A 1.5 liter (1500ml) bottle of soda will make about ? servings of 0.25 liter (250ml).

Answer:

h ≤ 6

Step-by-step:

Let h be the height of the triangular pennant in inches.

The formula for the area of a triangle is:

A = 1/2 * base * height

We know that the base of the triangle is 10 inches, so we can substitute this value into the formula:

A = 1/2 * 10 * h

Simplifying this equation, we get:

A = 5h

We also know that the area of the pennant must be at most 30 square inches. So we can write:

A ≤ 30

Substituting the formula for the area, we get:

5h ≤ 30

Dividing both sides by 5, we get:

h ≤ 6

Therefore, the possible heights of the triangle must be at most 6 inches in order for the area of the pennant to be at most 30 square inches.

The inequality that describes the possible heights of the triangle is:

h ≤ 6

If Erin and Christa were working on cubing binomials for math homework. Erin's shortcut does not always work and should not be used.

Erin's shortcut is not correct. To see why, let's expand (a+b)³ using the binomial theorem: :

(a+b)³ = (a+b)(a+b)(a+b)

= (a+b)(a²+2ab+b²)

= a³ + 2a²b + ab² + a²b + 2ab² + b³

= a³ + 3a²b + 3ab² + b³

So the correct expansion of (a+b)³ has four terms, not two as Erin's shortcut suggests. Therefore, Erin's shortcut does not always work and should not be used.

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Simplified expression of (5t^(3))/(-25t)| is (1/5)( [tex]t^{2}[/tex] ).

To simplify the expression |(5[tex]t^{(3))}[/tex]/(-25t)|, we need to follow these steps:

1. Start by simplifying the fraction inside the absolute value signs.

2. The 5 in the numerator and the 25 in the denominator can be reduced to 1/5.

3. The t in the denominator can be reduced with one of the t's in the numerator, leaving us with [tex]t^{2}[/tex] in the numerator. This gives us |(1/5)( [tex]t^{2}[/tex]))|.

5. The absolute value signs mean that we need to take the positive value of whatever is inside.

6. Since both 1/5 and [tex]t^{2}[/tex] are positive, we can remove the absolute value signs.

   This leaves us with (1/5)( [tex]t^{2}[/tex])) as our simplified expression.

So the final answer is (1/5)( [tex]t^{2}[/tex] ).

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The solution to the system of equations: x = 1 and y = 3

We have that the equations are;

y = −x^2 + 4     (1)

y = 2x + 1         (2)

Now, equate the two equations, we get;

-x² + 4 = 2x + 1

collect the like terms

-x² - 2x + 4 - 1

subtract the values

-x² - 2x + 3

Make into standard form

x² + 2x - 3 = 0

Solve the quadratic equation

Find the pair factors of -3 that sums up to 2, we have;

x² +3x - x - 3 = 0

factor the terms

x(x + 3) - 1(x + 3) = 0

(x - 1) = 0

Make 'x' the subject

x = 1

x + 3 = 0

x = -3

Then, y = 2(1) + 1 = 3

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The least common multiple (LCM) of the factored polynomials x(x-2) and x(x+1) is x(x-2)(x+1).

The Least Common Multiple (LCM) of two or more polynomials is the smallest polynomial that is a multiple of all the given polynomials. To find the LCM of the factored polynomials x(x-2) and x(x+1), we need to follow these steps:

1. Identify the common factors of the polynomials. In this case, the common factor is x.
2. Multiply the common factor by the remaining factors of each polynomial. In this case, we have (x-2) and (x+1).
3. Multiply the remaining factors together to get the LCM. In this case, we have (x-2)(x+1).
4. Finally, multiply the common factor by the product of the remaining factors to get the LCM. In this case, we have x(x-2)(x+1).

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The complete proof that ΔUWX ≅ ΔWUV is explained below.

Congruent triangles are set of given triangles which have equal values of corresponding properties. Thus the triangles have equal dimensions and measure of internal angles.

The two column complete proof required are given below:

              STATEMENT                             REASONS

1. WX ║ UV                                               Given

2. < V ≅ <X                                               Given

3. <VUW ≅ <UWX                                    Definition of alternate angles.

4. UW ≅ UW                                             Reflexive property of congruence

5. ΔUWX ≅ ΔWUV                                   Angle-Side-Angle (AAS) property

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Step-by-step explanation:

A + B = 437

A - B = 415 (for that sequence it is important to know that A is larger than B).

A = 415 + B

that we use now in the first equation :

415 + B + B = 437

2B = 22

B = 11

A = 415 + B = 415 + 11 = 426 tons

B = 11 tons

Answer:

w=137°

Step-by-step explanation:

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The results of the SPSS INDEPENDANT T TEST can help to determine whether the effect of sugar on activity is statistically significant.

Yes, sugar has an effect on activity. In order to determine this, you can use an SPSS INDEPENDANT T TEST. This is a statistical test used to determine whether there is a significant difference between two groups of data, in this case, the control (water) and experimental (juice) groups.

The T Test requires that the two data sets are of equal length and are both normally distributed. The data provided above meet these criteria. The results of the T Test will tell us if the difference between the two sets is statistically significant at the 5% level (α=.05).

Therefore, the results of the SPSS INDEPENDANT T TEST can help to determine whether the effect of sugar on activity is statistically significant.

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Answer: 54 cups of sugar

Step-by-step explanation:

part A  add 3 each time

1=3

2=6

3=9

4=12

5=15

6=18

Part B

1x3 per pie

so if there is 70 pies do 70x3

Part C]

if you do 18 and since you need 3 cups of sugar per pie

do 18 times 3 to get a total of 54 cups of sugar

18x3=54

The sampling method of (a) is systematic sampling, and (b) is cluster sampling.

It is an advanced form of simple random sampling. In this form, from the population of members, sample members are selected randomly. In this random selection, a starting point is fixed. The sample is created when the members are selected from the fixed intervals.

a. The sampling method described is systematic sampling, where every 45th customer that purchases a movie ticket is selected for the survey.

b. The sampling method described is cluster sampling, where the theater is divided into sections (clusters) and one cluster is randomly selected for the survey. Then, every customer within that cluster is polled.

Therefore, the sampling method of (a) is systematic sampling, and (b) is cluster sampling.

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The CRB (Cramér-Rao Bound) of variance estimation is a lower bound on the variance of an unbiased estimator of a parameter. The CRB of variance estimation for this system is (02 +0)^2/(02 +0 + (02 +0)^2). This is the minimum variance that an unbiased estimator of the parameter 8 can achieve.

In this case, the parameter we are trying to estimate is 8. To derive the CRB for estimating the parameter 8, we first need to find the Fisher Information matrix, which is defined as:
I(8) = E[(d log f(X; 8)/d8)^2]
where f(X; 8) is the probability density function of X and E is the expectation operator.

Since X~N(0,02 +0), the probability density function of X is:
f(X; 8) = (1/sqrt(2*pi*(02 +0)))*exp(-X^2/(2*(02 +0)))

Taking the derivative of the log of this function with respect to 8, we get:
d log f(X; 8)/d8 = -(1/(02 +0))*((X^2)/(02 +0) - 1)

Squaring this and taking the expectation, we get:
I(8) = E[(1/(02 +0))^2*((X^2)/(02 +0) - 1)^2]

Simplifying and using the fact that E[X^2] = 02 +0, we get:
I(8) = (1/(02 +0))^2*(02 +0 + (02 +0)^2)

Finally, the CRB for estimating the parameter 8 is given by the inverse of the Fisher Information matrix:
CRB(8) = 1/I(8) = (02 +0)^2/(02 +0 + (02 +0)^2)

Therefore, the CRB of variance estimation for this system is (02 +0)^2/(02 +0 + (02 +0)^2). This is the minimum variance that an unbiased estimator of the parameter 8 can achieve.

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We can be 90% confident that the true population variance of unemployed workers' math test scores falls between 65.61 and 197.57.

The first step in finding a 90% confidence interval for the population variance is to find the degrees of freedom for the sample. In this case, the degrees of freedom is 18 - 1 = 17.

Next, we need to find the critical value for a 90% confidence interval with 17 degrees of freedom. We can do this using a chi-squared distribution table. The critical values for a 90% confidence interval with 17 degrees of freedom are 8.671 and 27.488.

Now we can use the formula for a confidence interval for the population variance:

CI = [(n-1) * s²] / X²

Where n is the sample size, s is the sample standard deviation, and X^2 is the critical value from the chi-squared distribution table.

Plugging in the values we have:

CI = [(17) * (10.4)²] / X²

For the lower bound of the confidence interval, we use the smaller critical value:

CI = [(17) * (10.4)²] / 8.671

CI = 197.57

For the upper bound of the confidence interval, we use the larger critical value:

CI = [(17) * (10.4)²] / 27.488

CI = 65.61

So the 90% confidence interval for the population variance is (65.61, 197.57).

In conclusion, we can be 90% confident that the true population variance of unemployed workers' math test scores falls between 65.61 and 197.57.

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[tex]~~~~~~ \textit{Continuously Compounding Interest Earned Amount} \\\\ A=Pe^{rt}\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill & \$5000\\ r=rate\to 2.5\%\to \frac{2.5}{100}\dotfill &0.025\\ t=years\dotfill &7 \end{cases} \\\\\\ A = 5000e^{0.025\cdot 7} \implies A=5000e^{0.175} A \approx 5956.23[/tex]

Answer:

The formula for calculating the value of an investment that grows continuously is:

A = Pe^(rt)

Where:

A is the final amount

P is the principal amount

e is Euler's number (approximately 2.71828)

r is the annual interest rate (as a decimal)

t is the time in years

In this case, P = 5000, r = 0.025 (2.5% expressed as a decimal), and t = 7. Plugging these values into the formula, we get:

A = 5000 * e^(0.025*7) = 5000 * e^0.175 = 5000 * 1.19128 = 5956.40

Therefore, Ryan's investment will be worth $5,956.40 after 7 years. Rounded to the nearest cent, the answer is $5,956.40.

(a) (i) Φ((x + iy) + (u + iv)) = Φ((x + u) + (y + v)i) = y + v

Φ(x + iy) + Φ(u + iv) = y + v

Since these two expressions are equal, Φ is a homomorphism.

(ii) Φ(x + iy) = 0

y = 0
(iii)  The image of Φ is the set of all real numbers, we have C/R ~= R.

(b)  The centralizer of S in G is the set {1, r^2}.

This means that the kernel of Φ is the set of all complex numbers with an imaginary part of 0, which is the set of all real numbers. That is, ker Φ = R.

The fundamental theorem of Group Hormomorphisms states that for any group homomorphism Φ: G -> H, the kernel of Φ, ker Φ, is a normal subgroup of G, and the image of Φ, im Φ, is a subgroup of H. Furthermore, G/ker Φ is isomorphic to im Φ.

(a) (i) To show that Φ is a homomorphism, we need to show that it preserves the group operation. That is, for any two elements x + iy and u + iv in C, we need to show that Φ((x + iy) + (u + iv)) = Φ(x + iy) + Φ(u + iv). Using the definition of Φ, we have:

Φ((x + iy) + (u + iv)) = Φ((x + u) + (y + v)i) = y + v

Φ(x + iy) + Φ(u + iv) = y + v

Since these two expressions are equal, Φ is a homomorphism.

(ii) The kernel of Φ, ker Φ, is the set of all elements in C that are mapped to the identity element in R, which is 0. That is, ker Φ = {x + iy : Φ(x + iy) = 0}. Using the definition of Φ, we have:

Φ(x + iy) = 0

y = 0

This means that the kernel of Φ is the set of all complex numbers with an imaginary part of 0, which is the set of all real numbers. That is, ker Φ = R.

(iii) To prove that C/ker Φ is isomorphic to R, we can use the fundamental theorem of Group Hormomorphisms. Since ker Φ = R, we have C/R ~= im Φ. Since the image of Φ is the set of all real numbers, we have C/R ~= R.

(b) The centralizer of a subset S of a group G, denoted by C_G(S), is the set of all elements in G that commute with every element in S. That is, C_G(S) = {g : g in G, gs = sg for all s in S}. For the given groups G = D6 and S = {1,r^2}, we have:

C_G(S) = {g : g in G, g1 = 1g and gr^2 = r^2g}

= {g : g in G, g = g and gr^2 = r^2g}

= {1, r^2}

Thus, the centralizer of S in G is the set {1, r^2}.

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The sοlutiοn tο the given prοblem οf the triangle cοmes οut tο be triangle side lengths are multiples οf triangle side lengths is untrue.

A triangular is a pοlygοn because it has twο οr maybe mοre additiοnal sectiοns. It has a straightfοrward square fοrm. Only the edges A, B, but alsο C distinguishes a triangular frοm a parallelοgram. When the sides are nοt exactly cοllinear, Euclidean geοmetry prοduces a singular plane instead οf a cube. If a shape has three edges and three angles, it is said tο be triangular.

Here,

A cοllectiοn οf the three pοsitive integers a, b, and c knοwn as a Pythagοrean triple fulfill the fοrmula a² + b² = c², where c is the hypοtenuse length οf a right triangle οf legs οf length a and b.

Triangle has sides that are 6, 8, and 10 in length. The Pythagοrean Theοrem is satisfied by these three numbers:

6² + 8² = 36 + 64 = 100 = 10²

Triangle is a right triangle as a result, and its side lengths make a Pythagοrean triple.

Hence, The statement that triangle side lengths are multiples οf triangle side lengths is untrue.

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The volume of a rectangular prism is -

V = Length x width x height

V = L x B x H

Volume is a collection of two - dimensional points enclosed by a single dimensional line. Mathematically, we can write -

V = ∫∫∫ F(x, y, z) dx dy dz

Given is a rectangular prism.

The volume of a rectangular prism is the measurement of the total space inside it. Since the image of the prism is not given, we can write the volume of a rectangular prism as -

V = Length x width x height

V = L x B x H

Therefore, the volume of a rectangular prism is -

V = Length x width x height

V = L x B x H

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The transformation is (a) Graph opens down, it moves 2 units left and 5 units down.

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

y = -|x + 2| - 5

Where the parent function is

y = |x|

When we modify the equation to y = -|x + 2| - 5, we are applying several transformations to the parent graph:

So the resulting graph (a)

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The algebraic expression indicating the number of cookies remaining to be baked is 144 - x.

To find the number of cookies Pat still needs to bake, we need to subtract the number of cookies she has already baked from the total number of cookies she needs to bring to the party. In this case, the total number of cookies is 144 and the number of cookies she has already baked is represented by the variable x.

Therefore, the algebraic expression for the number of cookies Pat still needs to bake is:

This expression tells us that to find the number of cookies Pat still needs to bake, we need to subtract the number of cookies she has already baked (x) from the total number of cookies she needs to bring (144).

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a) is linearly dependent and set b) is linearly independent.

By inspection, we can determine whether the given sets are linearly dependent by looking at the relationship between the vectors in each set. If one vector can be written as a linear combination of the other vectors in the set, then the set is linearly dependent.

For set a) \( \left\{\left[\begin{array}{l}1 \\ 3\end{array}\right],\left[\begin{array}{l}2 \\ 6\end{array}\right]\right\} \), we can see that the second vector is simply 2 times the first vector. This means that the second vector can be written as a linear combination of the first vector, and therefore the set is linearly dependent.

For set b) \( \left\{\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right],\left[\begin{array}{l}0 \\ 1 \\ 0\end{array}\right],\left[\begin{array}{l}0 \\ 0 \\ 1\end{array}\right]\right\} \), we can see that none of the vectors can be written as a linear combination of the other vectors in the set. This means that the set is linearly independent.

Therefore, by inspection, we can determine that set a) is linearly dependent and set b) is linearly independent.

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To factor the expression[tex]9n^(4)-16p^(2)[/tex], we need to recognize that it is a difference of squares. A difference of squares can be factored into the product of two binomials, one with a sum and one with a difference. In this case, we can factor [tex]9n^(4)-16p^(2)[/tex] into [tex](3n^(2)+4p)(3n^(2)-4p)[/tex].

The final factored form of  [tex]9n^(4)-16p^(2)[/tex] is [tex](3n^(2)+4p)(3n^(2)-4p).[/tex]

So, the factors of the expression are the two binomials [tex](3n^(2)+4p[/tex]) and [tex](3n^(2)-4p)[/tex].

Here is the step-by-step explanation:

1. Recognize that the expression is a difference of squares.
2. Write the expression as the product of two binomials, one with a sum and one with a difference.
3. Simplify the binomials if necessary.
4. The final factored form is [tex](3n^(2)+4p)(3n^(2)-4p).[/tex]

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Answer:

1/4

Step-by-step explanation:

The area of the shaded segment is:= 16π - 32 cm² (exact value)

= 30.849 cm² (approximate value, rounded to three decimal places)

Area is the measurement of the surface inside a two-dimensional figure. It is expressed in square units, such as square meters or square inches.

To find the area of the shaded segment, we need to subtract the area of the triangle formed by the two radii and the chord from the area of the sector.

formula:

A = (θ/360)πr²

where A is the area of the sector, θ is the central angle in degrees, π is pi (approximately 3.14), and r is the radius of the circle.

Here, the central angle is 90 degrees and the radius is 8 cm.

area of sector is:

sector = (90/360)π(8)²

= 16π cm²

To find the area of the triangle, we need to find its base and height. The base is the chord, which is also the diameter of the circle and has a length of 16 cm (twice the radius). The height is half of the length of the chord (since the central angle is 90 degrees), which is 4 cm.

So the area of the triangle is:

A_triangle = (1/2)bh

= (1/2)(16)(4)

= 32 cm²

the area of shaded segment is:

shaded = sector - triangle

= 16π - 32

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The solution(s) of the equation are x = 0 and the solutions of the cubic equation x^(3)+3x^(2)+25x+75 = 0.

The solution(s) of the equation can be found by rearranging the terms and then factoring. Here are the steps:

Step 1: Rearrange the terms to have all the x terms on one side of the equation:
-4x^(4)+25x^(2)+75x+5x^(4)+3x^(3) = 0

Step 2: Combine like terms:
x^(4)+3x^(3)+25x^(2)+75x = 0

Step 3: Factor out the common factor of x:
x(x^(3)+3x^(2)+25x+75) = 0

Step 4: Use the zero product property to find the solutions:
x = 0 or x^(3)+3x^(2)+25x+75 = 0

The first solution is x = 0. The other solutions can be found by solving the cubic equation x^(3)+3x^(2)+25x+75 = 0. This equation does not have any rational solutions, so the solutions will be irrational or complex.

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The triangle solved  using the Law of Sines. D=10,∠A=30 4, and, C=120 4 where a = 5.77 c = 10 i = 30°

To solve the triangle, we need to use the formula:
a/sinA = b/sinB = c/sinC

Given that D=10, ∠A=30°, and C=120°, we can plug in these values into the formula and solve for the unknown sides:

a/sin30° = 10/sin120°

Cross-multiplying and solving for a gives us:

a = 10*sin30°/sin120°

a = 5.77

Similarly, we can solve for the other unknown side, c:

c/sinC = 10/sin120°

c = 10*sinC/sin120°

c = 10*sin120°/sin120°

c = 10

Finally, we can solve for the unknown angle, i, using the fact that the sum of the angles in a triangle is 180°:

i = 180° - 30° - 120°

i = 30°

Therefore, the solution to the triangle is:

a = 5.77
c = 10
i = 30°

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Total number of books in the library 1920

The linear equations in one variable is an equation which is expressed in the form of ax+b = 0, where a and b are two integers, and x is a variable and has only one solution.

Let the total number of books be x

50 % of the total books are of Marathi = x*50/100 = x/2

Books of English are 1/3rd of Marathi = (x/2)*(1/3)

                                                              = x/6

Books on Mathematics are 25 % of the English books = (x/6)*(25/100)

                                                                                          = x/24

Remaining books = 560

Now, according to the question.

⇒ x/2 + x/6 + x/24 + 560/1 = x/1

⇒ (12x + 4x + x + 13440)/24 = x/1

⇒ 17x + 13440 = 24x

⇒ 24x - 17x = 13440

⇒ 7x = 13440

⇒ x = 13440/7

⇒ x = 1920

So, there are 1920 books in the the library.

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The polynomial expression in standard form is -13x^(4)+5x.

A polynomial is an expression consisting of variables (or indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

The polynomial expression in standard form is -13x^(4)+6x-x. To write a polynomial expression in standard form, we need to rearrange the terms in descending order of their degree (exponent). We also need to combine any like terms.

First, we can rearrange the terms in descending order of their degree:
-8x^(4)-5x^(4)+6x-x

Next, we can combine the like terms:
-13x^(4)+6x-x

Finally, we can simplify the expression:
-13x^(4)+5x

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Answer:

D. 432

Step-by-step explanation:

To find out this question you need to find out how many cookies you have of each type. we know we have 120 chocolate chip cookies. There are 42 fewer oatmeal cookies than chocolate chip cookies so we take 120 and minus 42.

120 - 42 = 78

There are 78 oatmeal cookies. now we want to find out how many peanut butter cookies we have. There are 3 times as many peanut butter cookies than oatmeal cookies so we times 77 with 3.

78 × 3 = 234

Now we know how many cookies are in all the types, we need to find the total number of all cookies in the grocery store.

120 + 78 + 234 = 432

therefore the answer will be D. 432

Hope this helps :)

The equation that is equivalent to 52 = -4 (2n + 1) is C) -52/4 = 2n + 1.

Starting with the given equation:

52 = -4 (2n + 1)

First, we can simplify the right-hand side of the equation by distributing the -4:

52 = -8n - 4

Then, we can isolate the variable (n) by adding 4 to both sides of the equation:

56 = -8n

Finally, we can solve for n by dividing both sides by -8:

-7 = n

Therefore, we have found that the solution to the equation 52 = -4 (2n + 1) is n = -7.

Now, let's check each of the answer choices to see if they are equivalent to this solution:

A) 13 = -2 (2n + 1)

If we simplify the right-hand side of this equation, we get:

13 = -4n - 2

Then, if we add 2 to both sides and divide by -4, we get:

-3.75 = n

This solution is not equivalent to n = -7, so this equation is not equivalent to the original equation.

B) -7 = -2 (2n + 1)

If we simplify the right-hand side of this equation, we get:

-7 = -4n - 2

Then, if we add 2 to both sides and divide by -4, we get:

1.25 = n

This solution is not equivalent to n = -7, so this equation is not equivalent to the original equation.

C) -52/4 = 2n + 1

If we simplify the left-hand side of this equation, we get:

-13 = 2n + 1

Then, if we subtract 1 from both sides and divide by 2, we get:

-7 = n

This solution is equivalent to n = -7, so this equation is equivalent to the original equation.

D) -13 = -4 (2n + 1)

If we simplify the right-hand side of this equation, we get:

-13 = -8n - 4

Then, if we add 4 to both sides and divide by -8, we get:

1.875 = n

This solution is not equivalent to n = -7, so this equation is not equivalent to the original equation.

Therefore, the equation that is equivalent to 52 = -4 (2n + 1) is C) -52/4 = 2n + 1.

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