Read each question. Then write the letter of the correct answer on your paper.

Which of the following is the compound inequality that describes the range of the following function?

(F) -3
(H) 0 ≤ y ≤3 (I) 5 ≤y ≤6

The value of g(27) function is 67.

Since we know that g is a periodic function with a period of 24, we can determine the function value by considering the equivalent point within one period. To find g(27), we need to find equivalent point within one period. Since the period is 24, we can subtract the multiples of 24 from 27 to obtain a value within one period.

g(27) = 27 - 24 = 3

g(3)=67 ------ (given)

g(27) = g(3) = 67

Therefore, the value of g(27) is similar to g(3) which is 67.

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The surface area of the hemisphere after rounding to the nearest tenth is 161.5 [tex]cm^2[/tex].

We are given the circumference of the great circle of the hemisphere and we have to find the surface area of the hemisphere. The circumference of the great circle of the hemisphere is given as 26 cm. Now, to find the surface area, we will first determine the radius of the hemisphere and then apply the formula for surface area.

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

2[tex]\pi[/tex]r = 26

[tex]\pi[/tex]r = 13

r = 13/[tex]\pi[/tex]

Now, we know the radius and we will apply the formula for the area of the hemisphere.

A = 1/2(4[tex]\pi[/tex][tex]r^2[/tex]) + [tex]\pi[/tex][tex]r^2[/tex]

A = 1/2(4[tex]\pi[/tex]([tex]\frac{13}{\pi}[/tex][tex])^2[/tex]) + [tex]\pi[/tex]([tex]\frac{13}{\pi }[/tex][tex])^2[/tex]

= 2(169/[tex]\pi[/tex]) + (169/[tex]\pi[/tex])

= 3(169/[tex]\pi[/tex])

= 161.46 [tex]cm^2[/tex]

Therefore, the surface area of the hemisphere after rounding to the nearest tenth is 161.5 [tex]cm^2[/tex].

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A. The sum of the angle measures of triangle XYZ is always equal to 180 degrees.

B. Triangle XYZ is a two-dimensional geometric shape formed by three line segments, XY, YZ, and XZ, which connect three points, X, Y, and Z.

In any triangle, the sum of the interior angles is always equal to 180 degrees.

This property is known as the angle sum property of triangles.

Therefore, regardless of the specific values of the angles in triangle XYZ, their sum will always be 180 degrees.

This property holds true for all triangles in Euclidean geometry.

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The exact value of cos 75° is (√6 - √2)/4 or 0.2588190.

The sides and angles of a right-angled triangle are dealt with in Trigonometry. The ratios of acute angles are called trigonometric ratios of angles. The six trigonometric ratios are sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant (sec).

We have to find the exact value of cos 75.

So, The value of cos 75 degrees in decimal is 0.258819045.

Then,

cos 75°

= cos (1.3089)

= (√6 - √2)/4 or 0.2588190

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The graph of the inverse variation y = -16/x is attached below

An equation is an expression that shows the relationship between two or more numbers and variables.

An inverse variation is in the form:

y ∞ 1/x

y = k/x

Where k is the constant

Given that y = 8, when x = -2, hence:

8 = k / (-2)

k = -16

The equation becomes y = -16/x

The graph of the equation is attached

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The correct solution is: α = {π/3, 2π/3, 4π/3, 5π/3}

This is because the equation tan(α) = √3 has a period of π, and the tangent function repeats every π radians. In the interval [0, 2π), we find all the angles that satisfy tan(α) = √3, which are π/3, 2π/3, 4π/3, and 5π/3.

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In conclusion, the formula an = 3n(n+1) is an explicit formula, and the first five terms of the sequence are 6, 18, 36, 60, and 90.

The formula an = 3n(n+1) represents an explicit formula for the sequence. The first five terms of the sequence can be determined by substituting values of n from 1 to 5 into the formula.

An explicit formula directly expresses the nth term of a sequence in terms of n, without reference to previous terms. In the given formula an = 3n(n+1), the value of the nth term can be determined by substituting the value of n into the formula.

To find the first five terms of the sequence, we substitute values of n from 1 to 5 into the formula:

a1 = 3(1)(1+1) = 6

a2 = 3(2)(2+1) = 18

a3 = 3(3)(3+1) = 36

a4 = 3(4)(4+1) = 60

a5 = 3(5)(5+1) = 90

Therefore, the first five terms of the sequence are 6, 18, 36, 60, and 90, respectively.

In conclusion, the formula an = 3n(n+1) is an explicit formula, and the first five terms of the sequence are 6, 18, 36, 60, and 90.

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Expanding (2+t)⁴ using Pascal's Triangle gives the result 16 + 32t + 24t² + 8t³ + t⁴.

To expand (2+t)⁴ using Pascal's Triangle, we can utilize the binomial theorem. The fourth row of Pascal's Triangle is 1 4 6 4 1.

These numbers represent the coefficients of each term in the expansion. The general formula for expanding a binomial raised to the power of n is:

(2+t)⁴ = 1*(2)⁴*(t)⁰ + 4*(2)³*(t)¹ + 6*(2)²*(t)² + 4*(2)¹*(t)³ + 1*(2)⁰*(t)⁴

= 1*(16)(1) + 4(8)(t) + 6(4)(t)² + 4(2)(t)³ + 1(1)*(t)⁴

= 16 + 32t + 24t² + 8t³ + t⁴

Simplifying this expression gives the expanded form of (2+t)⁴. In this case, it is 16 + 32t + 24t² + 8t³ + t⁴.

Each term is obtained by multiplying the corresponding coefficient from Pascal's Triangle with the appropriate powers of 2 and t.

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There are no valid solutions within the specified range of 0 ≤ θ < 2π for the given equation sinθ = -sinθ cosθ.

The equation sinθ = -sinθ cosθ does not have a solution for θ within the specified range of 0 ≤ θ < 2π. This equation leads to a contradiction and does not satisfy any valid values of θ.

Let's analyze the given equation sinθ = -sinθ cosθ:

We can rearrange the equation to isolate the terms involving θ:

sinθ + sinθ cosθ = 0

Factor out sinθ from the left side:

sinθ(1 + cosθ) = 0

To solve this equation, we set each factor equal to zero:

sinθ = 0   or   1 + cosθ = 0

For the first factor, sinθ = 0, the solutions lie at θ = 0 and θ = π since sinθ is equal to zero at these values within the given range.

For the second factor, 1 + cosθ = 0, we can solve for cosθ by subtracting 1 from both sides:

cosθ = -1

The solution for cosθ = -1 lies at θ = π.

However, when we substitute these values back into the original equation sinθ = -sinθ cosθ, we find that it does not hold true. Therefore, there are no valid solutions within the specified range of 0 ≤ θ < 2π for the given equation sinθ = -sinθ cosθ.

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the derivative of y = -[tex]x^2[/tex] + 5x + 2 is  dy/dx = -2x + 5. . For y = 2[tex]x^2[/tex] - 8x + 10 the  dy/dx = 4x - 8. for y = -2[tex]x^2[/tex]+ 9x - 1 the  dy/dx = -4x + 9, after differentiation.

a. To find the derivative of y = -[tex]x^2[/tex] + 5x + 2, we differentiate each term with respect to x. The power rule states that for a term of the form [tex]x^n,[/tex] the derivative is n*[tex]x^(n-1)[/tex]. Applying this rule, we get:

dy/dx = -2x + 5

b. For y = 2[tex]x^2[/tex]- 8x + 10, we again differentiate each term using the power rule:

dy/dx = 4x - 8

c. Lastly, for y = -2[tex]x^2[/tex]+ 9x - 1, we differentiate each term:

dy/dx = -4x + 9

In each case, we obtain the derivative of y with respect to x. The resulting derivatives represent the instantaneous rate of change of y with respect to x at any given point on the curve. They also indicate the slope of the tangent line to the curve at that point. By finding the derivatives, we gain insight into the behavior and characteristics of the functions, such as the direction of increasing or decreasing values and the presence of maximum or minimum points.

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When two variables have a relationship of inverse variation, their product remains constant. In the given relationship, the product of xy is constant, indicating that the relationship is an inverse variation.

Inverse variation is a relationship between two variables where the product of their values remains constant. In this relationship, one variable increases while the other decreases.

For example, if y is inversely proportional to x, then the product xy is constant. The question asks to determine whether the given relationship is an inverse variation or not. Given: y 424 280 210 x 2 3 4

The product xy can be calculated as follows: 2 * 424 = 8483 * 280 = 8404 * 210 = 840.

Since the product of xy is constant, we can conclude that the relationship between x and y is an inverse variation.

Therefore, the answer is "The product xy is constant, so the relationship is an inverse variation."

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To model the total cost of the group's tickets, we can use an equation that relates the number of tickets to the cost per ticket. Let's assume the group consists of "n" friends. The equation to represent the total cost (C) of the group's tickets can be written as:

C = 8.00n

Here, "C" represents the total cost, and "n" represents the number of friends in the group. Since each ticket costs $8.00, multiplying the number of tickets by the cost per ticket gives us the total cost of the group's tickets. For example, if there are 5 friends in the group, substituting n = 5 into the equation yields:

C = 8.00 * 5

C = 40.00

Thus, the total cost of the group's tickets would be $40.00.

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The given statement that the geometric mean for consecutive positive integers is the mean of the two numbers is never true. The geometric mean and the mean have different mathematical definitions and yield different results for consecutive positive integers.

The given statement states that the geometric mean for consecutive positive integers is equal to the mean of the two numbers.

To determine the validity of this statement, let's consider the definitions of the geometric mean and the mean.

The geometric mean of two numbers is the square root of their product. So, for consecutive positive integers, if we have two consecutive integers, n and n+1, their product is n(n+1), and the geometric mean is √(n(n+1)).

The mean of two numbers is the sum of the numbers divided by 2. For two consecutive positive integers, the mean would be (n + (n+1))/2 = (2n+1)/2 = n + 0.5.

Now, let's compare the geometric mean and the mean for consecutive positive integers:

Geometric Mean: √(n(n+1))

Mean: n + 0.5

We can see that the geometric mean and the mean are not equal for consecutive positive integers. The geometric mean involves the square root of the product, while the mean is simply the sum divided by 2.

Therefore, the given statement that the geometric mean for consecutive positive integers is the mean of the two numbers is never true. The geometric mean and the mean have different mathematical definitions and yield different results for consecutive positive integers.

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The product of matrices A and B, AB, is -4.

To find the product AB of matrices A and B, we need to perform matrix multiplication. Matrix multiplication involves taking the dot product of each row in matrix A with each column in matrix B.

Given:

A = [3 -1 2 0]

B = [1 3 -2 2]

To calculate AB, we multiply each element of each row in matrix A by the corresponding element in each column of matrix B, and then sum up the results.

Matrix A has dimensions 1x4 (1 row and 4 columns), and matrix B has dimensions 1x4 as well. Therefore, the resulting matrix AB will have dimensions 1x1 (1 row and 1 column).

Calculating AB:

AB = (3 * 1) + (-1 * 3) + (2 * -2) + (0 * 2)

= 3 - 3 - 4 + 0

= -4

Therefore, the product of matrices A and B, AB, is -4.

The resulting matrix AB is a 1x1 matrix, meaning it has only one entry. In this case, that entry is -4.

It's important to note that the order of matrix multiplication matters, and in this case, since both A and B are 1x4 matrices, the result is a scalar (single value) rather than a matrix.

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Standard form of quadratic equation : y = ax² + bx + c

Example of quadratic equation : x² - 7x + 10

Let us take the quadratic equation in x,

Standard form of quadratic equation : y = ax² + bx + c

Here,

a = coefficient of x²

b = coefficient of x

c = constant term

Now

Let us take an example of quadratic equation ,

Equation : x² - 7x + 10

To get the value of x factorize the above quadratic equation ,

x² - 2x -5x + 10 = 0

x(x-2) -5(x-2) = 0

(x-5)(x-2) = 0

Thus the values of x are 5 , 2 .

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An appropriate method to solve the system of equations is the substitution method. The solution to the system of equations is x = -6 and y = 4.

To solve the system using substitution, we can start by solving one of the equations for one variable and then substitute that expression into the other equation.

Let's solve the first equation for x:

x + 3y = 6

x = 6 - 3y

Now, substitute this expression for x in the second equation:

4(6 - 3y) - 2y = -32

Simplify:

24 - 12y - 2y = -32

Combine like terms:

-14y = -56

Divide both sides by -14:

y = 4

Now, substitute the value of y back into the first equation to solve for x:

x + 3(4) = 6

x + 12 = 6

x = 6 - 12

x = -6

Therefore, the solution to the system of equations is x = -6 and y = 4.

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The number of students that can go on the field trip is at most 40 x ≤ 40 students.

Let's denote the number of students as "x."

According to the given information, the cost of the field trip is $220 plus $7 per student. Therefore, the total cost can be expressed as:

Total cost = $220 + $7x

The problem states that the school can spend at most $500. To represent this as an inequality, we can set up the following equation:

Total cost ≤ $500

Substituting the expression for the total cost:

$220 + $7x ≤ $500

Now, let's solve the inequality for the number of students, x:

$7x ≤ $500 - $220

$7x ≤ $280

Divide both sides of the inequality by 7:

x ≤ $280 / $7

x ≤ 40

Therefore, the number of students that can go on the field trip is at most 40 students.

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The percentage increase per year in the winner's check over the given period. If the winner's prize increases at the same rate, it will be $1,454,735,139.69 in 2040.

To calculate the percentage increase per year in the winner's check over the period from 1895 to 2007, we can use the following formula:

Percentage Increase = (Final Value - Initial Value) / Initial Value * 100

a. Calculating the percentage increase:

Initial Value = $140

Final Value = $1,171,000

Percentage Increase = (1,171,000 - 140) / 140 * 100 ≈ 835,714.29%

b. To estimate the winner's prize in 2040, we can assume the same annual percentage increase will continue. We need to calculate the number of years from 2007 to 2040 and apply the percentage increase to the 2007 prize.

Number of years = 2040 - 2007 = 33 years

Estimated prize in 2040 = 1,171,000 * (1 + (Percentage Increase / 100))^33

Estimated prize in 2040 = 1,171,000 * (1 + (835,714.29 / 100))^33 ≈ $1,454,735,139.69

Therefore, if the winner's prize increases at the same rate, it is estimated to be approximately $1,454,735,139.69 in 2040.

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The length of the  springboard which has 6-inch coils and forms an angle of 14.5° with the base is 24.77  inches.

Sine function of an angle is the ratio between the opposite side length to that of the hypotenuse.

Let's denote the length of the springboard as "L."

Consider a  trigonometric function

to find the value, consider a Sine function

[tex]sin(14.5^0) = \dfrac{6 inches} { L}[/tex]

The value of the unknown variable [tex]L[/tex] is

[tex]L =\dfrac{6 inches }{ sin(14.5^0)}[/tex]

[tex]L = 24.77 inches[/tex]

Therefore, the length of the springboard, rounded to two decimal places, is approximately [tex]24.77[/tex] Inches.

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Answer:

Correct option is C)

Given, sides of prism are 3 cm, 4 cm and 5 cm and height =10 cm

Let s be the semi-perimeter of the triangular base of the prism.

Then S=

2

3+4+5

=6 cm

Therefore, the area of the prism =

s(s−a)(s−b)(s−c)

=

6(6−3)(6−4)(6−5)

=

6×3×2×1

=

36

=6 sq. cm.

Then volume of the prism =area of base×height

= 6×10

= 60 cu.cm

The triangle is a right triangle with sides 3cm, 25cm, and 26cm. By using the formula for the area of a right triangle, we find that the area of the cross section of the right prism is 37.5 cm squared.

In this problem, you are asked to find the area of a cross section of a right prism, where the base is a triangle. The sides of the triangle given are 3 cm, 25 cm, and 26 cm. Based on those measurements, we can identify that this is a right triangle.

A right triangle can be identified when the square of the largest side (in this case 26 cm) is equal to the sum of the squares of the other two sides (3 cm and 25 cm). This is known as the Pythagorean theorem. So, 26^2 = 3^2 + 25^2, which is 676 = 9 + 625, thus confirming that these side lengths form a right triangle.

Now, to find the area of a right triangle, we use the following formula: (1/2) * base * height. Here, we can use 3 cm as the base and 25 cm as the height. Substituting those values in the formula gives: (1/2) * 3 * 25 = 37.5 cm2. So, the area of the cross section of the right prism is 37.5 cm2.

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The standard form of the equation of the circle passing through (-8,14) with the center at the origin is x² + y² = 320.


In the standard form of the equation of a circle, the equation is given as (x – h)² + (y – k)² = r², where (h, k) represents the coordinates of the center of the circle and r represents the radius. In this case, since the center is at the origin (0, 0), the equation simplifies to x² + y² = r².

To find the radius, we can use the distance formula between the origin and the given point (-8, 14). The distance is sqrt((-8 – 0)² + (14 – 0)²) = sqrt(64 + 196) = sqrt(260) = 2sqrt(65). Squaring this radius gives r² = (2sqrt(65))² = 260. Therefore, the equation of the circle is x² + y² = 260.

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The total number of questions on the survey is given as follows:

60 questions.

The total number of questions on the survey is obtained applying the proportions in the context of the problem.

We have that 35% of the total number of questions x is equivalent to 21 questions, hence the total number of questions on the survey is obtained as follows:

0.35x = 21

x = 21/0.35

x = 60 questions.

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Answer:

The geometric mean of a and b is d.

Step-by-step explanation:

To find the y-coordinate of the point of intersection of the two lines, we need to solve the system of equations formed by (1) and (2).

Given the equations:

(1) ax + by + c = 0

(2) bx + cy + d = 0

We are also given the conditions: E = F = 0.

To solve for the point of intersection, we can eliminate one variable (either x or y) by multiplying one equation by a suitable constant to make the coefficients of either x or y equal in magnitude but opposite in sign.

Let's eliminate x by multiplying equation (1) by b and equation (2) by -a:

b(ax + by + c) = 0

-a(bx + cy + d) = 0

Simplifying, we get:

abx + b^2y + bc = 0

-abx - acy - ad = 0

Adding these two equations together, we have:

(b^2 - ab)x + (bc - ac)y + (bc - ad) = 0

Since E = F = 0, we can conclude that (bc - ad) = 0. This condition implies that either b = 0 or c = 0.

If b = 0, then the line represented by (1) is a vertical line. In this case, we cannot find the point of intersection as it does not exist.

Therefore, the correct option is:

The geometric mean of a and b is d.

If the expression is undefined,  sin (-330°) = 1/2.

The value of sin (-330°) can be evaluated using the unit circle or the periodicity property of the sine function.

First, let's convert -330° to its equivalent angle within one revolution:

-330° = -360° + 30°

Since the sine function has a periodicity of 360°, we can rewrite -330° as:

-330° = 30°

Now, we can evaluate the sine of 30°, which is a well-known value:

sin(30°) = 1/2

Therefore, sin (-330°) = 1/2.

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The \underline{center} of a regular polygon is the distance from the middle to the circle circumscribed around the polygon.

The sentence is false.

Here, we have,

The center of a regular polygon is the point equidistant from all the vertices of the polygon, not the distance from the middle to the circle circumscribed around the polygon.

A revised true sentence would be:

The center of a regular polygon is the point equidistant from all the vertices of the polygon.

Hence, The \underline{center} of a regular polygon is the distance from the middle to the circle circumscribed around the polygon.

The sentence is false.

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The expanded and simplified form of f(x) is 3ln(x) + 2ln(x − 2).

To expand and simplify the expression f(x) = ln(x^5 − 4x^4 + 4x^3), we'll start by factoring the expression inside the natural logarithm:

f(x) = ln(x^5 − 4x^4 + 4x^3)

    = ln(x^3(x^2 − 4x + 4))

Next, we'll simplify the expression inside the logarithm using the fact that x^2 − 4x + 4 is a perfect square trinomial: f(x) = ln(x^3(x − 2)^2)

Now, we can use the properties of logarithms to expand the expression further. The property we'll use is ln(a * b) = ln(a) + ln(b)

f(x) = ln(x^3) + ln((x − 2)^2)

Finally, applying the power rule of logarithms, which states that ln(a^b) = b * ln(a): f(x) = 3ln(x) + 2ln(x − 2)

So the expanded and simplified form of f(x) is 3ln(x) + 2ln(x − 2).

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The determinant of the matrix [-1 3 5 2] is -17.To evaluate the determinant of the matrix [-1 3 5 2], we can use the formula for a 2x2 matrix.

| a  b |

| c  d |

The determinant of the matrix is calculated as ad - bc.

In this case, the matrix is [-1 3 5 2], so we have:

a = -1

b = 3

c = 5

d = 2

Substituting these values into the determinant formula:

|-1  3 |

| 5  2 |

The determinant is (-1 * 2) - (3 * 5) = -2 - 15 = -17.

Therefore, the determinant of the matrix [-1 3 5 2] is -17.

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The estimator proposed by Alice for estimating the unknown parameter c is the maximum of the observed data points. We need to determine whether the bias of this estimator, denoted as Ĉ, is nonnegative, nonpositive, or zero.

To analyze the bias of the estimator, we compare its expected value, E(Ĉ), with the true value of the parameter c. Since the sample space of Y is a closed interval [0, c], the maximum value among the n data points cannot exceed c. Therefore, we have Ĉ ≤ c with probability one.

Using the fact that if a random variable is smaller than or equal to a number with probability one, its expectation is also smaller than or equal to that number, we can conclude that E(Ĉ) ≤ c.

From this inequality, we can infer that the bias of the estimator, which is given by E(Ĉ) - c, is nonpositive or zero. If the expected value of the estimator is equal to c, the bias is zero. If the expected value is less than c, the bias is nonpositive.

In conclusion, the bias of Alice's estimator, Ĉ, is nonpositive or zero.

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The type of bias that is exemplified in the case above is: Sampling bias

Sampling bias occurs when a given section of a population is more likely to be chosen over others. This means that the sample chosen for the test is not representative of the entire population.

So, this is the case with the teacher who only selects the students that attended her 9 a.m., class instead of the entire students.

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Complete Question:

A health teacher wishes to do research on the weight of college students. She obtains the weights for all the students in her 9 A.M. class by looking at their driver's licenses or state ID's.

What is the type of bias?

- nonresponse bias

- sampling bias

- response bias

-4 is less than √-4 or 2i, with -4 being a real number and √-4 being an imaginary number.


When comparing -4 and √-4, we need to consider that √-4 is the square root of -4, which is a complex number.

The square root of a negative number involves the use of imaginary numbers.

In this case, √-4 can be written as 2i, where i is the imaginary unit (√-1).

Comparing -4 and 2i, we can see that -4 is a real number, while 2i is an imaginary number.

In the real number system, -4 is less than any positive number, including imaginary numbers.

Therefore, we can conclude that -4 is less than √-4 or 2i.

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