Fill in the blank, so that the ordered pair is a solution of the equation.
y = 15 - 3x; (__,6)

We can write P(x) as P(x) = (x - 2)(x + 3)Q(x), where Q(x) is another polynomial.

This is the factored form of P(x), and it can be used to find other roots or to simplify the polynomial equation.

The factor theorem is a useful tool in finding the factors of a polynomial equation. The theorem states that if k is a root of the polynomial equation P(x) = 0, then x - k is a factor of P(x).

In our problem, we have P(2) = 0 and P(-3) = 0. This means that 2 and -3 are both roots of the polynomial equation P(x) = 0. According to the factor theorem, this means that x - 2 and x + 3 are both factors of P(x).

Therefore, we can write P(x) as P(x) = (x - 2)(x + 3)Q(x), where Q(x) is another polynomial. This is the factored form of P(x), and it can be used to find other roots or to simplify the polynomial equation.

In conclusion, the factor theorem is a useful tool in finding the factors of a polynomial equation. In our problem, we used the factor theorem to find the factors x - 2 and x + 3 of the polynomial equation P(x) = 0.

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Answer: 0.125 and 12.5%

Step-by-step explanation:

To convert any fraction to decimal form, we just need to divide its numerator by the denominator.

Here, the fraction is 1/8 which means we need to perform 1 ÷ 8.

This gives the answer as 0.125. So, 1/8 as a decimal is 0.125

To get its percentage form, you simply multiply the obtained decimal value by 100:

0.125 x 100 = 12.5

In conclusion, the mapping in part B informs us that the two circles are equation congruent, i.e., that their size and shape are the same and that the mapping preserves all of their geometrical characteristics.

A mathematical equation links two statements and utilises the equals sign (=) to indicate equality. In algebra, an equation is a mathematical assertion that proves the equality of two mathematical expressions. For instance, in the equation 3x + 5 = 14, the equal sign separates the numbers by a gap. A mathematical formula may be used to determine how the two sentences on either side of a letter relate to one another. The logo and the particular piece of software are usually identical. like, for instance, 2x - 4 = 2.

The mapping in part B informs us that the points on the two circles correspond exactly to one another. In other words, there is exactly one point on the bigger circle for every point on the smaller circle, and vice versa.

In conclusion, the mapping in part B informs us that the two circles are congruent, i.e., that their size and shape are the same, and that the mapping preserves all of their geometrical characteristics.

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The statement  "If G is connected with 2 or more nodes, then all nodes must have a degree bigger or equal to 1." is true.

This is because a connected graph is defined as a graph in which there is a path between any two nodes. If a node has a degree of 0, that means it is not connected to any other nodes, and therefore the graph cannot be connected. So, in order for G to be connected, all nodes must have a degree of at least 1.

In conclusion, the answer is true.

A node is a point where several elements are connected.

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One way to label the tick marks on the number lines is to use whole numbers for the longer tick marks and decimals for the shorter tick marks.

For example, on the first number line, you could label the longer tick marks as 0, 1, 2, 3, 4, and 5, and the shorter tick marks as 0.1, 0.2, 0.3, etc.

This fits with the structure of the base-ten system because each whole number is ten times greater than the corresponding decimal.

Another way to label the tick marks is to use fractions for the shorter tick marks. For example, on the second number line, you could label the longer tick marks as 0, 1, 2, 3, 4, and 5, and the shorter tick marks as 1/10, 2/10, 3/10, etc.

This also fits with the structure of the base-ten system because each fraction is one-tenth of the corresponding whole number.

Both of these labeling methods are accurate and fit with the structure of the base-ten system, so either one would be an appropriate way to label the tick marks on the number lines.

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An equivalent expression for the total length of fencing is 11y+10 inches.

The whole distance encircling a shape is referred to as its perimeter. It is the length of any two-dimensional geometric shape's boundary or outline. Depending on the size, the perimeter of several figures can be the same.

From the given figure, length of rectangle is (3y+5) inches and breadth is y inches. Sides of equilateral triangle measures y inches.

a) We know that, the perimeter of a rectangle is 2 (Length+Breadth)

= 2(3y+5+y)

= 2(4y+5)

= 8y+10 inches

Perimeter of equilateral triangle is 3×Side

= 3×y

= 3y inches

So, total length = 8y+10+3y inches

b) Equivalent expression is

8y+10+3y

= 11y+10 inches

c) Factors of the expression is 1 and 11y+10.

Therefore, an equivalent expression for the total length of fencing is 11y+10 inches.

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The cord length is k* theta radian, where theta radian is the central angle and k is the proportionality constant. This means that the rope length is proportional to the central angle in radians.

The complete cord length for a full central angle of 2pai is 2pai * r if the cord arc is a segment of a circle of radius r, where k=2pai is the proportionality constant.

What is arc?

In mathematics, an arc is a portion of the boundary of a circle or curve. It is sometimes referred to as an open curve.

The measurement around a circle that determines its edge is called the circumference, often known as the perimeter. As a result, the distance between any two locations along an arc's circumference is how it is defined.

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Answer:

60 pages

Step-by-step explanation:

64 + 56 = 120

180-120 = 60 pages

By applying synthetic division concept, it can be concluded that f(x) = 4x³ + x² + x - 9 can be expressed as f(x) = (x + 1)(4x² -3x + 4) - 13

Synthetic division is a shorthand way of dividing polynomials where we can divide the coefficients of the polynomial by omitting variables and exponents. As a result, we get the coefficient of the quotient and the remainder.

Polynomial remainder theorem states that the value of p in argument b is equal to the remainder of the polynomial division p(x) / (x - b). Specifically, p(x) is divided by x - b with a remainder of zero if, and only if, b is a root of p.

We have the following polynomial:

f(x) = 4x³ + x² + x - 9

and we have k = -1

We will divide the polynomial f(x) by (x + 1). The steps are as follows:

1. Put the coefficients in a row and multiply the outside coefficient by the divisor: 4(-1)= -4

2. Add the inside coefficient to the product from the previous step: -4 + 1 = -3

3. Multiply the result from the previous step by the divisor: -3(-1) = 3

4. Add the next coefficient to the product from the previous step: 3 + 1 = 4

5. Multiply the result from the previous step by the divisor: 4(-1) = -4

6. Add the last coefficient to the product from the previous step: -4 - 9 = -13

Now we take the first coefficient and the result of the sum steps, we get the following numbers: 4, -3, 4, -13

Then we can write the expression as follows: 4x³ + x² + x - 9 = (x + 1)(4x² - 3x + 4) - 13

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The transformations that would occur to the parent function are a vertical translation, a horizontal translation to the right, and a vertical compression with a reflection over the x-axis.

The parent function of f(x)=-(1)/(3)(x-5)^(3)+3 is f(x)=x^(3). There are several transformations that would occur to the parent function in order to obtain the given function.

1. Vertical translation: The "+3" at the end of the given function indicates that the parent function is shifted up 3 units.

2. Horizontal translation: The "(x-5)" inside the parentheses indicates that the parent function is shifted to the right 5 units.

3. Vertical stretch/compression: The "-(1)/(3)" in front of the parentheses indicates that the parent function is vertically compressed by a factor of 1/3 and reflected over the x-axis.

Therefore, the transformations that would occur to the parent function of f(x)=-(1)/(3)(x-5)^(3)+3 are a vertical translation of 3 units up, a horizontal translation of 5 units to the right, and a vertical compression by a factor of 1/3 with a reflection over the x-axis.

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Answer: The surface area of a cylinder is given by the formula:

S = 2πr^2 + 2πrh

where r is the radius of the base, h is the height of the cylinder, and π is the mathematical constant pi.

In this case, the radius is 28 mm and the height is 30 mm.

Substituting these values into the formula, we get:

S = 2π(28)^2 + 2π(28)(30)

Simplifying the equation:

S = 2π(28)(28 + 30)

S = 2π(28)(58)

S = 32176π

Using a calculator, we can approximate the value of S to 3 significant figures:

S ≈ 101285.744 mm^2

Therefore, the surface area of the cylinder is approximately 101286 mm^2 (rounded to 3 significant figures).

Step-by-step explanation:

Answer:

Either x/9 or 0.75 per cookie

Step-by-step explanation:

Let the number of cookies in a box be x.

The cost of x=9$

Then, the cost of one cookie= x/9

alternatively:

Considering one box contains 12 cookies,

The cost of a box of cookies= 9$

Then, the cost of one cookie= 9/12=0.75

The true statement is all irrational and rational numbers have decimal expansions. (second option)

A rational number is a number that can be expressed as a fraction of two integers . Rational numbers have terminating decimals and they have decimal expansions. Examples of rational numbers are 1.5, 2 and 4.25

A irrational number is a number that cannot be expressed as a fraction of two integers . Irrational numbers have repeating decimals. Examples of irrational numbers are 22/7, 1/3.

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Answer:

Let's assume that the original price of petrol is x per liter.

After the increase of R1 per liter, the new price of petrol is (x+1) per liter.

According to the problem, with the new price, you can buy 7 liters less for R504, so we can write:

7(x+1) = 504

Simplifying this equation, we get:

7x + 7 = 504

Subtracting 7 from both sides, we get:

7x = 497

Dividing both sides by 7, we get:

x = 71

Therefore, the original price of petrol was Rx71 per liter.

Answer: cos B-sin²A = cos²A-sin-B

Step-by-step explanation:

Answer: cos B-sin²A = cos²A-sin-B

Explanation:

Using the double angle identity, we can express the left side of the equation as:

cos(A + B).cos(A-B) = cos A.cos B - sin A.sin B

Now, using the Pythagorean identity, we can rewrite the right side of the equation as:

cos²A-sin-B = cos A.cos B - sin A.sin B

Hence, we can conclude that cos(A + B).cos(A-B) = cos B-sin²A = cos²A-sin-B

Answer: Read the step by step explanation for the answer.

Step-by-step explanation:

We can prove this trigonometric identity using the product-to-sum and Pythagorean identities as follows:

Starting with the left-hand side (LHS):

cos(A + B).cos(A - B)

Using the product-to-sum identity:

= [cos A cos B - sin A sin B][cos A cos B + sin A sin B]

= cos²A cos²B - sin²A sin²B

Using the Pythagorean identity:

= cos²A(1 - sin²B) - sin²A(1 - cos²B)

= cos²A - cos²A sin²B - sin²A + sin²A cos²B

= cos²A - sin²A + sin²A cos²B - cos²A sin²B

Using the Pythagorean identity:

= cos²A - sin²A + sin²A(1 - sin²B) - cos²A sin²B

= cos²A - sin²A + sin²A - sin²A sin²B - cos²A sin²B

= cos²A - sin²A + sin²A cos²B - sin²B cos²A

= cos²A - sin²A + cos²A sin²B - sin²B cos²A

Using the Pythagorean identity:

= cos²A - sin²A + cos²A - sin²B

= 2cos²A - sin²A - sin²B

= cos²A + cos²A - sin²A - sin²B

= cos²A + sin²A cos²B - sin²B

= RHS

Therefore, we have shown that (b) cos(A + B).cos(A-B) = cos B-sin²A = cos²A-sin-B.

The first step is to recall the equation for an arithmetic sequence: an = a1 + (n - 1)d, where a1 is the first term, an is the nth term, and d is the common difference.

Given the sum of three consecutive terms of the sequence, we know that a3 + a2 + a1 = 6. We can also recall that the product of the terms is a1 x a2 x a3 = -42.

Now, using the two equations, we can solve for the three terms:
a3 = (6 - a2 - a1)/2,
a2 = 6 - a1 - 2a3,
a1 = 6 - a2 - 2a3

Substituting the second and third equation into the first, we get:
a3 = -14/2 = -7

Therefore, the three consecutive terms of the arithmetic sequence are: a1 = -5, a2 = 2, a3 = -7.

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Interest rates vary between lenders and can drastically change the monthly payment.

The interest rate is the cost of borrowing money from a lender. It determines how much extra you will pay on top of the principal amount borrowed.

The interest rate is a crucial factor to consider when shopping for loans because it directly affects the total cost of the loan.

Higher interest rates mean that you will pay more over the life of the loan, resulting in a higher monthly payment.

Lower interest rates mean that you will pay less over the life of the loan, resulting in a lower monthly payment.

Thus, choosing a loan with a lower interest rate can save you a significant amount of money in the long run.

Hence, Interest rates vary between lenders and can drastically change the monthly payment.

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156 customers are expected to wear a black shoe.

Probability can be defined as the ratio of the number of favorable outcomes to the total number of outcomes of an event.

We know that, probability of an event = Number of favorable outcomes/Total number of outcomes

As per the given data:

Customers with black shoes = 13 out of 30

Customers with brown shoes = 17 out of 30

Total customers in the mall = 360

Probability that a customer wears black shoes:

P(B) = 13/30 = 0.433

Number of customers out of 360 expected to wear black shoes:

= P(B) × 360

= 0.433 × 360

= 156 customers.

Hence, 156 customers are expected to wear a black shoe.

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i. The joint distribution for the coin and dice events is given by P(C,D) = P(C|D)P(D) = P(D|C)P(C), where C is the event that the coin lands heads and D is the event that the dice lands six.

ii. The prior probabilities for the coin and dice events are P(C) = 0.5 and P(D) = 1/6, since they are both fair.

iii. The full conditional distribution for the event that the coin is heads or the dice is six is given by P(C∪D) = P(C) + P(D) - P(C∩D) = 0.5 + 1/6 - (0.5)(1/6) = 0.5833.

iv. The full conditional distribution for the event that the coin is heads and the dice is six is given by P(C∩D) = P(C|D)P(D) = P(D|C)P(C) = (0.5)(1/6) = 0.0833.

v. The distribution for the coin and dice events given the event that the coin is heads or the dice is six is given by P(C,D|C∪D) = P(C∩D|C∪D)P(C∪D) = (0.0833)(0.5833) = 0.0486.

vi. The probability of observing that the coin is heads or the dice is six is given by P(C∪D) = 0.5833.

vii. The posterior distribution for the joint probability of the coin and dice events given the event that the coin is heads or the dice is six is given by P(C,D|C∪D) = P(C∩D|C∪D)P(C∪D) = (0.0833)(0.5833) = 0.0486.

viii. The marginal distribution for the event that the coin is heads and the dice is six given we know the event heads or six has occurred is given by P(C∩D|C∪D) = P(C,D|C∪D)/P(C∪D) = 0.0486/0.5833 = 0.0833.

ix. The marginal probability that the coin is heads and the dice is six given we know the event heads or six has occurred is given by P(C∩D|C∪D) = 0.0833.

x. The probability of you winning the prize is the same as the probability that the coin is heads and the dice is six, which is given by P(C∩D) = 0.0833.

Therefore, the probability of you winning the prize is 0.0833.

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The solution of the equation 1/2x² + 3/8x - 2 using completing the square method are: 1.66 or -2.41.

To solve the equation 1/2x² + 3/8x -2 using completing the square method, follow these steps:

Step 1: Move the constant term to the right-hand side

1/2x² + 3/8x = 2

Step 2: Multiply both sides by 2 to clear the fraction

x² + 3/4x = 4

Step 3: Add the square of half of the coefficient of x to both sides of the equation

x² + 3/4x + (3/8)² = 4 + (3/8)²

The left-hand side is now a perfect square, specifically (x + 3/8)².

Step 4: Simplify the right-hand side of the equation

x² + 3/4x + 9/64 = 256/64 + 9/64

x² + 3/4x + 9/64 = 265/64

Step 5: Take the square root of both sides

x + 3/8 = ±√(265)/8

Step 6: Solve for x

x = -3/8 ± √(265)/8

x = 1.66 or - 2.41

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Using expression, the following provides one explanation for why there are two ways to add up the symbols on the tablet.

A statement, at least two variables or integers, and one or more arithmetic operations make up a mathematical expression. This mathematical procedure makes it possible to multiply, divide, add, or subtract quantities.

The hieroglyphic number system of ancient Egypt employed a decimal system with hieroglyphs to denote the powers of ten. It is challenging to give a precise response, though, without being able to see the precise symbols on the tablet.

It is feasible that there are two distinct ways to add the numbers represented by the symbols on the tablet, presuming that the tablet is employing a fundamental arithmetic operation like addition and that each symbol represents a different number.

Finding two different sets of integers that, when added together, yield the same answer is one way to approach this issue.

As an illustration, if the symbols stand for 2, 3, and 5, one technique to add them up would be:

2 + 3 + 5 = 10

They might also be added up as follows:

3 + 7 = 10

In this instance, the number 2 would be represented by the sign 2, and the number 5 by the symbol 3. This could be one reason for why the symbols on the tablet can be added in two different ways.

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Step-by-step explanation:

red gumballs=6 blue gumballs=6

yellow gumballs=6 green gumballs=6

purple gumballs=6

total number of gumballs=30

probability of a student selecting a red gumballs

no of red gumballs 6 1

=------------------------ = ----- = -----

no of total gumballs 30 5

The options which satisfy the given diagram are

m∠5 + m∠6 = 180°

m∠2 + m∠3 = m∠6

m∠2 + m∠3 + m∠5 = 180°

The sum of the interior angles of a triangle is 180°.

The exterior angle of a triangle is equal to the sum of its interior opposite angles.

Let the Triangle be ABC and the exterior angle be ∠D , then

∠A + ∠B + ∠C =  180°

∠A + ∠B = ∠D , ∠A and ∠B are interior opposite angles of the triangle

So, the Options that satisfied

m∠5 + m∠6 = 180° { linear pair}

m∠2 + m∠3 = m∠6 {exterior angle theorem}

m∠2 + m∠3 + m∠5 = 180° {Angle Sum Property}

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The number of comic books is given as follows:

P/100 x 560.

In which P is the percentage equivalent to comic books on the circle graph.

The number comic books sold at the book fair is given by the proportions in the context of the problem.

The circle graph gives the percentage out of each type of book, hence the percentage of comic books is given as follows:

P/100.

Then the amount of comic books is obtained multiplying the percentage by the total amount of 560 as follows:

P/100 x 560.

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The reason that should be used to complete statement 2 "∠2 and ∠3 are supplementary" is linear pair postulate.

In Mathematics, the linear pair theorem is sometimes referred to as linear pair postulate and it states that the measure of two angles would add up to 180° provided that they both form a linear pair.

This ultimately implies that, the measure of the sum of two adjacent angles would be equal to 180° when two parallel lines are cut through by a transversal.

According to the linear pair theorem, we have the following supplementary angles:

∠2 + ∠3 = 180°

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The polynomial in factored form is: (x-5)(x+6)(x-(3-2i))(x-(3+2i))

Since complex roots always come in conjugate pairs, we know that if 3-2i is a root, then 3+2i must also be a root. Therefore, we can write the polynomial in factored form by multiplying together the factors (x-5), (x+6), (x-(3-2i)), and (x-(3+2i)).

Alternatively, we can write the polynomial in factored form using the formula for a polynomial with complex roots:

p(x) = (x-r1)(x-r2)(x-r3)(x-r4)

where r1, r2, r3, and r4 are the roots of the polynomial. In this case, r1 = 5, r2 = -6, r3 = 3-2i, and r4 = 3+2i. Substituting these values into the formula gives us:

p(x) = (x-5)(x+6)(x-(3-2i))(x-(3+2i))

Therefore, the polynomial in factored form is written in the following form: (x-5)(x+6)(x-(3-2i))(x-(3+2i)).

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The ratio of strawberries to bananas in the basket is 12:18 or 2:3. To find this ratio, you simply take the number of strawberries and place it over the number of bananas. In this case, there are 12 strawberries and 18 bananas, so the ratio is 12:18.

However, this ratio can be simplified by dividing both numbers by their greatest common factor, which is 6. So, the simplified ratio of strawberries to bananas is 2:3. This means that for every 2 strawberries, there are 3 bananas in the basket.

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a)z-score

b)z-score > 1.64

c)comparing the z-score to a z-table

d)a majority of the population of soft drink consumers prefer Pepsi over Coke.

To test the hypothesis that a majority of the population of soft drink consumers prefer Pepsi over Coke, we will use a significance level of α = .10.

H0: The proportion of soft drink consumers who prefer Pepsi is ≤ 0.5.
Ha: The proportion of soft drink consumers who prefer Pepsi is > 0.5.

Test statistic: z-score

Rejection Region: z-score > 1.64

We will obtain the P value by comparing the z-score to a z-table.

Conclusion: Based on the P value, if it is less than or equal to the significance level of 0.10, then we reject the null hypothesis. We can conclude that there is sufficient evidence to suggest that a majority of the population of soft drink consumers prefer Pepsi over Coke.

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The solution set for the inequality 6x + 1 ≤ 4x + 1 is x ≤ -3/2 can be written in interval notation as (-∞, 0].

To solve the inequality 6x + 1 ≤ 4x + 1, we need to isolate the variable on one side of the inequality. Here are the steps to do so:

1. Subtract 4x from both sides of the inequality: 6x + 1 - 4x ≤ 4x + 1 - 4x
2. Simplify: 2x + 1 ≤ 1
3. Subtract 1 from both sides of the inequality: 2x + 1 - 1 ≤ 1 - 1
4. Simplify: 2x ≤ 0
5. Divide both sides of the inequality by 2: 2x/2 ≤ 0/2
6. Simplify: x ≤ 0

The solution set in interval notation is (-∞, 0]. This means that any value of x less than or equal to 0 will satisfy the inequality.

"

Prper question:

Solve the inequality. Write the solution set in interval notation.

6x + 1 ≤ 4x + 1

"

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