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Therefore, the quotient when x³ - x² - 3x + 2 is divided by x - 2 is x² - 2x - 5 with a remainder of -3.

To find the quotient when x³ - x² - 3x + 2 is divided by x - 2, we will use the long division method. Here is the solution:  

Step 1: The first term of the quotient is x². Multiply x² by x - 2 to get x³ - 2x². Subtract this product from x³ - x² to get: x² - 3x² = -2x²

Step 2: Bring down the next term of the dividend, which is -3x. The new dividend is -2x² - 3x.

Step 3: The second term of the quotient is -2x. Multiply -2x by x - 2 to get -2x² + 4x. Subtract this product from -2x² - 3x to get -5x.

Step 4: Bring down the last term of the dividend, which is +2. The new dividend is -5x + 2. Step 5: The third term of the quotient is -5. Multiply -5 by x - 2 to get -5x + 10. Subtract this product from -5x + 2 to get -3.

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The algebra class of 14 students in the Algebra 2 class of 23 students do not play basketball or baseball.

The number of students in a certain Algebra 2 class of 23 students who play basketball and baseball is 9, given that 7 of them play basketball and 2 of them play baseball.

The number of students in the class who do not play basketball or baseball is given by the number of students who do not play basketball plus the number of students who do not play baseball and the students who do not play basketball or baseball.

However, since each student is either playing basketball or baseball or neither, we can say that the total number of students who do not play basketball or baseball is given by the number of students in the class minus the total number of students who play basketball or baseball.

Thus, the number of students who do not play basketball or baseball is given by:

23 - 9 = 14

Therefore, 14 students in the Algebra 2 class of 23 students do not play basketball or baseball.

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Answer:

The given triangles APSR and APQR are congruent by S.A.S. criteria.

Step-by-step explanation:

The S.A.S. (Side-Angle-Side) congruence criteria states that two triangles are congruent if their corresponding two sides and included angles are congruent to one another's corresponding two sides and included angle.

Here, we can observe that both triangles share the same side AP.

PR and PQ on the side are congruent.

- Both triangles share Angle P.

The supplied triangles APSR and APQR are therefore congruent according to S.A.S. requirements.

so we can conclude: S.A.S. for the triangles being congruent.

The two considered triangles JKL and MNP are similar by the AA rule of similarity as the two angles that is ∠K = ∠N and ∠L = ∠P are there.

Similar triangles are triangles that have the same shape, but their sizes may vary. All equilateral triangles, squares of any side lengths are examples of similar objects. In other words, if two triangles are similar, then their corresponding angles are congruent and corresponding sides are in equal proportion. We denote the similarity of triangles here by ‘~’ symbol.

It is given that two triangles that are JKL and MNP are considered in which:

Now, from ΔJKL and ΔMNP, we have

∠K = ∠N (Given in the question)

∠L = ∠P(Given in the question)

Thus, by AA rule of similarity,

ΔJKL is similar to ΔMNP.

Therefore, the two considered triangles JKL and MNP are similar by the AA rule of similarity as the two angles that is ∠K = ∠N and ∠L = ∠P are there.

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Answer:

Δ ABC [tex]\bold{\sim}[/tex]  ΔPQR is similar.

Step-by-step explanation:

Similar triangles are two or more triangles that have the same shape, but their sides are in proportion.

For Question:

In Δ ABC and ΔPQR

AB:PQ =8:6=4:3

BC: QR=12:9=4:3

AC: PR=12:9=4:3

Since the length of any side of one triangle is by the corresponding side of another triangle, you will get the same number.

Therefore,

Δ ABC [tex]\bold{\sim}[/tex]  ΔPQR is similar.

Hence Proved:

Answer:

ΔABC ~ ΔPQR

Step-by-step explanation:

In similar triangles, corresponding sides are always in the same ratio.

Therefore, if triangle ABC is similar to triangle PQR then:

[tex]AB : PQ = BC : QR = AC : PR[/tex]

Substitute the side lengths into the ratio equation:

[tex]AB : PQ = BC : QR = AC : PR[/tex]

   [tex]8 : 6 \;\;\;\:= \;\;12 : 9 \;\;\;\:= \;\;12 : 9[/tex]

Simplify each ratio by dividing all parts of the ratio by their highest common factor:

[tex]\dfrac{8}{2}:\dfrac{6}{2}=\dfrac{12}{3}:\dfrac{9}{3}=\dfrac{12}{3}:\dfrac{9}{3}[/tex]

[tex]4:3=4:3=4:3[/tex]

As the corresponding sides of triangles ABC and PQR are in the same ratio, this proves that the two triangles are similar.

Let's arrange the steps for solving the expression using the order of operations (also known as PEMDAS/BODMAS):

Subtract: (42-13) = 29

Multiply: 4 × 3 = 12

Exponentiation: 5² = 25

Multiply: 8 × 3 = 24

Divide: 24 ÷ 29 (result from step 1) = 0.8276 (rounded to 4 decimal places)

Add: 0.8276 (result from step 5) + 25 (result from step 3) = 25.8276 (rounded to 4 decimal places)

Add: 25.8276 (result from step 6) + 12 (result from step 2) = 37.8276 (rounded to 4 decimal places)

The steps should be performed in the following order:

Subtract -> Multiply -> Exponentiation -> Multiply -> Divide -> Add -> Add

So the steps should be performed in this order to evaluate the expression correctly.

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Answer:

4,200

Step-by-step explanation:

70s = b

70(60) = b

4200 = b

Helping in the name of Jesus.

The compensation equation for the employee may be stated as follows:

A equation is a mathematical statement that claims the equivalence of two expressions. Equations are used in mathematics, science, engineering, and many other professions to express connections between variables and to solve problems.

If s denotes income, x denotes years of experience, and d denotes years of education.

The employee's remuneration is computed as the basic wage of $35,000 + $2,000 for each year of experience plus $3,000 for each year of study. In deciding an employee's wage, this equation considers both their experience and education.

Therefore, [tex]\text{S}=35000+2000\text{x}+3000\text{D}[/tex] is the compensation equation for the employee.

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Work Shown:

y = 5x+8

0 = 5*8+8

0 = 40+8

0 = 48

The last equation is false, so the original equation is false when x = 8 and y = 0. This means the point (8,0) is NOT found on the line.

Visual confirmation is shown below.

When r is less than 5 and we substitute r = 4 into the expression [3r-15], the result is -3.

The expression [3r-15] represents an algebraic expression that depends on the value of r. The condition given is that r is less than 5. To evaluate this expression, we substitute the value of r into the expression and simplify it.

Given that r is less than 5, let's substitute r = 4 into the expression:

[3(4) - 15]

= [12 - 15]

= -3

Therefore, when r is less than 5 and we substitute r = 4 into the expression [3r-15], the result is -3.

It's important to note that this answer is specific to the given condition that r is less than 5. If the condition changes or if r is greater than or equal to 5, the result of the expression may be different.

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(a) To compute the experimental probability of rolling a 1 or 6 from Latoya's results, we need to determine the number of times a 1 or 6 was rolled and divide it by the total number of rolls Latoya made.

Let's assume that Latoya rolled the dice 100 times and obtained 20 1's and 10 6's. The total number of successful outcomes (rolling a 1 or 6) is 20 + 10 = 30.

Therefore, the experimental probability of rolling a 1 or 6 is 30/100 = 0.3 or 30%.

(b) Assuming the cube is fair, the theoretical probability of rolling a 1 or 6 can be determined by considering the favorable outcomes (rolling a 1 or 6) divided by the total possible outcomes (rolling any number from 1 to 6).

Since there are two favorable outcomes (1 and 6) out of six possible outcomes (1, 2, 3, 4, 5, 6), the theoretical probability is 2/6 = 1/3 or approximately 0.3333.

(c) The correct statement is B. As the number of rolls increases, we expect the experimental and theoretical probabilities to become closer, though they might not be equal. This is due to the law of large numbers, which states that as more trials are conducted, the experimental probability tends to converge towards the theoretical probability. However, it is not necessary for them to be exactly equal. Random variations can cause some discrepancy, but with a larger number of rolls, the experimental probability should approach the theoretical probability more closely.

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The statement "there exists x ∈ S ∖ m" means that there exists an element x that belongs to the set S but does not belong to the set m. In other words, x is an element that is present in S but is not present in m.

The phrase "m ⊊ m ∪ {x} ∈ X" states that the set m is a proper subset of the set m ∪ {x}, and this union belongs to the set X. This implies that the set m ∪ {x} contains all the elements of m along with the additional element x.

The phrase "a contradiction" indicates that the statement or assumption being made leads to a logical inconsistency or contradiction. In this context, the contradiction arises from the fact that the assumption that x is not in m contradicts the statement that m ∪ {x} is a proper superset of m.

Overall, the given statement implies that the existence of an element x in S, which is not in m, leads to a contradiction when considering the relationship between m and m ∪ {x}.

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Isabella's possible numbers can be represented by the compound inequality 15 ≤ 5n - 5 ≤ 50, which in interval notation is [4, 11].

Based on the given information, we can set up a compound inequality to represent the range of numbers that Isabella might be thinking of.

Let's denote the number Isabella is thinking of as 'n'.

The clue states that "5 less than 5 times his number is at least 15 and at most 50."

We can express this as:

15 ≤ 5n - 5 ≤ 50

To solve this compound inequality, we add 5 to all three parts of the inequality:

15 + 5 ≤ 5n - 5 + 5 ≤ 50 + 5

20 ≤ 5n ≤ 55

Finally, dividing all parts of the inequality by 5:

20/5 ≤ n ≤ 55/5

4 ≤ n ≤ 11

Therefore, Isabella's number, 'n', lies in the range [4, 11] in interval notation. In summary, the compound inequality 15 ≤ 5n - 5 ≤ 50 represents the range of numbers that Isabella might be thinking of. The interval notation [4, 11] indicates that her number could be any value between 4 and 11, inclusive.

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Answer:

x < –18 or x > 10

Step-by-step explanation:

|x + 4| – 2 > 12

x + 4 - 2 > 12

x + 2 > 12

x > 10

-x - 4 - 2 > 12

-x - 6 > 12

-x > 18

x < 18

So, the answer is x < –18 or x > 10

Answer: D:  x < –18 or x > 10

Step-by-step explanation:

In 5 years from 2020, the two colleges will have the same enrollment and At that time, both colleges will have an enrollment of 21,200 students.

To determine when the two colleges will have the same enrollment, we can set up an equation based on the given information.

Let's assume "x" represents the number of years from 2020.

For College A, the projected enrollment can be expressed as:

Enrollment_A = 18,700 + 500x

For College B, the projected enrollment can be expressed as:

Enrollment_B = 26,200 - 1000x

To find when the two enrollments will be equal, we set Enrollment_A equal to Enrollment_B and solve for "x":

18,700 + 500x = 26,200 - 1000x

Combining like terms:

1500x = 7,500

Dividing both sides by 1500:

x = 5

Therefore, in 5 years from 2020, the two colleges will have the same enrollment.

To find the enrollment at that time, we substitute "x = 5" into either of the enrollment equations. Let's use Enrollment_A:

Enrollment_A = 18,700 + 500(5)

Enrollment_A = 18,700 + 2500

Enrollment_A = 21,200

At that time, both colleges will have an enrollment of 21,200 students.

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Answer: 4 pages

Step-by-step explanation:

To solve this problem, we need to use the formula:

Rate = Output/Time

Let's use "p" to represent the number of pages Ethan can type and "t" to represent the time he has before the meeting starts.

Rate = 2 pages/(1/8 hour) = 16 pages/hour

Since the meeting starts 3/4 hour later than the scheduled time, Ethan has t = 1 - 3/4 = 1/4 hour to type pages before the meeting starts.

Output = Rate * Timep = (16 pages/hour) * (1/4 hour) = 4 pages

Therefore, Ethan can type 4 pages before the meeting starts.

Answer: 4 pages

Answer:

The interval from -12 to -7

The interval from 6 to 12.

Step-by-step explanation:

The equation is log(x + 9) + log(x 9) = log[(x + 9)(x 9)]. = log(x² − 81)

Here is the equation:

log(x² - 81) = 0.47712

This can be expressed in exponential form:

x² - 81 = 10^(0.47712)

x² = 81 + 3.02343

x² = 84.02343

x ≈ ± 9.1658

The answers are therefore x -9.1658 and x 9.1658.

These intervals are where these solutions occur:

The answer, x -9.1658, can be found in the range from -12 to -7.

- The answer x = 9.1658 is found in the range of 6 to 12.

The appropriate choices are:

- The range between -7 and -12; - The range between 6 and 12.

1) The probability distribution of the average weekly earnings for employees in general automotive repair shops is the sampling distribution of the sample mean. According to the Central Limit Theorem, if the sample size is large enough, the sampling distribution of the sample mean is approximately normal, with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size.

2) To find the probability that the average weekly earnings is less than $445, we can standardize the sample mean and use a z-table. The z-score for $445 is calculated as follows: z = (445 - 450) / (50 / sqrt(100)) = -1. Using a z-table, we find that the probability that the average weekly earnings is less than $445 is approximately 0.1587.

3) Since we are dealing with a continuous distribution, the probability that the average weekly earnings is exactly equal to any specific value is zero.

4) To find the probability that the average weekly earnings is between $445 and $455, we can subtract the probability that it is less than $445 from the probability that it is less than $455. The z-score for $455 is calculated as follows: z = (455 - 450) / (50 / sqrt(100)) = 1. Using a z-table, we find that the probability that the average weekly earnings is less than $455 is approximately 0.8413. Therefore, the probability that it is between $445 and $455 is approximately 0.8413 - 0.1587 = 0.6826.

5) In answering questions 2-4, we made an assumption about the population distribution based on the Central Limit Theorem. We assumed that since our sample size was large enough (n=100), our sampling distribution would be approximately normal.

6) If we assume that weekly earnings for employees in all general automotive repair shops are normally distributed with a mean of $450 and a standard deviation of $50, then we can calculate the z-score for an employee earning more than $480 in a given week as follows: z = (480 - 450) / 50 = 0.6. Using a z-table, we find that the probability that an employee will earn more than $480 in a given week is approximately 1 - 0.7257 = 0.2743.

a) The experimental probability of landing on red or yellow is:

P(red or yellow) = 33/40

b) The theoretical probability can be computed as follows:

P(red or yellow)  = 8/10

c) As the number of spins increases: Option A: we expect the experimental and theoretical probabilities to become closer, though they might not be equal.

Theoretical probability represents the likelihood of an event occurring. The theoretical probability of getting heads is 1/2, because we know that the probability of heads and tails on a coin is equal. Experimental probability describes how often an event actually occurred in an experiment.  

a) The experimental probability of landing on red or yellow is:

P(red or yellow) = (20/40) + (13/40) = 33/40

b) The theoretical probability can be computed as follows:

P(red or yellow) = (4/10) + (4/10) = 8/10

c) As the number of spins increases, we expect the experimental and theoretical probabilities to become closer, though they might not be equal.

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Therefore, BC is equal to 4 cm.

To solve this problem, we can use the Angle Bisector Theorem and the Pythagorean Theorem.

Let's start by applying the Angle Bisector Theorem. According to the theorem, the ratio of the segments of the hypotenuse formed by the altitude is equal to the ratio of the corresponding sides of the triangle.

In triangle ABC, we have:

AD/DB = AC/CB

Given that AD = 8 cm, we need to find DB. Let's denote DB as x.

8/x = AC/CB

Since AC is the altitude, it can be determined by applying the Pythagorean Theorem in right triangle ACH.

AC^2 = AH^2 + HC^2

AC^2 = (AB - BH)^2 + HC^2

AC^2 = (BC - 4)^2 + HC^2

Now, let's apply the Pythagorean Theorem in right triangle BCH.

BC^2 = BH^2 + HC^2

BC^2 = 4^2 + HC^2

Since AC = BC - 4, we can substitute these expressions into the equation:

(BC -4)^2 + HC^2 = BC^2

Expanding and simplifying this equation, we get:

BC^2 - 8BC + 16 + HC^2 = BC^2

Simplifying further, we have:

-8BC + 16 + HC^2 = 0

Now, let's substitute the value of HC = AD - AH = 8 - 4 = 4 into the equation:

-8BC + 16 + 4^2 = 0

-8BC + 16 + 16 = 0

-8BC + 32 = 0

-8BC = -32

BC = -32 / -8

BC = 4

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The equation that represents the total price of Michigan State University per semester is determined by the number of classes taken (c) and the total cost for the semester, which includes a one-time fee for room and board (t).

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The probability that a person will purchase no more than one costume is given as follows:

93%.

The parameters that are needed to calculate a probability are listed as follows:

Then the probability is calculated as the division of the number of desired outcomes by the number of total outcomes.

The total number of outcomes for this problem is given as follows:

187 + 228 + 29 = 444.

Only 29 people purchased more than one costume, hence the probability of at most one costume is calculated as follows:

(444 - 29)/444 = 0.93 = 93%.

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The value of x is 1.08

A logarithm is the power to which a number must be raised in order to get some other number.

For example log 100 = 2 . This means that the base 10 logarithm of 100 is 2 and we can also say that 10² = 100

Similarly, solving the equation;

log3(x+25) -log(x - 1) = 3

using law of logarithm

log( 3(x+25)/x-1) = log 1000

therefore;

3x + 75= 1000x -1000

1075 = 1000x -3x

1075= 997x

x = 1075/997

x = 1.08

Therefore the value of x is 1.08

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(1) The full payment required on January 22 to reduce the balance on the invoice to zero is $136,294.57.

(2)The full payment required on October 19 to reduce the balance on the invoice to zero is $98,482.75.

(3)The full payment required on April 3 to reduce the balance on the invoice to zero is $48,190.38.

The calculation of the full payment required takes into account the terms specified on the invoice. In the first scenario, the invoice terms are 2/10, n/30, which means that a 2% discount can be applied if payment is made within 10 days, otherwise, the full payment is due within 30 days.

Therefore, the full payment is calculated as the invoice amount minus the discount. Similarly, for the second scenario, the invoice terms are 3/10, 2/20, 1/30, and n/50, which involve multiple discounts based on different payment dates.

The full payment is determined by applying the applicable discounts. Finally, in the third scenario, the invoice terms are ROG (Receipt of Goods), and the full payment is due by April 15. Hence, the full payment required on the specified date is calculated.

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The length of CF, considering the Pythagorean Theorem in this problem, is given as follows:

CF = 26.08 m.

The Pythagorean Theorem states that in the case of a right triangle, the square of the length of the hypotenuse, which is the longest side,  is equals to the sum of the squares of the lengths of the other two sides.

Hence the equation for the theorem is given as follows:

c² = a² + b².

In which:

The length of EC is given as follows:

(EC)² = 10² + 24²

[tex]EC = \sqrt{10^2 + 24^2}[/tex]

EC = 26.

Then the length of CF is given as follows:

CF² = 26² + 2²

[tex]CF = \sqrt{26^2 + 2^2}[/tex]

CF = 26.08 m.

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When given that x + y = 4, the algebric expression 2(x + y) simplifies to 8.

To find the value of 2(x + y) when given that x + y = 4, we can substitute the value of x + y into the expression.

We are given x + y = 4.

To solve for 2(x + y), we multiply both sides of the equation x + y = 4 by 2:

2(x + y) = 2 * 4

This simplifies to:

2(x + y) = 8

Therefore, the value of 2(x + y) is 8.

To explain the reasoning behind this, let's break it down step by step:

1. We start with the equation x + y = 4.

2. To find 2(x + y), we distribute the 2 to both terms inside the parentheses: 2 * x + 2 * y.

3. This simplifies to 2x + 2y.

4. Since x + y = 4, we can substitute 4 for x + y in the expression 2x + 2y.

5. Therefore, 2(x + y) becomes 2 * 4, which equals 8.

In summary, when given that x + y = 4, the expression 2(x + y) simplifies to 8.

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The element of A n B is { 7, 8}

A set is the mathematical model for a collection of different things; a set contains elements or members, which can be mathematical objects of any kind.

For example, if the element of P is even numbers from 1 to 20 and element of Q is factor of 6 from 1 to 20 then we can say that set Q is a subset of set P.

The sign 'n' means intersection and this means what is common to two or more set.

If set A = { 1,2,5,7,8}

set B = { 6,7,8,9}

then we can see that 7 and 8 are common to both sides, then

A n B = { 7,8}

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Answer:

Step-by-step explanation:

g'(x) = 0.5*[(3x)^(-0.5)]*3 (By the power and chain rule)

And then just plug in 3, 0, and 2 into the given equation for g'(x)

[tex]\underline{\underline{\purple{\huge\sf || ꪖꪀᦓ᭙ꫀ᥅}}}[/tex]

First, we can simplify each expression:

(10⁰x10¹x1¹⁰) = 10¹¹

(10x10¹) = 100

(10⁰+10¹+1¹⁰) = 1 + 10 + 1,000,000,000 = 1,000,000,011

(10⁰+10¹x1¹⁰) = 10¹⁰

Now we can arrange them in increasing order:

10¹¹ < 10¹⁰ < 100 < 1,000,000,011

So the correct order from smallest to largest is:

(10⁰x10¹x1¹⁰) (10⁰+10¹x1¹⁰) (10 x 10¹) (10⁰+10¹+1¹⁰)

or

(10¹¹) (10¹⁰) (100) (1,000,000,011)

Therefore, the coordinates of point B are (-5.5, -10).

To find the coordinates of point B, we can use the midpoint formula. The midpoint formula states that the coordinates of the midpoint between two points (x1, y1) and (x2, y2) are given by:

Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)

We are given the midpoint coordinates as (-3, -5) and the coordinates of point A as (-0.5, 0). Let's assume the coordinates of point B as (x, y). Plugging the known values into the midpoint formula, we have:

(-3, -5) = ((-0.5 + x) / 2, (0 + y) / 2)

Simplifying, we get:

-3 = (-0.5 + x) / 2 ... (1)

-5 = y / 2 ... (2)

From equation (2), we can solve for y:

y = -5 * 2

y = -10

Now, substituting this value of y in equation (1), we can solve for x:

-3 = (-0.5 + x) / 2

-6 = -0.5 + x

x = -6 + 0.5

x = -5.5

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