Solve each equation using any method. When necessary, round real solutions to the nearest hundredth. x² = 11 x-10 .

The property of real numbers illustrated by the equation -(2t - 11) = 11 - 2t is the commutative property of addition.

Given that an expression we need to determine which property does it follow,

The commutative property of addition states that the order of numbers can be changed without affecting the result when adding them together. In other words, for any real numbers a and b, the sum of a and b is the same regardless of the order in which they are added.

In the given equation, we can observe that the terms on both sides of the equation involve addition and subtraction. By rearranging the terms, we can rewrite the equation as 11 - 2t = -(2t - 11).

This shows that the terms 11 and 2t have been swapped in their positions without altering the equality of the equation. This swap of terms demonstrates the commutative property of addition.

So, the commutative property of addition is illustrated in the equation -(2t - 11) = 11 - 2t by the interchangeability of the terms without affecting the solution or outcome.

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The polynomial equation in which 1+√2 is the only root is:

x² - 2x - 1

To find a polynomial equation in which 1+√2 is the only root, we can use the concept of conjugate pairs.

Since 1+√2 is a root, its conjugate, 1-√2, must also be a root.

This is because the conjugate of a root of a polynomial with rational coefficients is always another root.

To construct the polynomial equation, we can start by setting up two factors using the roots:

(x - (1 + √2))(x - (1 - √2))

Expanding these factors:

(x - 1 - √2)(x - 1 + √2)

Using the difference of squares formula, (a - b)(a + b) = a² - b²:

((x - 1)² - (√2)²)

Simplifying further:

(x² - 2x + 1 - 2)

Combining like terms:

x² - 2x - 1

Therefore, the polynomial equation in which 1+√2 is the only root is:

x² - 2x - 1

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While Marsha, Jan, and Cindy individually have transitive preferences over three goods, it is possible that the group's preferences might not be transitive when deciding on the "fruit of the month."

This scenario arises due to the aggregation of individual preferences and the potential conflicts that can emerge during the voting process.

When individuals vote on their preferred fruit of the month, the group's preference is determined by aggregating individual preferences. However, the aggregation process can lead to inconsistencies in transitivity. For example, let's assume Marsha prefers oranges to apples, Jan prefers apples to pears, and Cindy prefers pears to oranges.

Individually, their preferences are transitive. However, when their preferences are aggregated, conflicts arise. If the group votes between oranges and apples, Marsha's preference would favor oranges, Jan's preference would favor apples, and the group might choose apples as the fruit of the month. Similarly, if the group votes between apples and pears, Jan's preference would favor apples, Cindy's preference would favor pears, and the group might choose pears.

Now, if the group votes between oranges and pears, Marsha's preference would favor oranges, Cindy's preference would favor pears, but there is no unanimous preference between apples and pears. In this case, the group's preference would not be transitive because oranges are preferred to apples, apples are preferred to pears, but oranges are not preferred to pears.

This example demonstrates that the aggregation of individual preferences in a voting process can lead to situations where the group's preferences are not transitive.

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To find the volume of the solid obtained by rotating the region bounded by the curves y = 2x and y = 2x about the line y = 2, the method of cylindrical shells can be used.

When rotating the region bounded by the curves y = 2x and y = 2x about the line y = 2, we can visualize the resulting solid as a collection of infinitesimally thin cylindrical shells.

The height of each shell is given by the difference between the lines y = 2 and the curve y = 2x, which is 2 - 2x. The circumference of each shell is given by 2πx since the shell is formed by rotating a line segment of length x.

Integrating the product of the height and circumference over the range of x where the curves intersect (from x = 0 to x = 1), we can find the volume of the solid using the formula V = ∫(2πx)(2 - 2x) dx. Evaluating this integral will yield the volume of the solid.

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The given data presents the marginal effects estimated from a probit model. Each row represents a variable, and the corresponding values show the marginal effect, standard error, z-statistic, and p-value. The marginal effect, represented as dF/dx, measures the change in the probability of the dependent variable (usually a binary outcome) resulting from a one-unit change in the independent variable.

In a probit model, the marginal effects provide insights into how changes in the independent variables affect the probability of the dependent variable. The estimated marginal effects indicate the direction and significance of these effects.

For example, a positive marginal effect indicates that an increase in the corresponding independent variable leads to a higher probability of the outcome occurring. Conversely, a negative marginal effect suggests a decrease in the probability. The standard error quantifies the uncertainty associated with the marginal effect estimate, and the z-statistic and p-value assess the statistical significance of the effect.

The significance codes provided (***, **, *, etc.) indicate the level of significance at which the null hypothesis (no effect) can be rejected. Lower p-values suggest higher significance. Researchers can use these results to understand the relative importance of different variables in influencing the probability of the outcome.

It's important to note that interpreting marginal effects requires considering the context of the model, the specific variables involved, and any assumptions made during estimation.

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Based on the available information, the exponential version appears to better constitute the fashion of U.S Federal Spending through the years. The vast increases in spending endorse an exponential increase pattern, indicating a higher probability of persevered exponential growth inside destiny.

Based on the given facts for U.S Federal Spending over time, let's evaluate two viable models:

Model 1: Linear Model

Model 2: Exponential Model

To decide which model great suits the information, we will take a look at the tendencies and styles inside the information factors.

Using the linear model, we are able to calculate the annual boom in federal spending:

From 1965 to 1980 (15 years), there may be an increase of $670 billion ($1,300 billion - $630 billion), averaging approximately $44.67 billion according to 12 months.

From 1980 to 1995 (15 years), there was a growth of $650 billion ($1,950 billion - $1,300 billion), averaging approximately $43.33 billion in line with yr.

From 1995 to 2005 (10 years), there may be an increase of $700 billion ($2,650 billion - $1,950 billion), averaging about $70 billion in step with year.

Using the exponential version, we can calculate the compound annual boom fee (CAGR):

From 1965 to 2005 (40 years), the spending expanded with the aid of $2,020 billion ($2,650 billion - $630 billion).

Calculating the CAGR, we discover that the average annual growth charge is approximately 5.18%.

Based on the information and evaluation, it seems that the exponential version is a better healthy for the United States Federal Spending over time. The data indicates a vast increase in spending through the years, suggesting an exponential boom sample as opposed to a linear one.

However, it's vital to notice that this analysis is based on a confined dataset, and further analysis can be required to decide the most accurate model for predicting future federal spending.

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he seven steps of Operations Research (OR) provide a systematic framework for problem-solving.

Operations Research (OR) involves seven key steps. First, the problem is defined, followed by gathering relevant data. A mathematical model is then formulated to represent the problem. Solutions are derived from the model, and the model is subsequently tested to ensure its validity. Once validated, preparations are made to apply the model in practical scenarios, and finally, the model is implemented.

Operations Research (OR) is a systematic approach to problem-solving that utilizes mathematical models and analytical methods. The first step in OR is defining the problem. This involves clearly understanding the issue at hand, identifying the objectives, and setting specific goals to be achieved.

After defining the problem, the next step is gathering the necessary data. Accurate and relevant data is crucial for building an effective mathematical model. This data can be collected through various means, such as surveys, interviews, or existing databases.

With the data in hand, the third step is to formulate a mathematical model. The model represents the problem in a structured and quantifiable manner. It incorporates various variables, constraints, and relationships that exist within the problem domain.

Once the mathematical model is formulated, the fourth step involves deriving solutions from the model. This is done through mathematical techniques like optimization, simulation, or queuing theory. The aim is to find the best possible solutions that meet the defined objectives.

To ensure the reliability and accuracy of the model, the fifth step involves testing it. This includes validating the model's results and assessing its performance under different scenarios. If necessary, adjustments and refinements are made to improve the model's effectiveness.

After the model is tested and validated, the sixth step is preparing to apply the model in practical situations. This involves considering factors like resource allocation, implementation strategies, and potential challenges that may arise during the application of the model.

The final step is the implementation of the model. This involves putting the solutions derived from the model into action. It requires effective communication, coordination, and collaboration among relevant stakeholders to ensure a smooth and successful implementation process.

In conclusion, the seven steps of Operations Research (OR) provide a systematic framework for problem-solving. By following these steps, organizations can optimize their decision-making processes and improve efficiency in various domains such as logistics, supply chain management, finance, and healthcare.

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The solution to the equation is p = -72.

The equation, we need to isolate the variable 'p' on one side of the equation. Let's go through the steps:

-p/12 = 6

To get rid of the fraction, we can multiply both sides of the equation by 12:

12 * (-p/12) = 12 * 6

This simplifies to:

-p = 72

To isolate 'p,' we can multiply both sides of the equation by -1:

(-1) * (-p) = (-1) * 72

This gives us:

p = -72

Therefore, the solution to the equation is p = -72.

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The function y = x² - 6x + 1 can be rewritten in vertex form as y = (x - 3)² - 8.

To rewrite the given quadratic function in vertex form, we'll complete the square. The vertex form of a quadratic function is given by:

y = a(x - h)^2 + k

where (h, k) represents the coordinates of the vertex.

Let's go ahead and rewrite the function y = x² - 6x + 1 in vertex form:

Step 1: Group the quadratic terms together.

y = (x² - 6x) + 1

Step 2: Complete the square for the x terms inside the parentheses.

y = (x² - 6x + 9 - 9) + 1

Step 3: Rearrange the equation to isolate the completed square term.

y = (x² - 6x + 9) - 9 + 1

Step 4: Factor the trinomial and simplify the expression.

y = (x - 3)² - 8

Therefore, the function y = x² - 6x + 1 can be rewritten in vertex form as y = (x - 3)² - 8.

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To find the equation of the parabola passing through the given points (3, 4), (-2, 9), and (2, 1), we can use the general form of a parabolic equation, which is y = ax² + bx + c.

By substituting the coordinates of the points into this equation, we can form a system of equations to solve for the coefficients a, b, and c. Using the point (3, 4), we get the equation 4 = 9a + 3b + c. From the point (-2, 9), we have 9 = 4a - 2b + c. Lastly, using the point (2, 1), we obtain 1 = 4a + 2b + c. This gives us a system of three linear equations.

By solving this system of equations, we find that a = -1/5, b = -9/5, and c = 18/5. Substituting these values back into the general form of the parabolic equation, we have y = (-1/5)x² - (9/5)x + (18/5). To express the equation in standard form, we need to remove fractions and put the equation in the form of ax² + bx + c = 0. By multiplying through by 5 to eliminate the fractions, we get -x² - 9x + 18 = 0.

Therefore, the equation of the parabola passing through the given points (3, 4), (-2, 9), and (2, 1) is -x² - 9x + 18 = 0, written in standard form.

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The remainder when n is divided by 36 is equal to n.

To find the remainder when the integer [tex]$n$[/tex] is divided by 36, we can consider the divisibility rule for 36. A number is divisible by 36 if it is divisible by both 4 and 9.

First, let's look at the divisibility by 4. A number is divisible by 4 if the last two digits of the number form a multiple of 4. In this case, the last two digits of n are 73. Since 73 is not divisible by 4, we can conclude that n is not divisible by 4.

Next, let's examine the divisibility by 9. A number is divisible by 9 if the sum of its digits is divisible by 9. In this case, the sum of the digits in n is 13 \times 7 + 17 \times 3 = 91 + 51 = 142. Since 142 is not divisible by 9, we can conclude that n is not divisible by 9.

Since n is not divisible by either 4 or 9, it will not be divisible by 36. The remainder when n is divided by 36 is equal to n itself.

The remainder when n is divided by 36 is equal to n.

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In a function, each input should have a unique output. But in this case, the input 8 is associated with multiple outputs, violating the definition of a function.

To determine if a relation is a function, we need to check if each input (x-value) is associated with only one output (y-value). In this case, let's analyze the given relation:

(8,4),(8,3),(8,-1),(8,6)

The x-value is always 8 for each ordered pair. However, the y-values associated with 8 are different in each case. Since one input (8) is mapped to multiple outputs (4, 3, -1, 6), this relation is not a function.

In a function, each input should have a unique output. But in this case, the input 8 is associated with multiple outputs, violating the definition of a function.

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Both g(x) and h(x) are functions involving more than just the variable x, satisfying the condition that neither of them is solely x.

Given: f(x) = 2 / (5x - 1)

Let's start by identifying g(x) and h(x) separately.

We can see that the outer function g(x) involves dividing a constant (2) by a quantity.

Therefore, g(x) = 2 / x can be a suitable candidate.

Now, let's consider the inner function h(x). The expression within the denominator, 5x - 1, can be a good candidate for h(x) as it includes the variable x.

Therefore, h(x) = 5x - 1.

Now, we can rewrite f(x) as g(h(x)):

f(x) = g(h(x)) = 2 / h(x)

               = 2 / (5x - 1)

So, g(x) = 2 / x and h(x) = 5x - 1.

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Based on the given information, we cannot determine whether the measures define 0, 1, 2, or infinitely many triangles.

To determine the number of triangles that can be formed using the given measures, we need to apply the triangle inequality theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

However, the given measures only include angle measures (m\angle A = 41, m\angle B = 68), and they do not provide any information about side lengths. Angle measures alone are not sufficient to determine the lengths of the sides of a triangle.

Without knowing the lengths of the sides, we cannot apply the triangle inequality theorem, and therefore, we cannot determine the number of triangles that can be formed.

In conclusion, based on the given information, we cannot determine whether the measures define 0, 1, 2, or infinitely many triangles.

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The statement "In an isosceles triangle ABC, m∠B = 90" is sometimes true.

An isosceles triangle is a triangle that has at least two sides of equal length. In the given statement, it is stated that ∠B (angle B) measures 90 degrees. However, this information alone does not determine whether the triangle is isosceles or not.

To determine if ΔABC is isosceles, we need additional information about the lengths of its sides. If we have additional information stating that two sides of the triangle are congruent, then we can conclude that the triangle is isosceles.

Therefore, without any information about the side lengths, we cannot definitively say that the triangle ΔABC is isosceles based solely on the statement that ∠B = 90 degrees.

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Answer: $75

Step-by-step explanation: The explaination is:-

Mrs Olivia paid $45

She bought 15 Markers

Price of each Marker = Total Price/Number of Markers

Hence Each marker will cost

= $45/15

=$3

So the cost of 25 Markers will be

25 x Price of each Marker

= 25 x $3

=$ 75

The term Rk2 refers to multiple concepts related to regression analysis. It can refer to the R-square statistic evaluated at the kth observation in the data and an R-square statistic obtained by regressing the kth explanatory variable on all other explanatory variables. However, it does not refer to the sum of the residuals obtained by regressing the kth explanatory variable on all other explanatory variables or the adjusted R-square statistic.

The R-square statistic measures the proportion of the variation in the dependent variable that can be explained by the independent variables in a regression model. When evaluated at the kth observation, it provides a measure of how well the model fits that specific data point. It is computed by squaring the correlation between the observed and predicted values at that particular observation.

On the other hand, Rk2 can also refer to an R-square statistic obtained by regressing the kth explanatory variable on all other explanatory variables. This measures the proportion of the variation in the kth explanatory variable that can be explained by the remaining explanatory variables in the model.

However, Rk2 does not represent the sum of the residuals obtained by regressing the kth explanatory variable on all other explanatory variables. The residuals represent the differences between the observed and predicted values in a regression model.

Similarly, Rk2 does not represent the adjusted R-square statistic, which adjusts the R-square statistic for the number of variables in the model and the sample size, providing a more robust measure of the model's goodness of fit.

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In SPSS, the "Values" option under variable view is used to provide labels or numeric codes for categorical variables.

Categorical variables are variables that have distinct categories or groups, such as gender (male/female) or education level (high school/college/graduate). By specifying the values for a categorical variable, SPSS allows users to assign meaningful labels or numeric codes to each category.

When defining a categorical variable in SPSS, the "Values" field in variable view allows users to define the labels or numeric codes for each category. For example, if the variable "gender" has two categories, we can assign the value 1 to represent "male" and the value 2 to represent "female". These values will be displayed in the data editor and in any output or analysis that involves the variable.

It is important to note that the "Values" option under variable view is not used to describe continuous variables. Continuous variables are those that can take on any numerical value within a given range, such as age or income. The description of variables, including their labels, measurement scales, and other attributes, is typically done using the "Variable Labels" option in SPSS.

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The distance between each pair of points is given by the distance formula which is stated below; Distance formula. The distance formula is used to find the distance between two points in the coordinate plane. The distance formula is derived from the Pythagorean theorem. So, the distance between each pair of points is √101`

The distance formula is given as; d = √((x2-x1)² + (y2-y1)²), Where; x1, x2 are the x-coordinates of points 1 and 2. y1, y2 are the y-coordinates of points 1 and 2

Applying the distance formula to the given pair of points, (0, 6) and (-1, -4), we have;`x1 = 0`,`x2 = -1`,`y1 = 6`, and `y2 = -4`. Therefore, the distance between each pair of points is; d = √((-1 - 0)² + (-4 - 6)²)d = √((-1)² + (-10)²)d = √(1 + 100)d = √101 Answer: `√101`

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The distance between the points W(7,3) and Z(-4,-1) is approximately 13.93 units.

To find the distance between two points in a coordinate plane, we can use the distance formula:

Distance = √[tex]((x2 - x1)^2 + (y2 - y1)^2)[/tex]

In this case, the coordinates of point W are (7,3) and the coordinates of point Z are (-4,-1).

Plugging the values into the distance formula, we get:

Distance = √[tex]((-4 - 7)^2 + (-1 - 3)^2)[/tex]

        = √[tex]((-11)^2 + (-4)^2)[/tex]

        = √(121 + 16)

        = √137

        ≈ 13.93

Therefore, the distance between points W(7,3) and Z(-4,-1) is approximately 13.93 units.

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Answer: 13 FEET

Step-by-step explanation:

Given that: window is rectangular in shape,

                width = 5 feet

               length = 12 feet

    one diagonal = 13 feet

To find: length of other diagonal

Solution: As one of the rectangle's property says that: length of both the diagonals of rectangle is same

Therefore, length of other diagonal will be 13 feet.

The mean price of nonfiction books on the best-sellers list is $25.07, with a standard deviation of $2.62. The individual prices of the books are $26.95, $22.95, $24.00, $24.95, $29.95, $19.95, $24.95, $24.00, $27.95, and $25.00.

The mean price of $25.07 represents the average price of the nonfiction books on the best-sellers list. It is calculated by summing up all the prices and dividing the total by the number of books (10 in this case).

The standard deviation of $2.62 measures the variability or spread of the prices around the mean. It provides a measure of how much the prices deviate from the average. A lower standard deviation indicates that the prices are closer to the mean, while a higher standard deviation suggests greater variability.

Looking at the individual prices, we can see that they range from $19.95 to $29.95. These prices contribute to the overall mean and standard deviation. If a price is significantly higher or lower than the mean, it will have a greater impact on the standard deviation.

In this case, the prices appear to be relatively close to the mean, with some variation. This suggests that the prices of nonfiction books on the best-sellers list are centered around $25.07, but there are some books priced slightly higher or lower.

The mean and standard deviation provide valuable information about the distribution of prices on the best-sellers list, allowing us to understand the central tendency and variability of the data.

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Answer:

look at attachment

Step-by-step explanation:

The answer is:

Work/explanation:

First, I will use the slope formula to find slope:

[tex]\sf{m=\dfrac{y_2-y_1}{x_2-x_1}}[/tex]

where:

m = slope;

(x₁, y₁) and (x₂, y₂) are points on the line.

Plug in the data:

[tex]\sf{m=\dfrac{-2-3}{4-2}}[/tex]

[tex]\sf{m=\dfrac{-5}{2}}[/tex]

[tex]\sf{m=-\dfrac{5}{2}}[/tex]

So far, the equation is y = -5/2x + b.

Now, we'll use the first point (2,3) and plug it into the equation to solve for b.

3 = -5/2 (2) + b

3 = -5 + b

-5 + b = 3

b - 5 = 3

b = 3 + 5

b = 8

The length of the side of the original square cardboard, x, is 13 inches.

Let's solve the problem step by step:

We start with a square piece of cardboard with side length x.

We cut 4-in squares from each corner. This reduces the dimensions of the cardboard by 8 inches in both length and width. Therefore, the dimensions of the resulting box will be (x - 8) inches by (x - 8) inches.

We fold up the sides to create the box.

The volume of a rectangular box is given by the formula V = length × width × height.

In this case, the height is 4 inches because we have folded up the sides.

According to the problem, the box should hold 100 cubic inches, so V = 100 cubic inches.

Plugging in the values, we have (x - 8) × (x - 8) × 4 = 100.

Simplifying the equation, we get (x - 8)^2 = 25.

Taking the square root of both sides, we have x - 8 = ±5.

Solving for x, we get two possible solutions: x - 8 = 5 or x - 8 = -5.

If x - 8 = 5, then x = 13.

If x - 8 = -5, then x = 3.

However, we must consider that the box needs to have positive dimensions. Therefore, the valid solution is x = 13.

Thus, the length of the side of the original square cardboard, x, is 13 inches.

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The zeros of f(x) are 1, -10, and -2.

To find the zeros of the function[tex]f(x) = x^3 + 11x^2 + 8x - 20[/tex] algebraically, we need to solve the equation f(x) = 0.

There are several methods to find the zeros, and one commonly used approach is the factor theorem and polynomial long division.

Begin by trying possible integer values as potential zeros.

In this case, we can use the rational root theorem to narrow down the options.

The possible rational roots are factors of the constant term (-20) divided by factors of the leading coefficient (1).

Possible rational roots: ±1, ±2, ±4, ±5, ±10, ±20

Test each potential zero using synthetic division or polynomial long division until a zero is found.

Trying x = 1:

Applying synthetic division:

1 | 1 11 8 -20

| 1 12 20

Copy code

1   12  20    0

Since the remainder is zero, x = 1 is a zero of the function.

After finding one zero, we can factor the polynomial by dividing it by (x - 1) using polynomial long division or synthetic division.

The result is a quadratic equation:

[tex](x - 1)(x^2 + 12x + 20) = 0[/tex]

Solve the quadratic equation [tex]x^2 + 12x + 20 = 0[/tex] using factoring, completing the square, or the quadratic formula.

Factoring: (x + 10)(x + 2) = 0

Setting each factor to zero:

x + 10 = 0 or x + 2 = 0

x = -10 or x = -2

The zeros of the function [tex]f(x) = x^3 + 11x^2 + 8x - 20[/tex] are x = 1, x = -10, and x = -2.

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When the horizontal and the vertical lines crosses each other forming a square, the structure so formed is called pattern .

Let's consider a simple pattern where each figure consists of a square grid, and the number of squares in each row and column increases by 1 with each figure. They help to form the base layout for the designers.

In the first figure (n = 1), we have a 1x1 grid, which contains 1 square.

For n= 1  ;

where we have a grid 0f [tex]1 \times 1[/tex] .

The figure  so formed taking [tex]1\times 1[/tex] Grid is attached below in the image form .

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The given quadratic inequality is [tex]x^2 - 32 \geq 4x(x+6)(x-3) > -8\)[/tex]. To solve this inequality, we need to find the values of [tex]\(x\)[/tex] that satisfy the given conditions.

To solve the quadratic inequality[tex]\(x^2 - 32 \geq 4x(x+6)(x-3) > -8\)[/tex], we can break it down into two separate inequalities and solve them individually.

1)[tex]\(x^2 - 32 \geq 4x(x+6)(x-3)\)[/tex]:

We start by simplifying the expression on the right side:

[tex]\(4x(x+6)(x-3) = 4x(x^2 + 3x - 18) = 4x^3 + 12x^2 - 72x\).[/tex]

The inequality becomes:

[tex]\(x^2 - 32 \geq 4x^3 + 12x^2 - 72x\).[/tex]

Next, we rearrange the terms to form a quadratic equation:

[tex]\(4x^3 + 12x^2 - x^2 - 72x - 32 \geq 0\).[/tex]

Simplifying further:

[tex]\(4x^3 + 11x^2 - 72x - 32 \geq 0\).[/tex]

2) [tex]\(4x(x+6)(x-3) > -8\):[/tex]

Following the same process as before, we simplify the expression:

[tex]\(4x(x+6)(x-3) = 4x^3 + 12x^2 - 72x\)[/tex].

The inequality becomes:

[tex]\(4x^3 + 12x^2 - 72x > -8\)[/tex].

Finally, by solving each inequality separately, we can determine the values of [tex]\(x\)[/tex]that satisfy the given conditions.

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The conjecture is true that is ∠1 and ∠2 are supplementary angles, then ∠1 and ∠2 form a linear pair.

Given that,

We have to determine whether each conjecture is true or false. If ∠1 and ∠2 are supplementary angles, then ∠1 and ∠2 form a linear pair.

We know that,

Supplementary angle is defined as the sum of the any two angles should be 180°.

Linear pair is nothing but the two angles which are lies on the same line.

So,

From the figure ∠1 and ∠2 are supplementary angles because sum of the two angles is 180° and ∠1 and ∠2 form a linear pair because they lie on the same line.

Therefore, the conjecture is true.

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The equation of a hyperbola given its foci and vertices, we need to determine the centre and key parameters of the hyperbola  is [tex]\dfrac{x^{2} }{4} } -\dfrac{y^{2} }{{2} } = 1[/tex]

The vertices are (±2, 0), the centre is at (0, 0).

The distance between the centre and each vertex is known as the "semi-major axis" (a). In this case, a = 2.

The distance between the centre and each focus is known as the "c" value. In this case, c = √5.

The equation of a hyperbola with its centre at the origin (0,0) is given by:

[tex]\dfrac{x^{2} }{a^{2} } -\dfrac{y^{2} }{b^{2} } = 1[/tex]

Where b represents the "semi-minor axis."

The relationship between a, b, and c in a hyperbola is ;

[tex]c^2 = a^2 + b^2[/tex]

Squaring both sides and substituting the known values, we have:

[tex](\sqrt {5)}^2 = (2)^2 + b^2[/tex]

[tex]5 = 4 + b^2\\b^2 = 5 - 4\\b^2 = 1\\b = 1[/tex]

Now we have all the required values to write the equation of the hyperbola:

[tex]\dfrac{x^{2}}{4} - y^{2} = 1[/tex]

[tex]\dfrac{x^{2} }{4} } -{y^{2} } = 1[/tex]

Therefore, the equation of the hyperbola with the given foci (± √5, 0) and vertices (± 2, 0) is:

[tex]\dfrac{x^{2} }{4} } -\dfrac{y^{2} }{{2} } = 1[/tex]

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The equation of the hyperbola with the given foci and vertices is: x²/4 - y² = 1

To write the equation of a hyperbola with the given foci and vertices, we can start by considering the standard form of a hyperbola:

(x - h)² / a² - (y - k)² / b² = 1

Given the foci (±√5, 0) and vertices (±2, 0)

Center:  the x-coordinate of the center is the average of the x-coordinates of the vertices, and the y-coordinate remains at 0.

Center = ((2 + (-2)) / 2, 0) = (0, 0)

Distance : The distance from the center to one of the vertices is given as 2.

Therefore, a = 2.

Distance from the center to the foci along the x-axis (c): The distance from the center to one of the foci is √5. Therefore, c = √5.

Using the relationship between 'a', 'b', and 'c' in a hyperbola (c² = a² + b²),

(√5)² = (2)² + b²

5 = 4 + b²

b² = 1

b = 1

Now,(x - 0)² / 2² - (y - 0)² / 1² = 1

x² / 4 - y² = 1

Therefore, the equation of the hyperbola with the given foci and vertices is: x²/4 - y² = 1

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There are no solutions to the equation |2x - 3| = -1.

The absolute value equation given is:

|2x - 3| = -1

Absolute values are always non-negative, so it is not possible for the absolute value of an expression to equal -1. Therefore, there are no solutions to this equation.

If we assume that the absolute value expression is positive, we can set it equal to the positive value on the right-hand side:

2x - 3 = 1

Adding 3 to both sides:

2x = 4

Dividing both sides by 2:

x = 2

However, upon checking this solution, we find that it does not satisfy the original equation:

|2(2) - 3| = |-1| = 1 ≠ -1

Therefore, there are no solutions to the equation |2x - 3| = -1.

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