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A system of inequalities can be solved by subtracting, adding, multiplying, or dividing both sides of the inequalities until a single variable is left on its own.

In Mathematics, an inequality can be defined as a mathematical relation that is typically used for comparing two (2) or more numerical data and variables in an algebraic equation based on any of the following inequality symbols:

For instance, the following inequality can be solved as follows;

0.75 < -1.5n

0.75 + 1.5n < 0

1.5n < -0.75

Next, we would divide both sides of the inequality by 1.5 as follows:

n < -0.75/1.5

n < -0.5

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The negative infinity, which is smaller than any real number, is the idea of infinity. For the representation of negative infinity, use the sign "-∞".

Given that,

We have to find what is negative infinity.

We know that,

The negative infinity, which is smaller than any real number, is the idea of infinity. For the representation of negative infinity, use the sign "-∞".

Look closely at this representation of infinity: opposite-pointing zip lines that travel along positive and negative infinity. To get closer to negative infinity on the zip line, they must turn left. On a number line, numbers move closer to positive infinity as they advance to the left and negative infinity as they move to the right. Positive infinity is always larger than any number, while negative infinity is always smaller.

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The required median of the first ten natural numbers is 5.5.

Natural numbers are a part of the number system, including all the positive numbers from 1 to infinity. Natural numbers are also called counting numbers because they do not include zero or negative numbers.

The median is the number in the middle of a list of numbers in ascending order, from lowest to highest.

The median is the middlemost element if the total number of elements in the list is odd, and the median is the average of the middlemost two values if the total number of elements in the list is even.

The list of the first 10 natural numbers is: 1,2,3,4,5,6,7,8,9,10

Here is the total number of elements in the list is even.

And the two middlemost elements are 5 and 6.

The required median is: 5+6 /2 = 11/2 = 5.5

Hence, the required median of the first ten natural numbers is 5.5.

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The list of the first 10 natural numbers is: 1,2,3,4,5,6,7,8,9,10

the total number of elements in the list is even.

the two middlemost elements are 5 and 6.

The required median is: 5+6 /2 = 11/2 = 5.5

the required median of the first ten natural numbers is 5.5.

The domain of the function y = 3x+1 is the set of all real values

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

Function: y = 3x + 1

The above function is a linear function

A linear function can take any real number as its input

This means that the domain of the function contains all real values

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The solutions of the cubic equation x³ + 6x² + 11x + 6 = 0 will be -1, -2, and -3.

The allocation of weights to the important variables that produce the calculation's optimum is referred to as a direct consequence.

It is a method for dividing a polynomial into pieces that will be multiplied together. At this moment, the polynomial's value will be zero.

The equation is given below.

x³ + 6x² + 11x + 6 = 0

Factorize the equation, then we have

x³ + 6x² + 11x + 6 = 0

x³ + x² + 5x² + 5x + 6x + 6 = 0

x²(x + 1) + 5x(x + 1) + 6(x + 1) = 0

(x + 1)(x² + 5x + 6) = 0

(x + 1)[x² + 3x + 2x + 6] = 0

(x + 1)(x + 3)(x + 2) = 0

x = -1, -2, -3

The solutions of the cubic equation x³ + 6x² + 11x + 6 = 0 will be -1, -2, and -3.

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Two months would pass before they received the same sum of money.

In mathematics, equivalence is a relationship that states that two quantities have the same value or that two mathematical expressions reflect the same mathematical object. A = B is the symbol for the equality of A and B, and it is also pronounced as A = B.

Total amount saved = amount started with + (amount deposited per month x number of months)

Amount saved by Tonya = $60 + $20m

Amount saved by Jull = $20 + $40m

The two aforementioned equations would be equivalent if they both saved the same amount of money:

60 + 20m = 20 + 40m

Take the below actions to find the value of m:

60 - 20 = 40m - 20m

40 = 20m

m = 40 / 20

m = 2

In the calculation that shows how much money Tonya has overall saved, replace m with:

$60 + $20(2)

$60 + $40 = $100

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The method to use when proving triangle congruence depends on the information given about the triangles.

Triangle congruence is a doctrine of mathematics which states that two triangles are congruent when all the corresponding sides and angles of both triangles have the same measure. This means that the two triangles are of the same size and shape, and each of their sides and angles correspond to each other in terms of size and measure. Triangle congruence is an important theorem in geometry, and is often used for solving various problems in mathematics.

If two triangles have three pairs of corresponding congruent sides, then the Side-Side-Side (SSS) method can be used. If two triangles have two pairs of corresponding congruent angles and a pair of corresponding congruent sides, then the Angle-Side-Angle (ASA) method can be used. If two triangles have two pairs of corresponding congruent sides and a pair of corresponding congruent angles, then the Side-Angle-Side (SAS) method can be used. If two triangles have three pairs of corresponding congruent angles, then the Angle-Angle-Side (AAS) method can be used.

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The statements that are true are the volume of the box is v = lwh = 864 in³ and the volume can be expressed in an equation as v = 6x² = 864in³

The volume of a square box is equal to the cube of the length of the side of the square box. The formula for the volume is V = l³, where "l" is the length of the side of the square box.

In the given problem, the volume of the box is 864 cubic inches.

However, we have to remember that the box isn't a square, but rather the base has a square shape.

The volume of the figure is

v = l * w * h

If we divide the volume by the height, we will have the area of the figure

Area = volume / height

Area = 864 / 6

Area = 144 square inches

Since the base has a square shape;

A = l²

144 = l²

l = √144

l = 12 in

The side length is 12 inches.

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Answer:

-14

Step-by-step explanation:

The roots of the quadratic equation 4x² + 4√3x - 3 = 0 are two distinct imaginary roots.

To find the roots of a quadratic equation, we have to use the quadratic formula. The formula is x = [-b ± √(b² - 4ac)]/2a. In this case, a = 4, b = 4√3, and c = -3. Plugging in the values, we get x = [-4√3 ± √(4√3² - 4*4*(-3))]/8. Simplifying and solving, we get x = [-2i ± √12]/4. Since both solutions contain an imaginary component, this means that the roots of the equation are two distinct imaginary roots.

x = [-4√3 ± √(4√3² - 4*4*(-3))]/8

x = [-4√3 ± √(48)]/8

x = [-2i ± √12]/4

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The vertex of the graph of f(x) = |x + 8| - 3 is (-8, -3). It lies in the third quadrant.

Therefore the answer is (-8, -3).

Simplifying the given graph to the form f(x) = a|x - h| + k

f(x) = |x + 8| - 3

f(x) = 1 × |x + 8| - 3

where a = 1, h = -8, k = -3

So f(x) = |x + 8| - 3 is symmetrical about the line x = -8. At x = -8, y = -3.

Therefore, the vertex of the graph is (-8, -3).

--The question is incomplete, answering to the question --

"What is the vertex of the graph of f(x) = |x + 8| - 3?"

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Answer:

-20

Step-by-step explanation:

if y varies jointly as x and z, then y = k * x * z

where k is the variation constant.

-40 = k * -2 * -1

-40 =2k

k = -20

(a) The probability of obtaining two rods of the desired length is (100/200) * (99/199) = 0.495.

This can be found by using the formula for independent events:

P(A and B) = P(A) * P(B)

Since the two draws are of rods without replacement,

the probability of the first-rod being of the desired length = 100/200 (there are 100 desired-length rods out of a total of 200).

The probability of the second-rod being of the desired length = 99/199 (since one of the desired length rods has already been drawn). Multiplying these two probabilities gives us the answer.

(b)The probability of obtaining exactly one rod of the desired length

=(100/200) * (100/199) + (100/200) * (100/199)

= 0.990.

This can be found by using the formula for the union of two events: P(A or B) = P(A) + P(B).

Since the two draws are rods without replacement, the first draw has a probability of 100/200 of being of the desired length (there are 100 desired-length rods out of a total of 200).

The probability of the second draw being of the desired length is 100/199 (since one of the desired length rods has already been drawn). Adding these two probabilities gives us the answer.

(c) The probability of obtaining none of the desired length is (50/200) * (50/199) = 0.015.

This can be found by using the formula for independent events

: P(A and B) = P(A) * P(B).

Since the two draws are of rods without replacement, the probability of the first-rod being of the undersized length is 50/200 (there are 50 undersized rods out of a total of 200).

The probability of the second-rod being of the undersized length is 50/199 (since one of the undersized rods has already been drawn). Multiplying these two probabilities gives us the answer.

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If the length of segment ef is 8 units and the length of segment ed is 10 units. The length of segment fa is 6 units.

Using Pythagoreans theorem formula to find the length of segment

c² = √a² + b²

Where:

c = Length

a = 10 units

b = 8 units

Let plug in the formula

c² = √10² - 8²

c² = √100 - 64

c² =  √36

c = 6 units

Therefore we can conclude that the length is 6 units.

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The possible rational zeros of f(x) = [tex]2x^{3} - 15x^{2} +9x-22[/tex] are ±1, ±2, ±11, ±22, ±1/2, ±11/2.

What is mean of rational zero ?

A rational zero is a rational number, which is a number that can be written as a fraction of two integers. An irrational zero is a number that is not rational, so it has an infinitely non-repeating decimal.

By Rational Zero Theorem,

If P(x) is a polynomial with integer coefficients and if is a zero of P(x) (P( ) = 0),

then p is a factor of the constant term of P(x)

q is a factor of the leading coefficient of P(x)

Possible value of rational zero is p/q

Given, f(x) = [tex]2x^{3} - 15x^{2} +9x-22[/tex]

Here, constant term, p = +22

Leading coefficient, q = +2

The factors of the constant term +22 are ±1, ±2, ±11, ±22.

Therefore , The possible rational zeros of f(x) = [tex]2x^{3} - 15x^{2} +9x-22[/tex] are ±1, ±2, ±11, ±22, ±1/2, ±11/2.

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By distributing the division, we will get the simplified expression:

(3/2)*x² + 2

Here we have the following quadratic expression:

(9x² + 12)/6

Remember that the division is distributive, then we can rewrite:

(9x² + 12)/6 = 9x²/6 + 12/6

Now we can just simplify the two fractions.

 (9x² + 12)/6 = (9/6)*x² + 2

                   = (3/2)*x² + 2

This is the most we can simplify the given expression.

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A function is increasing if the function value increases on increasing the input value and the function is decreasing if the function value decreases on increasing the input value.

We know that a function is a relation between input and output values where each input has exactly one output.

Consider a function y = f(x)

For a given function if the value of y increases on increasing the value of x, then the function is called as an increasing function

If the y-value decreases on increasing the value of x, then the function is called as a decreasing function.

Another method is that if a function f(x) is differentiable on an open interval, we calculate the derivative of function f(x)

If the derivative f′(x) > 0 on an open interval, then function f(x) is increasing on that interval.

If the derivative f′(x) < 0 on an open interval, then function f(x) is decreasing on that interval.

Therefore, if the output value of function increases on increasing the input value then the function is an increasing function and if the output value of function decreases on increasing the input value then the function is an decreasing function.

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The first Quartile of the given set of data is 13.5.

The value that splits a list of integers into quarters can be calculated with the help of the quartile formula. The information is separated into quartiles after being initially organized in ascending order. While quartiles divide data into four equal halves, the median divides it into two equal portions.

A set of observations can be divided into four equal halves using the quartile formula. Between the first term and the median is where the first quartile is located.

The quartiles are shown when the set of observations is organized in ascending order.

First Quartile (Q1) = ((n + 1)/4)thTerm

Given observations are

15, 18, 10, 20, 23, 28, 12,16

To find the First Quartile/Lower Quartile.

Arrange the given numbers in ascending order.

10, 12, 15, 16, 18, 20, 23, 28                                                        

Here there are 8 numbers

median is the = 4th term+5th term / 2 = 16+18/2 = 17

the median of the numbers below the median of the whole data set, 17. These numbers are 10,12,15,16. Since there are also an even number of numbers in this data set, the median is the average of its two middle numbers (12+15)÷2 = 27÷2 or 13.5.

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The mean absolute deviation (MAD) is a measure of forecast accuracy that is used to detect random factors in a data set. It is calculated by taking the mean of the absolute differences between the actual values and the forecast values.

The MAD can be used to eliminate forecast errors by identifying and removing random factors that are causing the errors. It is not typically used to estimate the trend line or measure forecast accuracy. Mean absolute deviation (MAD) is a measure of the variability or dispersion of a data set. It is calculated by taking the mean of the absolute differences between the individual values in the data set and the mean of the data set.

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The nature of root equation is depends of their types.

(1) When D=0, the roots in this situation are x = -b/2a.

(2) When D>0, the roots are real and unequal.

(3) When D<0, the roots are imaginary and unequal.

The type of roots depends on the discriminator:

We will talk about the following scenarios involving the nature of roots in accordance with the discriminates value.

Case: D=0

The roots of the quadratic equation ax^2 + bx + c = 0 are real and equivalent if the discriminate is equal to zero (b^2 - 4ac = 0), a, b, and c are real values, and a(0) . The roots in this situation are x = -b/2a. The X axis is the only location at which the equation's graph meets it.

Case: D>0

The roots of the quadratic equation ax2 + bx + c = 0 are real and unequal if the discriminate is bigger than zero (b2 - 4ac > 0), a, b, and c are real integers, and a0. The X-axis is intersected by the equation's graph twice.

Case: D<0

When a, b, and c are real numbers and the discriminate is smaller than zero (b2 - 4ac 0), the roots of the quadratic equation ax2 + bx + c = 0 are imaginary and unequal. There are conjugate pairings of roots. The equation's graph does not extend to the X-axis.

Case: D > 0 and perfect square

The roots of the quadratic equation are real, unequal, and rational if D > 0 and a perfect square.

Case: D < 0 and perfect square

The roots of the quadratic equation are real, unequal, and irrational if D > 0 and not a perfect square.

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An example of distributive property is 3 × (4 + 7) = (3 × 4) + (3 ×  7).

Distributive Property is a property of multiplication and it is related with the distribution of numbers.

It states that the product of three numbers do not change even if the distribution of numbers are changed.

The distributive property of multiplication states that:

a × (b + c) = (a × b) + (a ×  c).

Let us verify it with help of an example.

3 × (4 + 7) = (3 × 4) + (3 ×  7)

Taking L.H.S

3 × (4 + 7) = 3 × 11 = 33

Taking R.H.S

(3 × 4) + (3 ×  7) = 12 + 21 = 33

Therefore, L.H.S = R.H.S

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The median of the set of numbers 18, 16, 12, and 14 is 15. This is the average of the two middle numbers in the ordered set, and it is not affected by extreme values or outliers.

The median of a set of numbers is the value that is exactly in the middle of the set when it is ordered from least to greatest. In this case, the set of numbers is 18, 16, 12, and 14. When we order this set from least to greatest, we get 12, 14, 16, and 18. There are four numbers in this set, so the median is the average of the two middle numbers. The two middle numbers in this set are 14 and 16, so the median is (14 + 16) / 2 = 15.

To find the median, we first need to arrange the numbers in order from least to greatest. If there is an odd number of numbers in the set, the median is the middle number. If there is an even number of numbers in the set, the median is the average of the two middle numbers. In this case, there are four numbers in the set, so the median is the average of the two middle numbers.

In summary, the median of the set of numbers 18, 16, 12, and 14 is 15. This is the average of the two middle numbers in the ordered set, and it is not affected by extreme values or outliers.

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Answer:

  1282 miles

Step-by-step explanation:

You have a rectangle with sides 365 and 276 miles, and you want to know the perimeter. (This is an approximation of the state of Wyoming.)

The perimeter of a rectangle is the sum of its side lengths. The formula is ...

  P = 2(L +W)

  P = 2(365 +276) = 2(641)

  P = 1282

The perimeter of Wyoming is about 1282 miles.

If a new doughnut shop lost $624 during its first 13 days of business. If it lost the same amount of money each day, the loss each day is: A. (−624)÷13=−48; The doughnut shop lost $48 each day.

Given data:

Amount lost = $624

Days of business = 13 days

Now let determine the amount that the doughnut shop lost per day

Loss per day = Amount lost / Days of business

Loss per day  = - $624/13 days

Loss per day  = -$48

Therefore we can conclude that the correct option is A

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Answer:

x=-³√4-1

Step-by-step explanation:

x³+3x²+3x+5=0

(x+1)³+4=0

(x-1)³=-4

x+1=-³√4

x=-³√4-1

(1) 6/36

(2) 30/36

There are six possible outcomes. The probability of obtaining any side/number is 1/6,

Then for 2 dies it’s 2/36

Total possible outcomes=total numbers of the first die (6)× total numbers of the second die(6)= 36

The outcomes of rolling two dice are below:

 1 2 3 4 5 6

1 (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6)

2 (2, 1) (2, 2)(2, 3)(2, 4)(2, 5)(2, 6)

3 (3, 1) (3, 2)(3, 3)(3, 4)(3, 5)(3, 6)

4 (4, 1) (4, 2)(4, 3)(4, 4)(4, 5)(4, 6)

5 (5, 1) (5, 2)(5, 3)(5, 4)(5, 5)(5, 6)

6 (6, 1) (6, 2)(6, 3)(6, 4)(6, 5)(6, 6)

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The present age of the Son will be 10 years and Father's present age will be 40 years.

We will solve this problem with the concept of age.

Age is referred to as the length of time that a person or item has been alive. Problems based on age generally consist of information about the ages of two or more people and a relation between their ages in the present or future or past. Using this information it is asked to calculate the ages of one or more people in the present past or future.

The important thing to keep in mind while solving problems of ages.

If the present age is y, then n times the present age = ny.

If the present age is x, then age n years later/hence = x + n.

If the present age is x, then the age n years ago = x – n.

The ages in a ratio a: b will be ax and bx.

If the current age is y, then 1/n of the age is y/n.

According to our problem:

Let the present age of the son = x

The present age of the father = 4x

After 5 years

Son age = x+5.

Father age = 4x+5

According to the condition =

4x+5 = 3(x+5)

4x+5 = 3x + 15

x = 10

Therefore the present age of the son will be x = 10 years and the father will be 4x= 40 years.

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