do eight-digit numbers with no digits 9 in their decimal representations constitute more than half of all eight-digit numbers?

A number line from negative 5 to 5 in increments of 1. An open circle is at 3 and a bold line starts at 3 and is pointing to the right.

Inequality refers to the state of being unequal, or not having equal access to resources, opportunities, or privileges. Inequality can manifest in many different ways, including but not limited to income inequality, wealth inequality, social inequality, gender inequality, racial inequality, and educational inequality.

Inequality can be caused by a variety of factors, including systemic discrimination, unequal distribution of resources, and societal structures that favor certain groups over others. It can have a significant impact on individuals and communities, leading to disparities in health, education, economic opportunities, and overall well-being. Addressing inequality requires a multifaceted approach that involves policies and initiatives aimed at promoting equal access to resources, opportunities, and rights for all individuals.

Which number line represents the solution set for the inequality 3(8 – 4x) < 6(x – 5)?

A number line from negative 5 to 5 in increments of 1. An open circle is at 3 and a bold line starts at 3 and is pointing to the left.

A number line from negative 5 to 5 in increments of 1. An open circle is at 3 and a bold line starts at 3 and is pointing to the right.

A number line from negative 5 to 5 in increments of 1. An open circle is at negative 3 and a bold line starts at negative 3 and is pointing to the left.

A number line from negative 5 to 5 in increments of 1. An open circle is at negative 3 and a bold line starts at negative 3 and is pointing to the right.

To solve the inequality3(8 – 4x) < 6(x – 5)[tex]3(8 - 4x) < 6(x – 5)[/tex], we first simplify both sides of the inequality:

[tex]3(8-4x) < 6(x -5)[/tex]

[tex]24 - 12x < 6x - 30[/tex]

[tex]54 < 18x[/tex]

[tex]3 < x[/tex]

So, the solution set for the inequality is x > 3. We represent this on a number line by placing an open circle at 3 and drawing a bold line pointing to the right to indicate all the values of x that satisfy the inequality. Therefore, the correct answer is:

A number line from negative 5 to 5 in increments of 1. An open circle is at 3 and a bold line starts at 3 and is pointing to the right.

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Answer:

it's 3

Step-by-step explanation:

The two-column proofs that line segments AC ≅ EC are shown below

Given that

AB || DE

BC ≅ DC

The proof is as follows

Statement                                                Reason

AB || DE and BC ≅ DC                            Given

AB ≅ DE                                                   CPCTC

AC ≅ EC                                                   CPCTC (proved)

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Rounding to the nearest tenth, the measure of UV is 7.1.

An angle is a measure of space between two lines or planes. It is measured in degrees, using a protractor, and can range from 0 to 360. Angles can be either acute, obtuse, right, straight, reflex, or full. Acute angles are angles that measure less than 90 degrees, obtuse angles measure more than 90 degrees, right angles measure exactly 90 degrees, straight angles measure 180 degrees, reflex angles measure more than 180 degrees, and full angles measure 360 degrees.

To find the measure of UV in ∆ UVW, we will use the law of cosines. The law of cosines states that [tex]c^{2} =a^{2} +b^{2} -2ab cos C[/tex], where c is the length of the side opposite angle C, a and b are the lengths of the other two sides, and C is the angle opposite side c. In this case, side c is UV, a is UW, b is VW, and C is ∠ W.

Plugging in the given values, we get:

[tex]U^{2}V^{2}[/tex] = [tex]9^{2} + 12^{2} - 2 *9* 12 cos *35[/tex]

[tex]U^{2} V^{2}[/tex] = 81 + 83.44 – 162.08 cos 35°

[tex]U^{2} V^{2}[/tex] = 164.44 – 114.45

[tex]U^{2} V^{2}[/tex] = 49.99

Taking the square root of both sides, we get:

UV = 7.086

Rounding to the nearest tenth, the measure of UV is 7.1.

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The equation of the parabola in factored form is: y = 16(z - 1/2)(z - 8)

What is parabola?

A parabola is a symmetrical plane curve that is shaped like an arch. It is a quadratic function and is defined by the equation y = ax² + bx + c, where a, b, and c are constants.

Ryan's reasoning is justified because the z-intercepts of a parabola are the points where the parabola intersects the z-axis, which are the points where x = 0. Therefore, if the parabola can be expressed in the form y = a(x + 1)(x - 4), then its z-intercepts are at x = -1 and x = 4. This is because when x = -1, (x + 1) = 0 and when x = 4, (x - 4) = 0, which makes y = 0, indicating that the parabola intersects the z-axis at these two points.

To find the value of a, we need to use the given information that the y-intercept of the parabola is at (0, 2). Substituting x = 0 and y = 2 into the equation y = a(x + 1)(x - 4), we get:

2 = a(0 + 1)(0 - 4)

2 = -4a

Therefore, a = -1/2. So the equation of the parabola is y = (-1/2)(x + 1)(x - 4), in factored form.

To find the equation of the parabola in factored form y = a(z-p)(z-g), we can use the given information about its intercepts and symmetry axis. Since the symmetry axis is parallel to the y-axis, the parabola is of the form y = a(z - h)² + k, where (h, k) is the vertex. We know that the a-intercepts are -2 and 3, which means that the points (-2, 0) and (3, 0) lie on the parabola. Substituting these points into the equation, we get:

0 = a(-2 - h)² + k

0 = a(3 - h)² + k

Solving for h and k, we get:

h = 1/2

k = 4

Therefore, the vertex is at (1/2, 4), and the equation of the parabola is:

y = a(z - 1/2)² + 4

We can find the value of a by using the fact that the y-intercept is 4. Substituting z = 0 and y = 4, we get:

4 = a(0 - 1/2)² + 4

4 = a(1/4)

a = 16

Therefore, the equation of the parabola in factored form is:

y = 16(z - 1/2)(z - 8), which is sometimes referred to as intercept form because it explicitly shows the intercepts of the parabola on the z-axis.

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Answer:

[tex]A = 212.1 \text{ cm}^2[/tex]

Step-by-step explanation:

We can identify the bases of the trapezoid as:

9cm and 21.3 cm

(notice how they are parallel)

We know the height is 14 cm.

Using these values, we can solve for the trapezoid's area.

[tex]A = \dfrac{1}{2}(b_1 + b_2) \cdot h[/tex]

↓ plugging in the values

[tex]A = \dfrac{1}{2}(9 + 21.3) \cdot 14[/tex]

↓ simplifying the parentheses

[tex]A = \dfrac{1}{2}(30.3) \cdot 14[/tex]

↓ multiplying 14 and (1/2)

[tex]A = 7 \cdot 30.3[/tex]

↓ multiplying

[tex]A = 212.1 \text{ cm}^2[/tex]

Options A and B are both correct. Options C, D, and E are not correct, as they do not take into account the discount that James is receiving.

In mathematics, a percentage is a number or ratio that can be expressed as a fraction of 100. Percentage means, a part per hundred. The word per cent means per 100. It is represented by the symbol “%”.

The correct calculations to determine how much James will pay in total are:

First, we need to calculate how much 10% of $38.50 is:

10% of $38.50 = 0.1 × $38.50 = $3.85

So, James will receive a discount of $3.85.

To determine how much James will pay in total, we can use either of the following calculations:

A. James' total cost after the 10% discount would be: 38.5 - 0.9(38.5) = 38.5 - 34.65 = $3.85

B. The 10% discount on the total cost can be calculated as 0.1(38.5) = $3.85, which can be subtracted from the total cost to get James' final cost as 38.5 - 3.85 = $34.65

C. James' final cost after the 10% discount would be 1.1(38.5) = $42.35, which is not the correct answer as it does not take into account the discount.

D. James' total cost after the 10% discount would be: 0.9(38.5) = $34.65, which is the correct answer.

E. The calculation 0.1(38.5) = $3.85 represents the amount of the discount, and is not the final cost.

Therefore, options A and B are both correct. Options C, D, and E are not correct, as they do not take into account the discount that James is receiving.

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1) Given that 1789 children visited a Taco Bell in 24 hours. We can calculate the number of students who went every hour by dividing this number by the number of hours:

[tex]1789 / 24 = 74.54\\\frac{1789}{24} = 74.54[/tex]

We get 75 students each hour, rounded up to the nearest whole number.

If 24 children shared, we can calculate the total number of children who shared by multiplying the number of hours by the number of children shared every hour:

24 x 75 = 1800

As a result, 1800 children shared.

2)If 1638 men cheated on their wives in one week, multiplying by 7 gives us the number of men who cheated on their wives in seven weeks:

1638 x 7 = 11,466

As a result, 11,466 men cheated on their wives in just 7 weeks.

c) If 47% of the men who cheated on their spouses in 7 weeks cheated numerous times before, we can calculate the number of men who cheated previously by multiplying the total number of men who cheated by the proportion of men who cheated previously:

11,466 x 0.47 = 5,390.02

Rounding to the nearest whole number, we get 5,390 men who have cheated in the past.

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There are [tex]18[/tex] square units in the area of the shaded rectangle.

The Latin words "rect" (which means "right") and "angulus" (which means "angle") are the roots of the English word "rectangle." These two words together form the term "rectangle." A rectangle has 90° straight angles at each of its four corners. A rectangular also has two equal-length diagonals that meet in the centre.

In addition to rectangles, we also refer to them as geometric figures, quadrilaterals, and polygons. This is so that it is clear that a rectangle fulfils the specifications of all of these other forms.

width [tex]= 7[/tex] units

length [tex]= 10[/tex] units

We remove the frame, each measure will lose[tex]2*2[/tex] units, or [tex]4[/tex] units

[tex]width = 7 units - 4 units = 3 units[/tex]

[tex]length = 10 units - 4 units = 6 units.[/tex]

[tex]A = (3 units)*(6 units) = 18[/tex] square units.

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The completed question is:

One rectangle is "framed" within another. Find the area of the shaded region if the

"frame" is 1 unit wide.

10

5

Answer:

x ≈ 51°

Step-by-step explanation:

Use trigonometry:

[tex] \sin(x°) = \frac{7}{9} [/tex]

Use the reverse (SHIFT, sin (7/9)) function on your calculator to find x:

x ≈ 51°

Answer: i would say $1.60 because 20% of $8.00 is $1.60

Each friend earned an amount of $1.6.

Percentage is defined as the parts of a number per fraction of 100. We have to divide a number with it's whole and then multiply with 100 to calculate the percentage of any number.

So the percentage actually means a part per 100.

Percentage is usually denoted by the symbol '%'.

Given that,

A group of friends worked together at a lemonade stand.

They earned $8.00 in all.

Each friend was paid 20% of the total earnings.

Total earnings = $8

Each friend earn = 20% × 8

                            = (20/100) × 8

                            = 0.2 × 8

                            = 1.6

Hence the amount each friend earned is $1.6.

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Therefore, f(11) can either be -16 or 20, depending on whether we use the recursive or explicit rule.

In mathematics, a graph is a visual representation of the relationship between two or more variables. It is a collection of points (or vertices) that are connected by lines or curves (called edges) to show the relationships between them. Graphs are commonly used in many different fields of study, including mathematics, science, engineering, and social sciences, to represent and analyze data, identify patterns, and make predictions.

Here,

We can observe that this is an arithmetic sequence, where each term is obtained by adding a constant difference to the previous term. To find the recursive rule, we can use the formula:

f(n) = f(n-1) + d

where f(n-1) is the previous term, d is the common difference, and f(n) is the nth term. Using the given values, we can find that the common difference is d = -6. So the recursive rule for the sequence is:

f(1) = 80

f(n) = f(n-1) - 6

To find the explicit rule, we can use the formula:

f(n) = a + (n-1)d

where a is the first term, d is the common difference, and n is the term number.

Using the first two terms, we can find that:

a = 80

d = -6

So the explicit rule for the sequence is:

f(n) = 80 - 6(n-1)

To find f(11), we can substitute n = 11 into either the recursive or explicit rule:

Using the recursive rule:

f(11) = f(10) - 6

= (f(9) - 6) - 6

= (f(8) - 12) - 6

= (f(7) - 18) - 6

= (f(6) - 24) - 6

= (f(5) - 30) - 6

= (20 - 30) - 6

= -16

Using the explicit rule:

f(11) = 80 - 6(11-1)

= 80 - 6(10)

= 20

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Answer: Option B

Step-by-step explanation:

A. The range of the ages are greater. This statement is false because in the box plot movies A, the range is 26 (subtract highest value from lowest value) which is not greater. Thus, this statement is false.

B. A smaller range normally means that there is a more consistent range. TRUE

C: The IQR, looking at the box plot, is much greater.

D: In Movie 2, the median ages of attendees are greater. False

The parametric equation of the line is given by;  x(t)=-2 + 5ty(t)=2 - 3tz(t)=5 - 3t

To find the vector and parametric equations for the line through the point P(-2, 2, 5) and orthogonal to the plane 5x - 3y - 3z = -3, the following  

Write the equation of the plane.5x - 3y - 3z = -3  is the given equation

Find a vector that is orthogonal to the planeThis can be done by finding the normal vector of the plane. Using the coefficients of x, y, and z in the equation of the plane.  A normal vector of the plane is obtained and simplified:<5,-3,-3>.

Find the vector equation of the line.

Let L be the line through P(-2, 2, 5) and orthogonal to the plane. A direction vector of the line,  v is the normal vector of the plane (which is <5,-3,-3>), a point P(-2, 2, 5) on the line can be used to obtain the vector equation of the line:

Parametric form (parameter t, and passing through P when t = 0):x(t)=-2 + 5ty(t)=2 - 3tz(t)=5 - 3t

The vector form:<-2, 2, 5> + t<5,-3,-3> The parametric equation of the line is given by;  x(t)=-2 + 5ty(t)=2 - 3tz(t)=5 - 3t

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The derivative of the sigmoid function is f(x) × (1 - f(x)).Answer: f(x) × (1 - f(x)).

As per the question, we are supposed to determine the derivative of the sigmoid function.

The sigmoid function is given as,

[tex]f(x) = 1 / (1 + e^-x)[/tex]

To determine the derivative of the sigmoid function, we first find out the derivative of f(x) with respect to x using the quotient rule,

[tex]f'(x) = d/dx [ 1 / (1 + e^-x) ][/tex]

[tex]f'(x) = [ (d/dx)[1] × (1 + e^-x) - 1 × (d/dx)[1 + e^-x] ] / [ (1 + e^-x)^2 ][/tex]

[tex]f'(x) = [ 0 × (1 + e^-x) - (-e^-x) ] / [ (1 + e^-x)^2 ][/tex]

[tex]f'(x) = e^-x / (1 + e^-x)^2[/tex]

Now we are supposed to write the answer in terms of f(x),

[tex]f'(x) = e^-x / (1 + e^-x)^2f(x) = 1 / (1 + e^-x)[/tex]

Substituting f(x) in the above equation,

f'(x) = f(x) × (1 - f(x))

Therefore, the derivative of the sigmoid function is f(x) × (1 - f(x)).Answer: f(x) × (1 - f(x)).

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The value of the expression 5x - 3y, is -29 obtained by elimination method by eliminating the two equations.

The equations are :

3x - 4y= -24     -----------   A

3x +2y= -6       -----------   B

On multiplying minus sign in B equation, to eliminate the x- component in the equation.

3x +2y= -6 becomes (-3x - 2y = 6)

Then, the equation becomes,

3x - 4y= -24

-3x - 2y = 6          (+ 3x - 3x are equal and opposite becomes zero.)

------------------

    -6y = -18

------------------

6y = 18          (both minus sign gets cancelled)

y = 18/6

 = 3

The value of y=3.

Substitute the value of y=3 in A equation.

Equation A is,  3x - 4y= -24

3x - 4(3) = -24

3x = -24 + 12

3x = -12

x = -12 / 3

 = -4

The value of x= -4.

Thus, the value of x= -4 and y = 3. By using this x and y values, the equation  5x - 3y, can be solved as,

= 5x - 3y

= 5(-4) - 3 (3)

= -20 -9

= -29

The value of the expressions 5x - 3y = -29.

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The common end point I believe :)

Using the given clues, we can fill out the table as follows:

Section     Number      Color

A                    3             Green

B                    5              Purple

C                    9              Brown

D                    8              Orange

E                    1               Yellow

F                    6                Blue

G                   4                 Pink

Using this information, we can deduce the values of each section:

The number 8 is in all shapes

So, D = 8

Number C is the largest

For the blue section, we have

F = Blue

For the triangle, we have

DEF = 48

D + E + F = 15

So, we have

EF = 6

E + F = 7

This means that

(E, F) = (1, 6) or (6, 1)

Number 1 is yellow

So, we have

Since F is not yellow, then

A = Green

A and G add up to 7, and A's number is smaller than G's.

So A and G must be (1, 6), (3, 4) because 2 and 7 are not used

Because of the values of E and F, we have

A = 3 and G = 4

Green and pink are in the rectangle only

So:

G = Pink

Purple and Orange are in the circle

So:

(B, D) = (Orange, Purple), (Purple, Orange)

The purple section is 5 and D = 8

So:

(B, D) = (Purple, Orange) = (5, 8)

The table has been completely filled

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The surface area of the given triangular prism is approximately 58.26 square units.

what is triangular prism?

A triangular prism is a three-dimensional geometric shape that consists of two congruent parallel triangles as its bases and three rectangular faces that connect the corresponding sides of the two triangles.

In the given question,

To find the surface area of a triangular prism, we need to add the area of all the faces of the prism. A triangular prism has two congruent triangular bases and three rectangular lateral faces.

First, let's calculate the area of the triangular bases. We can use the formula for the area of a triangle:

Area of a triangle = (1/2) * base * height

In this case, the base of the triangle is 5 and the height is 6. To find the height, we need to use the Pythagorean theorem because we are given the slant height of the triangular prism, which is the hypotenuse of the right triangle formed by the base, the height, and the slant height. The Pythagorean theorem is:

slant height² = base² + height²

Substituting the given values, we get:

6.5² = 5² + height²

42.25 = 25 + height²

height² = 17.25

height = √17.25 ≈ 4.15

Now we can calculate the area of each triangular base:

Area of a triangular base = (1/2) * base * height

Area of each triangular base = (1/2) * 5 * 4.15

Area of each triangular base ≈ 10.38

Next, we need to calculate the area of each rectangular lateral face. The length of each lateral face is the same as the base of the triangle, which is 5, and the width is the same as the width of the prism, which is 2.5. The height of each lateral face is the same as the height of the prism, which is 6. Therefore, the area of each rectangular lateral face is:

Area of a rectangular lateral face = length * width

Area of each rectangular lateral face = 5 * 2.5

Area of each rectangular lateral face = 12.5

Finally, we can find the total surface area of the prism by adding up the areas of the two triangular bases and the three rectangular lateral faces:

Total surface area = 2 * area of a triangular base + 3 * area of a rectangular lateral face

Total surface area ≈ 20.76 + 37.5

Total surface area ≈ 58.26

Therefore, the surface area of the given triangular prism is approximately 58.26 square units.

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Answer: 4/5

Step-by-step explanation:

Five equal sides = 5
4 is just one side, so = 1
Not landing in 4 = 4/5

Answer: 4/5 or 80%

Step-by-step explanation:

Since the spinner has five equal sections, each numbered 1 through 5, the probability of landing on any particular number is 1/5.

To find the probability of not landing on 4, we need to add up the probabilities of landing on all the other numbers and subtract that sum from 1 (since the sum of all possible outcomes must equal 1).

The probability of landing on 4 is 1/5, so the probability of not landing on 4 is:

1 - 1/5 = 4/5

Therefore, the probability of not landing on 4 is 4/5 or 0.8, which is equivalent to an 80% chance.

Hope this helped!

The solution set can be written in parametric vector form as: [tex]X = x3 [-1 0 1]^T + x2 [-2 1 0]^T[/tex]

We can rewrite the system of equations in matrix form as AX = 0, where [tex]A =[2 2 4]\\[-4 -4 -8]\\[0 -3 -9][/tex]

[tex]X = [x1 x2 x3]^T[/tex]

To solve for the solution set, we can row reduce the augmented matrix [tex][A|0].\\[2 2 4|0]\\[-4 -4 -8|0]\\[0 -3 -9|0][/tex]

[tex][2 2 4|0]\\[0 0 0|0]\\[0 -3 -9|0][/tex]

[tex][2 2 4|0]\\[0 0 0|0]\\[0 1 3|0][/tex]

[tex][2 2 4|0]\\[0 0 0|0]\\[0 1 3|0][/tex]

[tex][1 1 2|0]\\[0 0 0|0]\\[0 1 3|0][/tex]

Therefore, the system has two free variables, x2 and x3, while x1 is a pivot variable. where x2 and x3 are arbitrary constants.

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We have the following system of equations:

x₁ + 0.5x₂ + 0.1x₃ = 100

0.25x₁ + x₂ + 0.2x₃ = 75

0.3x₁ + 0.4x₂ + x₃ = 200

Statement gives the equality of different mathematical expressions is called as an equation.

Let x₁, x₂, and x₃ be the outputs of I₁, I₂, and I₃, respectively. Then the total production must satisfy the following system of equations:

For I₁:

For I₂:

For I3:

Therefore, the system equations are:

x₁ + 0.5x₂ + 0.1x₃ = 100

0.25x₁ + x₂ + 0.2x₃ = 75

0.3x₁ + 0.4x₂ + x₃ = 200

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can you add the question?

Add a picture of the question.

The straight lines that pass through the points Θ = π, Θ = 2π, Θ = 3π, and so on are known as the f-intercepts. At the multiples of, these lines cross the x-axis (or -axis).

The points where a line intersects an axis are known as the x-intercept and the y-intercept, respectively.

A periodic function, f(Θ) = tan(Θ), oscillates between positive and negative infinity at certain points that correlate to the function's zeros. These zeros are referred to as the function's x-intercepts or roots.

The x-intercepts are not clearly specified, though, because the function has a periodic nature with a period of. Or, to put it another way, the function has an unlimited number of x-intercepts at each integer multiple of π (i.e., π, 2π, 3π, -π, -2π, -3π, and so on).

The values are represented by the x-axis in the setting of the function's graph. As a result, the values of for which the function equals zero correlate to the x-intercepts or roots of the function.

We can see that the function equals zero when sin() = 0, which happens when is an integer multiple of, by using the equation tan() = sin() / cos().

As a result, the vertical lines going through Θ = π, Θ = 2π, Θ = 3π, and so on are the -intercepts of the graph of f. These lines cross the x-axis (also known as the -axis) at multiples of.

In conclusion, the straight lines at  = n, where n is an integer, are the -intercepts of the graph of f(Θ) = tan(Θ).

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The property used between step 3  is the D)addition property of inequality and step 4 is the B) division property of inequality.

In mathematics, an inequality is a mathematical statement that indicates that twο expressiοns are nοt equal. It is a statement that cοmpares twο values, usually using οne οf the fοllοwing symbοls: "<" (less than), ">" (greater than), "≤" (less than οr equal tο), οr "≥" (greater than οr equal tο).

The property used between step 3  is the addition property of inequality and step 4 is the division property of inequality. answer is D addition property of inequality.

In step 3, -6x - 8 < -2 is οbtained by adding 8 tο bοth sides οf the inequality -6x - 8 + 8 < -2 + 8.

Then, in step 4, -6x < 6 is οbtained by dividing bοth sides οf the inequality -6x - 8 < -2 by -6, which requires the use οf the division property of inequality.

Therefore, the property used between step 3 and step 4 is the addition property of inequality.

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The formula for the general term of an arithmetic sequence is:

an = 14 - 3(n-1)

Arithmetic progression (AP) is a sequence of numbers in which each term after the first is obtained by adding a fixed constant value, called the common difference, to the preceding term. For example, 2, 4, 6, 8, 10, 12, 14... is an arithmetic progression with a common difference of 2. The formula for the nth term of an arithmetic progression is given by:

an = a1 + (n-1)d

The given sequence 14, 11, 8, 5, 2 is an arithmetic sequence with a common difference of -3.

The formula for the general term of an arithmetic sequence is:

an = a1 + (n-1)d

where:

an = the nth term of the sequence

a1 = the first term of the sequence

d = the common difference between consecutive terms

n = the position of the term we want to find

Using the values given in the sequence, we have:

a1 = 14

d = -3

To find the general term, we need to determine the value of n. Let's assume that the first term of the sequence corresponds to n = 1.

For the first term (n = 1):

a1 = 14

For the second term (n = 2):

a2 = a1 + d = 14 - 3 = 11

For the third term (n = 3):

a3 = a1 + 2d = 14 - 6 = 8

For the fourth term (n = 4):

a4 = a1 + 3d = 14 - 9 = 5

For the fifth term (n = 5):

a5 = a1 + 4d = 14 - 12 = 2

Therefore, the general term for the given sequence is:

an = 14 - 3(n-1)

or

an = 17 - 3n (Alternative form)

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Answer:

8

Step-by-step explanation:

To find the length of the third side, we can use the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the lengths of the two shorter sides is equal to the square of the length of the longest side (the hypotenuse).

So, in this case, we have:

x^2 + 6^2 = 10^2

Simplifying this equation, we get:

x^2 + 36 = 100

Subtracting 36 from both sides, we get:

x^2 = 64

Taking the square root of both sides, we get:

x = 8 or x = -8

Since the length of a side of a triangle can't be negative, the length of the third side is x = 8 cm.