There are three possible cases (or scenarios) for how many solutions that an absolute value equation could have. How many solutions are there for each case? Why are their differences in the number of solutions? Give a mathematical example in your explanation.
PLS HELP ASAP

The functions y(x) = [tex]e^{4x}[/tex] and y(x) = [tex]e^{8x}[/tex] are therefore solutions of the specified differential equation.The values of mi = 4 and 8.

A differential equation is an equation with one or more terms and one or more derivatives of the dependent variable with regard to the independent variable. (i.e., independent variable) dy/dx equals f(x) Here, the independent variable "x" and the dependent variable "y" are both present. So, dy/dx equals 5x, for instance.

We are given with differential equation

[tex]\frac{d^{2}y }{dx^{2} } - 12 \frac{dy}{dx} + 32y = 0[/tex]

We need to find the values of m1 and m2

[tex]y(x) = e^{m1x} and y(x) = e^{m2x}[/tex]are solutions to the differential equation.

First, we solve the differential equation for [tex]y(x) = e^{mx}[/tex]to obtain:

[tex]m^{2} e^{mx} - 12me^{mx} + 32 e^{mx} = 0[/tex]

By factoring out e(mx) from the left side, we obtain

[tex]e^{mx} (e^{mx} - 12m + 32) = 0[/tex]

The number enclosed in parentheses must be equal to zero for e(mx) to be a non-zero solution of the differential equation. To resolve the quadratic problem, we must:

[tex]m^{2} - 12m + 32 = 0[/tex]

The quadratic algorithm yields:

[tex]m = (12 ± √(144 - 4*32))/2\\m = (12 ± \sqrt{\frac{(144 - 4 * 32)}{2} } \\[/tex]

mi = 6 ± 2

Therefore, m1 and m2 have the following values:

m1 = 4

m2 = 8

The functions y(x) = [tex]e^{4x}[/tex]and y(x) = [tex]e^{8x[/tex]are therefore solutions of the specified differential equation.

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Complete question:

The complete question is in the image form.

So, the value of the car after 13 years is approximately $13,330.

In mathematics, an initial value refers to the value of a function or a variable at a specific starting point or initial condition. For example, when solving a differential equation, the initial value is the value of the function at the starting point or initial time.

In the context of calculus, the initial value of a function f(x) at a point x=a is denoted by f(a). In the context of differential equations, the initial value of a function y(x) at a point x=a is denoted by y(a) or y(a) = y0.

he given exponential function can be written as:

[tex]v(t) = 25,900(0.95)^t[/tex]

where t is the age of the car in years.

To find the initial value of the car, we need to evaluate v(0):

[tex]v(0) = 25,900(0.95)^0[/tex]

[tex]v(0) = 25,900(1)[/tex]

[tex]v(0) = 25,900[/tex]

So, the initial value of the car is $25,900.

To find the value of the car after 13 years, we need to evaluate v(13):

[tex]v(13) = 25,900(0.95)^{13}[/tex]

v (13) ≈ 13,330

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Answer:

A. 6, 7 , 8, 9 and 15

B. 1, 3, 5, 7 and 19

Step-by-step explanation:

please see the attached I put.

Answer:

Step-by-step explanation:

divide both by 100 to get

3072/1

answer: 3072

Answer:

-2x × 10^1 + 8 × 10^0

or

-2x × 10 + 8 × 1

Step-by-step explanation:

Multiply the following to put them in standard form

-4 (5x-2)=

-20x +8

standard form

-2x × 10^1 + 8 × 10^0

or

-2x × 10 + 8 × 1   (remember  PEMDAS)

Answer:1,208 square centimeters

Step-by-step explanation:

Step-by-step explanation:

because the circle arcs are equal, it also means that the lengths of the chords (WV and XY) are equal.

9x - 34 = 4x + 1

5x - 34 = 1

5x = 35

x = 35/5 = 7

For given dataset,  the p-value is greater than the significance level of 0.01, we fail to reject the null hypothesis.

What exactly is a data set?

A dataset is a collection of data points or observations that are related to each other in some way. Each data point in a dataset represents a single instance or measurement of some variable or phenomenon that is being studied.

A dataset can take many different forms depending on the type of data being collected and the purpose of the study.

Now,

To test the claim that 3% of users develop nausea, we will use a hypothesis test with a significance level of 0.01.

Let p = true proportion of users who develop nausea.

Null Hypothesis: H0: p = 0.03

Alternative Hypothesis: H1: p > 0.03 (one-tailed test)

We will use the normal approximation to the binomial distribution since np and n(1-p) are both greater than 10, where n is the sample size and p is the hypothesized proportion.

The test statistic is calculated as follows:

z = (P - p)/√(p(1-p)/n)

Here P = sample proportion and n = sample size.

The sample proportion is:

P = 160 / 6209 = 0.0258

Substituting the values, we get:

z = (0.0258 - 0.03) / √(0.03 x 0.97 / 6209) = -1.72

Using a z-table or calculator, we find that the p-value for this test is approximately 0.043. Since the p-value is greater than the significance level of 0.01, we fail to reject the null hypothesis.

Therefore, we do not have sufficient evidence to support the claim that more than 3% of users develop nausea. However, we should still be cautious about this adverse reaction and continue monitoring it.

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The area of the shaded region is 1.916 ft²

Area is a measure of the amount of space inside a two-dimensional shape, such as a square, rectangle, triangle, circle, or any other shape that has a length and a width. It is usually measured in square units, such as square inches, square feet, or square meters.

To calculate the area of the shaded region, we need to find the area of the entire circle and subtract the area of the unshaded portion.

The central angle corresponding to the shaded region is 110 degrees. The entire circle has a central angle of 360 degrees.

The unshaded portion of the circle is the sector that corresponds to the central angle of 360 - 110 = 250 degrees. The area of a sector of a circle with radius r and central angle θ is given by the formula:

A = (θ/360)πr²

Substituting θ = 110 degrees and r = 2 ft, we get:

A = (110/360)π(2)

A =  (0.305)π(2)

A =  (0.305)π(2)

A = 0.61π

A = 1.916 ft²

Therefore, the area of the shaded region is 1.916 ft².

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Complete question:

Calculate the area of the shaded region. the apothem is 2 ft, and the shaded area is 110 degrees. the figure is a circle

Graph with Numbers

The coordinate plane is a two-dimensional plane that is divided into four quadrants. The upper-right quadrant is known as quadrant one, and the following quadrants go in counter-clockwise order: quadrant two, quadrant three, quadrant four (see image below). Quadrants are typically labeled using Roman numerals (I, II, III, and IV). The coordinate plane is comprised of two axes that are perpendicular lines, meaning when they are joined together, they create four right angles. The x-axis is the horizontal axis, and the y-axis is the vertical axis. The point where the two axes are joined together is called the origin. The coordinates of the origin are (0, 0). This means that the x-coordinate is 0 and the y-coordinate is 0. Ordered pairs are always written in the format (x, y).

The coordinate plane has four quadrants, a horizontal x-axis and vertical y-axis, and the origin is the point in the center.

The left side of the x-axis contains negative numbers. Traveling to the left on the coordinate plane indicates the x-value is getting smaller. The right side of the x-axis contains positive numbers. Traveling to the right on the coordinate plane indicates the x-value is getting larger. The upper half of the y-axis contains positive numbers. Traveling upward on the coordinate plane indicates the y-value is getting larger. The lower half of the y-axis contains negative numbers. Traveling downward on the coordinate plane indicates the y-value is getting smaller.

In the following image, point A is graphed on the coordinate plane. The coordinates for point A are (3, 2). This means that the x-value is 3 and the y-value is 2. To graph point A on the coordinate plane, start at the origin and go three spaces to the right (because the x-value, 3, is positive) and two spaces upward (because the y-value, 2, is positive). Since the x-value and y-value are both positive, point A will be in quadrant I.

Graphing the point (3, 2)

In order to graph the point (3, 2), move left three spaces and up two spaces on the coordinate plane.

The solution of the equation which is a vertex, from plot of the equation is

In mathematics, a vertex (plural: vertices) is a point where two or more lines, curves, or edges meet. The term is commonly used in geometry, graph theory, and other areas of mathematics.

In geometry, a vertex is a point where two or more edges of a polygon or a polyhedron meet. For example, a triangle has three vertices, while a cube has eight vertices.

The graph is attached and other vertices shown

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We are 90% confident that the true proportion of surveys returned is between 0.4330 and 0.4882.

What is percentage?

Percentage is a way of expressing a number or proportion as a fraction of 100. It is represented by the symbol "%".

(a) First, we need to calculate the sample proportion, denoted by $\hat{p}$. The sample proportion is the number of surveys returned divided by the total number of surveys sent:

p = 1111/ 2410 ≈ 0.4606

(b) The standard error of the sample proportion is:

SE = p(1−p)/n = 0.4606(1−0.4606)/2410≈0.0168

(c) To find the margin of error at 90% confidence level, we need to use the z-score associated with a 90% confidence interval, which is 1.645 (obtained from a standard normal table or calculator). The margin of error is:

ME = z∗×SE = 1.645 × 0.0168 ≈ 0.0276

(d) Finally, we can construct the 90% confidence interval for the proportion of returned surveys by subtracting and adding the margin of error to the sample proportion:

p ± ME = 0.4606 ± 0.0276 =(0.4330,0.4882)

Therefore, we are 90% confident that the true proportion of surveys returned is between 0.4330 and 0.4882.

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The solution to the triangle is: a ≈ 22.4, b = 4 , c ≈ 77.2 ,A = 27° ,B ≈ 42.5°

C ≈ 110.5°  we got the answer by using law of sine

what is  law of sine?

The Law of Sines, also known as the Sine Rule, is a mathematical formula used in trigonometry to relate the sides and angles of a non-right-angled triangle. It states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle.

In the given question,

Using the law of sines and the fact that the angles of a triangle sum up to 180 degrees, we can solve for the missing parts of the triangle:

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

sin(27)/a = sin(B)/42 = sin(C)/72

We can use the first equation to solve for a:

a = b(sin(A)/sin(B)) = 42(sin(27)/sin(B)) ≈ 22.4161

Next, we can use the second equation to solve for sin(B):

sin(B) = b(sin(A)/a) = 42(sin(27)/a) ≈ 0.6822

Taking the inverse sine, we find that B ≈ 42.4827 degrees.

Now we can use the fact that the angles of a triangle sum up to 180 degrees to solve for C:

C = 180 - A - B ≈ 110.5173 degrees.

Finally, we can use the law of sines again to solve for the remaining side, c:

c = a(sin(C)/sin(A)) = 22.4161(sin(110.5173)/sin(27)) ≈ 77.2352

Therefore, the solution to the triangle is:

a ≈ 22.4

b = 42

c ≈ 77.2

A = 27°

B ≈ 42.5°

C ≈ 110.5°

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Answer: (-3, 0)

Step-by-step explanation:

         First, we will graph these equations. See attached. One has a y-intercept of 3 and then moves right one unit for every up down (we get this from the slope of 1). The other has a y-intercept of 4, and then moves down two units for every unit right (we get this from the slope of 1).

         The point of intersection is the solution, this is the point at which both graphed lines cross each other. Our solution is:

                         (-3, 0)     x = -3, y = 0

Answer:

(-3, 0)

Step-by-step explanation:

When we graph the equations:

y = x + 3,

y = -2 - 6,

we can see that they intersect at (-3, 0).

Therefore, that is the solution to the system.

— Extra note —

When graphing a line from a slope-intercept form equation (y = mx + b), we know the following:

m is the line's slope (rise over run)

b is the y-coordinate of the line's y-intercept

Answer:

the bird will need to fly 4feet closer to the flower

Answer:

2

Step-by-step explanation:

2 appears the most as we can see theres only

two 0

and few of the other numbers

mode is the most common number in the list of any given numbers like the one abpve

Expressing the resuIt in οne significant figure, we get: 1x10^22

The number οf digits in a particuIar vaIue οr measurement required tο determine the accuracy and precisiοn οf measurement are knοwn as significant figures (οr significant digits).

The resuIt οf[tex](5.01\times10^{21})* 3[/tex] is [tex]1.50\times10^{22}.[/tex]

Tο express the answer in οne significant figure withοut rοunding, we need tο determine the first significant figure in the resuIt.

In scientific nοtatiοn, the first significant figure is the first nοn-zerο digit. In this case, the first nοn-zerο digit is 1.

Therefοre, expressing the resuIt in οne significant figure, we get:

[tex]1\times10^{22[/tex]

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a)  Population proportion is the sample proportion: p' = 108/4319 ≈ 0.025. b) 0.007. c) (0.018, 0.032) d) Option A is correct statement.

In statistics, the standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. It is also called the Z distribution, where Z represents the standard normal variable.

The standard normal distribution is a continuous probability distribution that is symmetric around the mean of 0. The shape of the distribution is bell-shaped, with the majority of the data falling within 3 standard deviations from the mean.

The standard normal distribution is widely used in statistics and probability theory as a benchmark for comparing other normal distributions. It is particularly useful for converting other normal distributions into a standardized form, which allows for easier comparison and analysis.

a) The best point estimate of the population proportion is the sample proportion: p' = 108/4319 ≈ 0.025.

b) The margin of error E is given by E = z* ×√(p'(1-p')/n), where z* is the critical value of the standard normal distribution for a 99% confidence level. Using a table or calculator, we find that z* ≈ 2.576. Substituting the given values, we have E ≈ 0.007.

c) The confidence interval is given by p' ± E, or 0.025 ± 0.007. Using a calculator, we find that the lower bound is approximately 0.018 and the upper bound is approximately 0.032. Therefore, the 99% confidence interval for the proportion of adverse reactions is (0.018, 0.032).

d) A. One has 99% confidence that the interval from the lower bound to the upper bound actually does contain the true value of the population proportion.

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Answer: 28/52

Step-by-step explanation: its easy

Answer:

28/52. You will get it right.

Answer: -1 Is the answer.

The total area of construction paper needed for the project is:

180.25 + 127.5833... = 307.8333... square inches (rounded to four decimal places)

Area is the measure of the amount of surface inside a closed boundary, usually measured in square units. It is used to express the size or extent of a two-dimensional region or shape, such as a square, rectangle, circle, or triangle.

To find the total area of the construction paper needed for the project, we need to find the area of each rectangle and then add them together.

For the first rectangle with dimensions of 12 inches by 15 1/4 inches, the area is:

12 × 15 1/4 = 180 + 45/4 = 180.25 square inches

For the second rectangle with dimensions of 10 1/3 inches by 10 1/4 inches, the area is:

10 1/3 × 10 1/4 = (31/3) × (41/4) = 127.5833... square inches (rounded to four decimal places)

Therefore, the total area of construction paper needed for the project is:

180.25 + 127.5833... = 307.8333... square inches (rounded to four decimal places)

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Assuming you meant "x - 2(x - 2) * 3" as the expression, here is how you can simplify it:

= x - 2(3x - 6) // Distribute the 2

= x - 6x + 12 // Distribute the -2

= -5x + 12 // Simplify

Therefore, the simplified expression is -5x + 12.

To simplify the expression "x subtract 2 x subtract 2 multiply 3," we need to follow the order of operations, which is Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

First, we need to simplify the multiplication by 3 inside the parentheses:

Next, we can simplify the subtraction of 2x:

-x - 2(3)

Finally, we can simplify the multiplication of 2 and 3:

-x - 6

Therefore, the simplified expression is "-x - 6".

Therefore, the line Ф = [tex]\frac{\pi }{2}[/tex], but not with respect to the pole, is the answer to the given equation's problem.

Complex algorithms frequently employ variable words to demonstrate coherence between two opposing assertions. Equations are academic expressions that are used to demonstrate the equality of different academic figures.

In this case, levelling generates [tex]b+7[/tex] rather than another method that could divide [tex]12[/tex] into two parts, analyse the information provided by [tex]y+7[/tex]and generate [tex]y+7[/tex]

A curve is said to be symmetric with respect to the pole if it does not alter when the pole is reflected across the origin. To test for this symmetry, we replace [tex]r[/tex] with [tex]-r[/tex] and see if the answer is still correct.

∴[tex]r = -3+3cos[/tex]Ф[tex]- r = 3-3(cos[/tex]Ф[tex])[/tex]Ф

Regarding the lineФ[tex]= \frac{\pi }{2}[/tex] there is symmetry. We replace Ф with [tex]\pi[/tex]/Ф and verify to see if the solution holds to see if this symmetry holds.

[tex]r= 3-3cos(\pi -[/tex]Ф[tex])[/tex]

[tex]r= 3-3cos[/tex]Ф

Due to the constant expression, the curve is symmetric with respect to the line Ф = [tex]\pi[/tex]/[tex]2[/tex].

Therefore symmetric with respect to the polar axis and the line Ф = [tex]\pi[/tex]/[tex]2[/tex], but not with respect to the pole.

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Exact Form: 131/12

Decimal Form:10.916

Mixed Number Form: 10 11/12

Answer:

131/12 or 10.91666. or  [tex]10\frac{11}{12}[/tex]

Hope this helps!

Step-by-step explanation:

( [tex]\frac{8}{3}[/tex] × [tex]\frac{11}{2}[/tex] ) - [tex]\frac{15}{4}[/tex]

( [tex]\frac{4}{3}[/tex] × [tex]\frac{11}{1}[/tex] ) - [tex]\frac{15}{4}[/tex]

[tex]\frac{44}{3}[/tex] - [tex]\frac{15}{4}[/tex]

[tex]\frac{176}{12}[/tex] - [tex]\frac{45}{12}[/tex]

[tex]\frac{131}{12}[/tex]

The sides of the triangle are a ≈ 8.3, b = 7, and c = 10, and the angles are A = 51°, B ≈ 51.5°, and C ≈ 77.5°.

A triangle is a closed, double-symmetrical shape composed of three line segments known as sides that intersect at three places known as vertices. Triangles are distinguished by their sides and angles.

To solve the triangle, we need to find the values of the remaining sides and angles. We can start by using the Law of Sines, which states that:

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

where a, b, and c are the sides of the triangle, and A, B, and C are the angles opposite to those sides.

Using the given values, we can write:

a / sin 51 = 7 / sin B

a / sin 51 = 10 / sin C

To solve for a, we can use either of the two equations above. Let's use the first one and solve for sin B:

sin B = (7 / sin 51) * sin B

sin B = 7 / (sin 51 / sin B)

sin B = 7 / sin(180 - 51 - B)

sin B = 7 / sin(129 - B)

Using the sine rule, we can determine the angle B:

sin B / 7 = sin(129 - B) / 10

sin B = (7/10) * sin(129 - B)

sin B = (7/10) * (sin 129 * cos B - cos 129 * sin B)

sin B = (7/10) * sin 129 * cos B - (7/10) * cos 129 * sin B

(7/10 + (7/10) * cos 129) * sin B = (7/10) * sin 129 * cos B

sin B = (7/10) * sin 129 * cos B / (7/10 + (7/10) * cos 129)

sin B = sin 51.5

B = sin(sin B)

B = sin(sin 51.5)

B ≈ 51.5°

We can now find the remaining angle, C:

C = 180 - A - B

C = 180 - 51 - 51.5

C ≈ 77.5°

Finally, we can use the Law of Sines again to find the remaining side, a:

a / sin A = c / sin C

a / sin 51 = 10 / sin 77.5

a = (10 * sin 51) / sin 77.5

a ≈ 8.3

Therefore, the sides of the triangle are a ≈ 8.3, b = 7, and c = 10, and the angles are A = 51°, B ≈ 51.5°, and C ≈ 77.5°.

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Answer:

  "time and a half" workers were paid 50% more per day

Step-by-step explanation:

You want to know the relative daily pay rates of laborers making time and a half versus those who were not. 4 getting time and a half and 20 not earned a total of $2080, while 2 getting time and a half and 15 not earned a total of $1440.

We can let x and y represent the daily pay of time and a half workers, and those not earning time and a half. The two payrolls can be described by ...

  4x +20y = 2080

  2x +15y = 1440

Subtracting the first equation from 2 times the second, we have ...

  2(2x +15y) -(4x +20y)  = 2(1440) -(2080)

  10y = 800 . . . . . . . simplify

  y = 80 . . . . . . . . . . divide by 10

  2x +15(80) = 1440

  2x = 240 . . . . . . . . . . subtract 1200

  x = 120 . . . . . . . . divide by 2

Workers getting time and a half made $120 per day; workers not getting time and a half made $80 per day. Workers getting time and a half made half again as much as those who didn't: $40 per day more, or 50% more, or half-again as much.

__

Additional comment

"Time and a half" means the pay is 50% more. That's what we see here. You don't need any math to say those getting time and a half were getting 50% more per day.

The 2nd graph shows the line of best fit for the data (10, 2) and (6, 4) and (3, 6)

A scatter plot is a type of graph that is used to display the relationship between two sets of data. The data is represented by a collection of points on a two-dimensional plane, with one variable represented on the x-axis and the other variable represented on the y-axis. Each point on the graph represents an individual observation, and the position of the point on the x and y axes represents the values of the two variables for that observation.

Plot the data points on a scatter plot.

Visually identify the overall trend or pattern of the data. Does it appear to be a positive or negative correlation, or is there no correlation at all?

Draw a straight line that appears to fit the trend of the data as closely as possible. This line is your estimate of the line of best fit.

Calculate the equation of the line of best fit using statistical methods such as linear regression. This will give you the most accurate representation of the line of best fit.

Therefore, Plot the line of best fit on the scatter plot to see how well it fits the data.

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Complete question:

Look at the data graphed on the scatter plot below.

(10, 2) and (6, 4) and (3, 6)

Which graph shows the line of best fit for the data?

a. FPCF for this sample is FPCF = √(199/24) = 2.89.

b. The population cannot be considered effectively infinite, so the answer is no.

When the sample size is significant compared to the size of the community, the Finite Population Correction Factor, also known as the FPC factor, is applied. The population in most cases is so vast that typical sample sizes are insufficient to raise concerns about the need for the FPC.

The recommendation is to use the FPC when the sample size to population size ratio is higher than 5%. The FPCF must be used, for instance, if the population number is 300 and the sample size is 30, which results in a ratio of 10%.

a. To calculate the FPCF (finite population correction factor) for this sample, we can use the formula:

FPCF = √(N/n)

where N is the population size, and n is the sample size.

In this case, N = 199 (the number of blood samples in the racks), and n = 24 (the number of samples selected every seventh blood sample).

FPCF = √(199/24) = 2.89 (rounded to two decimal places)

b. To determine whether the population should be considered effectively infinite, we need to compare the sample size to the population size. A general rule of thumb is that the population can be considered effectively infinite if the sample size is less than 5-10% of the population size.

In this case, the sample size is 24, which is approximately 12% of the population size of 199. Therefore, the population cannot be considered effectively infinite, and the FPCF should be applied to adjust for the finite population size.

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Answer:16.7

Step-by-step explanation: So the first step is to add all of the numbers in the second column. (15,18,14,16,19,18). At the end of your problem it says 100. So you have 6 different numbers, and 100. You will do 100 divided by 6. Finally you will get 16.66 and you will just round it.

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Answer:  [tex]\textsf{B (-3), C (3), and D (24)}[/tex]

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Given:  [tex]\textsf{8(x - 3) = 2x + b + 6x}[/tex]

Find: [tex]\textsf{Which values for b are not a solution}[/tex]

Solution: In order to make it easier for us to determine the answer, let us first simplify the expression that was given to us.  We can do that by first combining like terms on the right side of the equation and distributing the 8 on the left side of the equation.

We can now subtract 8x from both sides of the equation.

Now that we know that b is equal to -24, that means that any value other than -24 would cause the equation to have no solutions.  Therefore, the correct answer would be B (-3), C (3), and D (24).