinates at which the tangent line to f(x) = (x-6/x)^8 is horizontal are (√6, f(√6)) and (-√6, f(-√6)), where f(x) is the given function.

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According to the given information, the absolute value of the difference between the y-intercepts of f(x) and g(x) is 0.

What is a function?

A function is a relation between a set of inputs and a set of possible outputs, with the property that each input is related to exactly one output.

To find the y-intercept of a function, we set x=0 and evaluate the function at that value.

For the function f(x) = (x + 4)2 - 3, we have:

f(0) = (0 + 4)2 - 3 = 13.

To find the y-intercept of the function g(x), we can use the given points and try to write it in the form y = ax² + bx + c, where a, b, and c are constants.

Using the given points, we can write three equations:

When x = -2, y = 17: 17 = 4a - 2b + c

When x = -1, y = 15: 15 = a - b + c

When x = 1, y = 9: 9 = a + b + c

Solving this system of equations, we get a = -1, b = 1, and c = 13. Therefore, the equation of the function g(x) is:

g(x) = -x² + x + 13.

To find the absolute value of the difference between the y-intercepts of f(x) and g(x), we can subtract the two y-intercepts and take the absolute value:

|f(0) - g(0)| = |13 - 13| = 0.

Therefore, the absolute value of the difference between the y-intercepts of f(x) and g(x) is 0.

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Thus, the polynomial at x=i needs to be evaluated: As a result, when x-1 is divided by [tex]-3x^4 - 2x^3 + 5x^2 - 7x,[/tex] the remaining is -8i - 5.

Using just the activities of addition, removal, multiplication, and non-negative decimal exponents, a polynomial is a mathematical equation made up of variables and coefficients. Polynomials can contain one or perhaps more variables, and they can be categorised based on their degree, which is the polynomial's highest exponent. The most familiar example of polynomial is the exponential, which has a rank of 2 and may be expressed in the form ax2 + bx + c. The shortest polynomials be monomials, which have only one term. Algebra, algebra, and number theory are just a few of the mathematical areas where polynomials are used.

given

The remainder theorem can be used to get the remaining when[tex]-3x^4, 2x^3, 5x^2[/tex], and 7x are divided by x-i.

The remainder is p when a polynomial p(x) is divided by (x-a), according to the theorem (a).

In this instance, we must determine the remaining after dividing [tex]3x^4[/tex] by x-i and adding [tex]2x^3 , 5x^2 ,7^x.[/tex]

Thus, the polynomial at x=i needs to be evaluated: As a result, when x-1 is divided by [tex]-3x^4 - 2x^3 + 5x^2 - 7x,[/tex] the remaining is -8i - 5.

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Answer:

Step-by-step explanation:

For question 3,

Simplified brackets -> (8+11)2-8+11

Open brackets -> 19*2-8+11

Multiply -> 38-8+11

Calculate -> 41

Solution = 41

For question 4,

Remove Brackets -> m+11+m+44

Put common numbers together -> m+n+44+11

Calculate -> m+55+n

Solution = m+55+n

-Your smart 6th grader

Answer:

[tex]v = \frac{28\pi}{3} [/tex]

Step-by-step explanation:

First, we can find the area of the cone's base:

[tex]a(base) = \pi \times {r}^{2} = 4\pi[/tex]

Now, let's find the volume:

[tex]v = \frac{1}{3} \times a(base)\times h[/tex]

[tex]v = \frac{1}{3} \times 4\pi \times 7 = \frac{28\pi}{3} [/tex]

Answer:

m∠B = 36.9 °

Step-by-step explanation:

SOH  -  CAH  -  TOA

Sine → Opposite/Hypotenuse

sin(θ) = 3/5

[tex]\theta = sin^-^1(3/5)\\[/tex]

θ = 36.86 degrees

Round to nearest tenth, so m∠B = 36.9 °

To determine the actual height of the climbing frame, we need to know the scale factor of the model. If the scale factor is, for example, 1:50, it means that every 1 cm on the model represents 50 cm in real life.

Assuming that we have the scale factor, we can use the following proportion:

model height / actual height = scale factor

We know that the model height is 10 cm, and we want to find the actual height. Let's say the scale factor is 1:100. Then we have:

10 cm / actual height = 1/100

Multiplying both sides by the actual height, we get:

actual height = (10 cm) x (100/1) = 1000 cm

Therefore, the actual height of the climbing frame in this example is 1000 cm, or 10 meters.

The correct option is- C) 0 <= x <= 12 and 16 <= y <= 48, is the reasonable constraints for the give graph of total number of patients showed up to nurse Jackie.

When variables are employed in equations to simulate real-world scenarios, constraints must be applied to set limits and bounds on those variables.

From the given graph

x-axis shows the time duration between 9 AM to 9 PM.

y-axis shows the number of patients visited.

Value shown on the graph;

Thus, 0 <= x <= 12 and 16 <= y <= 48, is the reasonable constraints for the give graph of total number of patients showed up to nurse Jackie.

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The probability that the first three cases are all pink is 0.015. In most cases, the probability is given as a ratio between the total number of outcomes in the sample space and the number of positive outcomes.

Probability is the potential for something to occur. The value is expressed in the range of 0 to 1 .In light of this, whenever we are unsure of how an event will turn out, we can talk about the probabilities of various outcomes, or how likely they are.

Statistics is a term that refers to the study of probability-based phenomena. Hence, the probability that an event will occur depends on both the quantity of favorable outcomes and the total number of outcomes.

Probability = no. of favorable cases/total number of cases

Given

Total No. of cases = 80

Black = 40

White = 40

Pink = 30

Probability = no. of favorable cases/total number of cases

For first draw

Pink cases = 10

Total cases = 80

P1  = 10/80

For second draw

Pink cases left = 9

Total cases = 79

P2  = 9/79

For second draw

Pink cases left = 8

Total cases = 78

P2  = 8/78

Now Total Probability is

P = P1× P2 × P3

P = 10/80 × 9/79 × 8/78 = 720/492,960

P = 3/2054 ≈ 0.015

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Answer: To determine the range of input (L) over which production should take place, we need to find the point at which the marginal product of labor (MPL) is zero:

MPL = 20 - 0.4L

0 = 20 - 0.4L

0.4L = 20

L = 50

So production should take place for values of L less than or equal to 50.

To find the range of output over this range of labor, we can use the average product of labor (APL) equation:

APL = 20 - 0.2L - 320/L

Substituting L = 50, we get:

APL = 20 - 0.2(50) - 320/50

APL = 20 - 10 - 6.4

APL = 3.6

So the range of output over the range of labor from 0 to 50 is approximately 0 to 3.6 units of output.

Step-by-step explanation:

Answer:

To find the solution to f(x)=g(x), we need to find the point(s) where the two curves intersect.

The graph is not provided, but we can find the solution algebraically by setting the two functions equal to each other:

f(x) = g(x)

x^2 = 12^(x-1)

To solve for x, we can take the logarithm of both sides:

log(x^2) = log(12^(x-1))

2log(x) = (x-1)log(12)

2log(x) = xlog(12) - log(12)

2log(x) - xlog(12) = -log(12)

log(x^2) - log(12^x) = -log(12)

log(x^2/12^x) = -log(12)

log(x/12) = -log(12)

log(x) - log(12) = -log(12)

log(x) = 0

x = 1

Therefore, the solution to f(x)=g(x) is x=1.

Step-by-step explanation:

Answer:

using the trigonometry identities

Cos ∅ = Adj/Hyp

where ∅ = 73°

Cos 73 = 8.6/x

X × Cos 73 = 8.6

x = 8.6/0.2923

x = 29.421 ≈ 29.4

Answer:

C)  decreasing then increasing.

Step-by-step explanation:

A function is said to be increasing if the y-values increase as the x-values increase.

A function is said to be decreasing if the y-values decrease as the x-values increase.

From inspection of the given graph of y = x², we can see that for the first half of the graph, the y-values are decreasing as the x-values increase. Therefore, the function is decreasing for this part of the graph.

Similarly, for the second half of the graph, we can see that the y-values are increasing as the x-values increase. Therefore, the function is increasing for this part of the graph.

So the description of the graph of the function is:

Hence, (2/n)* is the radical expression's abbreviated form (3n) as 12 and n have a common factor of 4.

In maths, an expressions is a set of digits, parameters, and operators that denotes a quantity or relationship. Aside from basic arithmetic operations like addition, reduction, multiplication, and division, expressions can also include more intricate operations like exponents, number theory, and trigonometric functions. Expressions might be basic, including a single variable and one operation, like 3x or 5 + 7, or complex, requiring several variables and actions, like (x + y)2 - 2x. Expressions can represent arithmetic, inequalities, and other scientific connections.

given

Due to the fact that 12 and n have a common factor of 4, the radical statement 12n/n can be further reduced.

We can rewrite 12 as 4 * 3 to simplify the expression, and then we can take the square root of 4 to get 2:

√12n/n = √(4 * 3 * n)/n = √(4/n) * √(3n) = (2/√n) * √(3n) (3n)

Hence, (2/n)* is the radical expression's abbreviated form (3n) as 12 and n have a common factor of 4.

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So the speed of the wind is 52 miles per hour. When the airplane is flying into a headwind, its ground speed (the speed at which it appears to be moving relative to the ground)

what is speed  ?

In physics, speed is the rate at which an object moves, or the distance traveled per unit of time. It is a scalar quantity, meaning that it has magnitude (a numerical value) but no direction.

In the given question,

Let's use "s" to represent the airspeed of the plane, and "w" to represent the speed of the wind.

When the airplane is flying into a headwind, its ground speed (the speed at which it appears to be moving relative to the ground) is s - w. We know that the airplane travels 1560 miles in 3 hours, so we can set up the equation

1560 = (s - w) * 3

Simplifying this equation, we get:

s - w = 520

When the airplane is flying with a tailwind (i.e., in the opposite direction of the headwind), its ground speed is s + w. We know that the airplane travels 1560 miles in 2.5 hours (since 2 hours and 30 minutes is equal to 2.5 hours), so we can set up the equation:

1560 = (s + w) * 2.5

Simplifying this equation, we get:

s + w = 624

Now we have two equations:

s - w = 520

s + w = 624

We can solve this system of equations by adding them together. When we add the left sides of the equations, we get:

2s = 1144

Dividing both sides by 2, we get:

s = 572

So the airspeed of the plane is 572 miles per hour.

Now we can use one of the equations we found earlier to solve for the wind speed. Let's use the equation:

s - w = 520

Substituting s = 572, we get:

572 - w = 520

Simplifying this equation, we get:

w = 52

So the speed of the wind is 52 miles per hour.

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Step-by-step explanation:

Teresa și sa se uite pe aici prin intermediul acestui an indirect bună am văzut pe net și sa se întâmple și sa ne contactați la adresa lui și sa ne întâlnim cu toții la

Solving a system of equations we will see that the values are:

x = 115

y = -38.25

We know that the sum of two adjacent angles is always 180°, then we can write two linaer equations:

8y + 4x + 26 = 180

4y - 12 + 3x = 180

We can simplify that to get the system of equations:

4x + 8y = 154

3x + 4y = 192

To solve this, we can take the difference between twice the second equation and once the first equation to get:

2*(3x + 4y) - (4x + 8y) = 2*192 - 154

6x + 8y - 4x - 8y = 230

2x = 230

x = 230/2

x = 115

Then the value of y is:

3*115 + 4*y = 192

4y = 192 - 3*115

y = (192 - 3*115)/4

y = -38.25

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A). The function is symmetric about the line x = 1 and continuous.

For 0 x 2, the function is positive. The highest value for the function is

(1, 1). The value of f(x) increases as x approaches positive infinity.

Each input value is given a distinct output value by a rule known as a function.

Functions can be shown using graphs, tables, mathematical notation, and other techniques.

The function is positive in the range 0 x 2, therefore we can limit the curve to that region. As a result, the curve may increase quickly as x moves away from 2. The produced graph might look like this:  

B). The function is continuous and symmetrical about the line x = 2. For the function, the bare minimum is (2, 3). As x approaches positive or negative infinity, f(x) approaches infinity.

Similar to how we can design a symmetric curve with a minimum point at x = 2 because the function is symmetric around that value. (2, 3). When x gets close to positive or negative infinity, the function moves towards infinity.  

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Answer:

o find MW, we need to use the fact that OS is perpendicular to WV, which means that OS is also perpendicular to MW since it bisects WV.

Let's label the midpoint of WV as point N. Then we can use the Pythagorean theorem to find MW.

First, we need to find the length of ON. Since OS is a radius of the circle, it is equal to the radius of the circle, which we can call r. Then, using the Pythagorean theorem, we have:

ON^2 = OS^2 - SN^2

ON^2 = r^2 - (WV/2)^2

ON^2 = r^2 - (MW/2)^2 (since NW = MV)

Next, we need to find the length of MN. We know that OM is half of WV, so OM = WV/2. Then, using the Pythagorean theorem again, we have:

MN^2 = ON^2 + OM^2

MN^2 = r^2 - (MW/2)^2 + (WV/2)^2

MN^2 = r^2 - (MW/2)^2 + (2MW/2)^2 (since WV = 2MW)

MN^2 = r^2 - (MW/2)^2 + MW^2

Finally, we can solve for MW by using the Pythagorean theorem one more time:

MW^2 = MN^2 + NW^2

MW^2 = (r^2 - (MW/2)^2 + MW^2) + (MW/2)^2

MW^2 = r^2 - (MW/2)^2 + MW^2/4 + MW^2/4

MW^2 = r^2 - (MW/2)^2 + MW^2/2

Multiplying both sides by 4 gives:

4MW^2 = 4r^2 - MW^2 + 2MW^2

3MW^2 = 4r^2

MW^2 = 4r^2/3

MW = 2r/sqrt(3)

Therefore, the length of MW is 2r/sqrt(3).

Answer:

Step-by-step explanation:

In the given figure, we have three circles arranged such that they touch each other. Let the centers of these circles be A, B, and C, with radii 5.4, 4.4, and 3.4, respectively.

We can see that triangle ABC is an equilateral triangle, since all sides are of equal length (the radii of the circles).

To find the angle A, we can use the law of cosines, which states that:

c^2 = a^2 + b^2 - 2ab cos(C)

where a, b, and c are the lengths of the sides of a triangle, and C is the angle opposite the side of length c.

Since triangle ABC is equilateral, we have a = b = c, and C = 60°. Therefore, we can rewrite the above equation as:

c^2 = 2a^2 - 2a^2 cos(60°)

Simplifying and solving for a, we get:

a = c / sqrt(3)

Substituting the given values, we have:

a = 4.4 / sqrt(3) ≈ 2.54

Therefore, angle A is:

A = 180° - 60° - 60° = 60°

And angle B is:

B = 180° - A = 120°

Finally, we can use the law of sines to find the length of side c:

sin(A) / a = sin(B) / b = sin(C) / c

Substituting the values we have found, we get:

sin(60°) / 2.54 = sin(120°) / c

Simplifying and solving for c, we get:

c = 2.54 / sqrt(3) / sin(120°) ≈ 3.71

Therefore, the length of side c is approximately 3.71, and angle B is 120°.

Angela's friend earned a salary of $950 plus a commission of $1,599.50 for a total earnings of $2,549.50 in the month they both sold $23,800 worth of merchandise.

To find out how much Angela's friend earned in the same month, we need to first calculate their commission earnings.

Angela's commission earnings can be found by multiplying the total sales by her commission rate of 11%:

Commission earnings = $23,800 x 0.11 = $2,618

Now, let's calculate her friend's commission earnings. First, we need to subtract the salary from the total sales:

Total sales - Salary = Commissionable sales
$23,800 - $950 = $22,850

Next, we can calculate the commission earnings by multiplying the commissionable sales by the commission rate of 7%:

Commission earnings = $22,850 x 0.07 = $1,599.50

Adding the commission earnings to the salary gives us the total earnings for the month:

Total earnings = $950 + $1,599.50 = $2,549.50

Therefore, Angela's friend earned $2,549.50 for the same month.
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Answer: The length is 8 yards

Step-by-step explanation: First, take the volume of the prism (115 cubic yards), divide it by the width (2 1/2), the divide that by the height (5 3/4) getting you the length: 8 yards

The statements that are true regarding a traditional individual retirement account are:

• People can contribute to the account until retirement age.

• Contributions to the account are limited each year.

• Contributions reduce taxable income.

Taxable income is the portion of an individual's income that is subject to taxation by the government. It is calculated by subtracting all allowable deductions, exemptions, and credits from an individual's gross income.

The other two statements are not true regarding a traditional individual retirement account:

Employers do not create them and match employee contributions. This describes a different type of retirement account, such as a 401  or a 403.

People cannot withdraw money penalty-free at any time. There are penalties for withdrawing money from a traditional IRA before the age of 59 ½, with certain exceptions.

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The equation of the line perpendicular to the given line passing through the point   [tex](-2, 3)[/tex] is  [tex]y = -3/2 x + 15/2[/tex] , which is not one of the options provided.

To find the equation of a line parallel to a given line, we need to use the fact that parallel lines have the same slope.

The given line has a slope of  [tex]2/3,[/tex]so the parallel line we're looking for will also have a slope of [tex]2/3[/tex]. Using the point-slope form of a line, we can write:

[tex]y - y_{1} = m(x - x_{1} )[/tex]

where m is the slope and [tex](x_{1} , y_{1} )[/tex] is the given point. Substituting the values we have:

[tex]y - 3 = (2/3)(x - (-2))[/tex]

[tex]y - 3 = 2/3 x + 4/3[/tex]

[tex]y = 2/3 x + 13/3[/tex]

So the equation of the line parallel to the given line passing through the point  [tex]t (-2, 3) is y = 2/3 x + 13/3[/tex], which is option A.

To find the equation of a line perpendicular to a given line, we need to use the fact that perpendicular lines have slopes that are negative reciprocals of each other.

The given line has a slope of 2/3, so the perpendicular line we're looking for will have a slope of -3/2. Using the point-slope form of a line again, we can write:

[tex]y - y_{1} = m(x - x_{1} )[/tex]

where m is the slope and  [tex](x_{1} , y_{1} )[/tex] is the given point. Substituting the values we have:

[tex]y - 3 = (-3/2)(x - (-2))[/tex]

[tex]y - 3 = -3/2 x - 9/2[/tex]

[tex]y = -3/2 x + 15/2[/tex]

Therefore,  the equation of the line perpendicular to the given line passing through the point [tex](-2, 3) is y = -3/2 x + 15/2,[/tex] which is not one of the options provided.

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At a 5% level of significance, we reject the null hypothesis and conclude that there is evidence to support the alternative hypothesis that the percentage of vegetarians in age group 18 to 35 years is higher than 5%.

To test the hypothesis that the percentage of vegetarians is higher in the age group 18 to 35 years at a 5% level of significance, we can use a hypothesis test with the following null and alternative hypotheses:

Null hypothesis (H0): The percentage of vegetarians in the age group 18 to 35 years is equal to 5%.

Alternative hypothesis (Ha): The percentage of vegetarians in the age group 18 to 35 years is greater than 5%.

We can conduct a one-tailed z-test to test this hypothesis, using the following formula:

z = (p - P0) / sqrt(P0 * (1 - P0) / n)

where:

p is the sample proportion of vegetarians in the age group 18 to 35 years

P0 is the hypothesized proportion (5%)

n is the sample size

We will reject the null hypothesis if the calculated z-value is greater than the critical z-value corresponding to a 5% level of significance (one-tailed test).

Assuming a sample of size n = 100, if we find that 10 people in the sample identify themselves as vegetarians, then the sample proportion is:

p = 10/100 = 0.1

Using the formula above, we can calculate the z-value:

z = (0.1 - 0.05) / sqrt(0.05 * 0.95 / 100) = 1.96

The critical z-value for a one-tailed test at a 5% level of significance is 1.645 .

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The required dimensions of the kennel are 17 feet by 8.5 feet.

Let the length of the kennel be L and the width be W.

We know that the total length of fence available is 25 feet. Since one side of length 9 feet does not need fencing, the total length of the other three sides that need fencing is (L + 2W - 9).

Therefore, we have:

25 = L + 2W - 9

Simplifying the equation, we get:

L + 2W = 34

We also know that the area of the kennel is given by:

Area = Length x Width

Substituting L = 34 - 2W from the first equation into the above equation, we get:

Area = (34 - 2W) x W

Simplifying the equation, we get:

Area = 34W - 2W²

To maximize the area, we differentiate the above equation with respect to W, set it equal to zero, and solve for W:

d(Area)/dW = 34 - 4W = 0

Solving for W, we get:

W = 8.5

Substituting this value of W back into the equation L + 2W = 34, we get:

L + 2(8.5) = 34

L + 17 = 34

L = 17

Therefore, the dimensions of the kennel are 17 feet by 8.5 feet.

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The conclusion about p using an absolute value inequality is in the range of 29.8% to 34.2%.

What is absolute value inequality?

An expression using absolute functions and inequality signs is known as an absolute value inequality.

We know that the absolute value inequality about p using an absolute value inequality is written as,

[tex]|p-\hat{p}|\leq E[/tex]

where E is the margin of error and is the sample proportion.

Now, it is given that the poll result of 72% with a margin of error of 4% indicates that p is most likely to be between 68% and 76%. Therefore, p can be written as,

[tex]|p-0.72|\leq 0.04\\(0.72-0.04)\leq p\leq (0.72+0.04)\\\\0.68 \leq p \leq 0.76[/tex]

Thus, the p is most likely to be between the range of 68% to 76%.

Similarly, the proportion of people who like dark chocolate more than milk chocolate was 32% with a margin of error of 2.2%. Therefore, p can be written as,

[tex]|p-0.32| \leq 0.022\\\\0.248 \leq p \leq 0.342[/tex]

Thus, the p is most likely to be between the range of 29.8% to 34.2%.

Hence, the conclusion about p using an absolute value inequality is in the range of 29.8% to 34.2%.

Learn more about Absolute Value Inequality:

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