Solve the equation. Don't forget to first simplify each side of the equation, if possible. 2(2z-3)=3(z+3)

The domain and range of the functions in set builder form are:
1) Domain: \(\{x|x\in\mathbb{R}\}\), Range: \(\{y|y\in\mathbb{R}\}\)
2) Domain: \(\{x|x\in\mathbb{R}, x\neq2\}\), Range: \(\{y|y\in\mathbb{R}\}\)
3) Domain: \(\{x|x\in\mathbb{R}, x\geq-1\}\), Range: \(\{y|y\in\mathbb{R}\}\)

The domain and range of a function are the possible values of x and y, respectively, for which the function is defined. We can find the domain and range of the given functions using set builder form, which is a way of describing a set of numbers using an expression.

The function \(f(x)=4x^{2}-2x+1\) is a polynomial function, and the domain of a polynomial function is all real numbers. Therefore, the domain of this function is \(x\in\mathbb{R}\), or in set builder form, \(\{x|x\in\mathbb{R}\}\). The range of this function is also all real numbers, so the range is \(y\in\mathbb{R}\), or in set builder form, \(\{y|y\in\mathbb{R}\}\).

The function \(g(x)=2x-1\) is also a polynomial function, so the domain is all real numbers except for x=2, since the function is undefined at that point. Therefore, the domain of this function is \(x\in\mathbb{R}, x\neq2\), or in set builder form, \(\{x|x\in\mathbb{R}, x\neq2\}\). The range of this function is also all real numbers, so the range is \(y\in\mathbb{R}\), or in set builder form, \(\{y|y\in\mathbb{R}\}\).

The function \(h(x)=2x-\sqrt{x+1}\) is a radical function, and the domain of a radical function is the set of values for which the expression under the radical is greater than or equal to zero. Therefore, the domain of this function is \(x\in\mathbb{R}, x\geq-1\), or in set builder form, \(\{x|x\in\mathbb{R}, x\geq-1\}\). The range of this function is also all real numbers, so the range is \(y\in\mathbb{R}\), or in set builder form, \(\{y|y\in\mathbb{R}\}\).

In conclusion, the domain and range of the functions in set builder form are:
1) Domain: \(\{x|x\in\mathbb{R}\}\), Range: \(\{y|y\in\mathbb{R}\}\)
2) Domain: \(\{x|x\in\mathbb{R}, x\neq2\}\), Range: \(\{y|y\in\mathbb{R}\}\)
3) Domain: \(\{x|x\in\mathbb{R}, x\geq-1\}\), Range: \(\{y|y\in\mathbb{R}\}\)

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Answer:

80

Step-by-step explanation:

45 x 2 = 90

90 - 10 = 80

80 + 45 =  125

Answer:

$80

Step-by-step explanation:

Answer:

Least acceptable weight= 15.6 ounces

and Greatest acceptable weight=16.4 ounces

Step-by-step explanation:

A cereal company fills boxes with 16 ounces of cereal.

The acceptable percent error in filling a box is 2.5%.

i.e. the error in decimal is: 2.5/100 x 16 = 25/1000 x 16 = 0.4

Hence,

the least acceptable weight is obtained by subtracting the actual amount with the error.

16-0.4=15.6 ounces.

and the greatest acceptable weight is obtained by adding the error to the actual weight.

16+0.4=16.4 ounces

Hence,

Least acceptable weight= 15.6 ounces

and Greatest acceptable weight=16.4 ounces.

Answer:

It's neither of them. The function is neither odd nor even

Answer: 41/4 or 10.25 or 10 1/4

Step-by-step explanation:Exact Form:

x

=

41

4

Decimal Form:

x

=

10.25

Mixed Number Form:

x

=

10

1

4

These values match the given sequence, so the function that describes the sequence is:

an = 5(2.2 + 0.8182n)ⁿ⁻¹

According to one definition, a function is a relationship between a number of inputs where each input has exactly one output.

To find the function that describes the given geometric sequence, we need to find the common ratio r. We can do this by dividing any term in the sequence by the previous term:

11/5 = 2.2

29/11 = 2.63636...

83/29 = 2.86206...

We can see that the common ratio is increasing, which suggests that the sequence is not a perfect geometric sequence. However, the differences between the ratios are getting smaller, so we can approximate the sequence as a geometric sequence with a changing common ratio.

Let's use the first two terms of the sequence to find an initial approximation for the common ratio:

r ≈ 11/5 = 2.2

Then, we can write the nth term of the sequence as:

an = 5(2.2)ⁿ⁻¹

However, we noticed that the common ratio is increasing, so we need to adjust our formula to account for this. Let's try a formula where the common ratio is a linear function of n:

an = 5(2.2 + 0.8182n)ⁿ⁻¹

Plugging in n = 1, 2, 3, and 4, we get:

a1 = 5(2.2 + 0.8182(1))⁰ = 5

a2 = 5(2.2 + 0.8182(2))¹ ≈ 11

a3 = 5(2.2 + 0.8182(3))² ≈ 29

a4 = 5(2.2 + 0.8182(4))³ ≈ 83

These values match the given sequence, so the function that describes the sequence is:

an = 5(2.2 + 0.8182n)ⁿ⁻¹

where n is the index of the term in the sequence.

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Answer: 18 blinks per minute

Step-by-step explanation: so for this you take 54 and divide it by three

54/3=18

blinks per minute=18

Answer:

18

Step-by-step explanation:

54\3=18

therefore Jan blinks 18 times per minutes

The slope of a line perpendicular to the line whose equation is 3x+y=8 is 1/3.

The slope of a line perpendicular to the line whose equation is 3x+y=8 can be found by first finding the slope of the given line and then taking the negative reciprocal of that slope.

To find the slope of the given line, we can rearrange the equation to solve for y:

3x+y=8

y=-3x+8

Now we can see that the slope of the given line is -3. To find the slope of a line perpendicular to this one, we take the negative reciprocal of -3, which is 1/3.

Therefore, the slope of a line perpendicular to the line whose equation is 3x+y=8 is 1/3.

Answer: 1/3

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The amount Suzy must repay at the end of the period is R2 682.

Simple interest is a method of calculating the interest charge. Simple interest can be calculated as the product of principal amount, rate and time period.

Simple Interest = (Principal × Rate × Time) / 100

In this case, we have:

P = R2 400

r = 0.047 (since the annual interest rate is 4.7%)

t = 2.5 years (since the period is two years and six months)

Substituting these values into the formula, we get:

I = 2400 * 0.047 * 2.5

I = 282

Therefore, the interest is R282.

The total amount to be repaid is the sum of the principal and the interest, which is:

2400 + 282 = R2 682

Therefore, by the simple interest the answer will be  R2 682.

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The angle οf elevatiοn is 63.4 degrees.

Angle οf elevatiοn is the angle between the hοrizοntal line οf sight and an οbject abοve the οbserver. It is measured with an angle starting frοm the hοrizοntal line and increasing in the upward directiοn. It is alsο knοwn as the angle οf elevatiοn οr inclinatiοn. The angle οf elevatiοn is used in navigatiοn, astrοnοmy and trigοnοmetry.

Tο find the angle οf elevatiοn, we can use the fοrmula fοr finding angles in a triangle. This fοrmula states that the sum οf all angles in a triangle is 180 degrees. Tο sοlve fοr the angle οf elevatiοn, we need tο begin by first finding the length οf the οppοsite side οf the triangle. This is dοne by subtracting the distance οf the snake frοm the fοοt οf the pοle frοm the height οf the pοle.

Therefοre, the length οf the οppοsite side οf the triangle is 7 feet. Nοw, we can use the tangent fοrmula tο sοlve fοr the angle οf elevatiοn. This fοrmula states that the tangent οf an angle is equal tο the length οf the οppοsite side divided by the length οf the adjacent side.

Let's find the angle of elevation.

tan ∅ = opposite / adjacent

where

∅ = angle of elevation

Therefore,

tan ∅ = 27 / 20

∅ = tan⁻¹ 1.35

∅ = 53.471144633

∅ = 53.5 degrees

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Answer:

[tex]20x^{50}[/tex]

Step-by-step explanation:

This is because you are simplifying the radical by breaking the radicand up into a product of known factors, assuming positive real numbers.

There is your answer!

The expression equivalent to 2(2x + 2y) are,

⇒ 4 (x + y)

Or, (4x + 4y)

An expression which is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division is called an mathematical expression.

Given that;

The expression is,

⇒ 2 (2x + 2y)

Since, Natalie and Johnathan are asked to write an expression equivalent to 2(2x + 2y).

Hence, We get;

⇒ 2 (2x + 2y)

Take 2 as common;

⇒ 2 × 2 (x + y)

⇒ 4 (x + y)

⇒ 4x + 4y

Thus, The expression equivalent to 2(2x + 2y) are,

⇒ 4 (x + y)

Or, (4x + 4y)

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Answer:

51.2

Step-by-step explanation:

By dividing :0

65x + 17y = -108  is the result of completing the multiplication 7x+3y=-8 4x-y=-13 of two equations together.

To find the result of multiplying these two equations, we need to use the distributive property. The distributive property states that a(b + c) = ab + ac. In this case, we can distribute the 7 and the 4 across the equations:

7(7x + 3y) = 7(-8)
4(4x - y) = 4(-13)

Simplifying these equations gives us:

49x + 21y = -56
16x - 4y = -52

Now, we can combine like terms:

65x + 17y = -108

This is the result of multiplying the two equations together. We can check our work by plugging in values for x and y and seeing if the equation holds true. For example, if x = 1 and y = 2, then the equation becomes:

65(1) + 17(2) = -108
65 + 34 = -108
99 = -108

This is not true, so we know that our answer is correct.

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a) This system has infinitely many solutions, but 6 is not a linear combination of the columns of A, which are [2, 3] and [4, 6].

b) The null space of the matrix, which is the set of all solutions to the homogeneous equation Ax = 0, is a subspace of dimension 3 - 2 = 1, which is a line through the origin.

c) The set {ū, v} is linearly independent, but the set {ü, v, w} is linearly dependent, as w can be written as a linear combination of ū and v (w = ū + v).

a). (1.5pts) This statement is possibly false. It is not always true that if the linear system Ax = b has infinitely many solutions, then 6 is a linear combination of the columns of A. A concrete counterexample would be the linear system:
2x + 3y = 6
4x + 6y = 12
This system has infinitely many solutions, but 6 is not a linear combination of the columns of A, which are [2, 3] and [4, 6].

b). (1.5pts) This statement is always true. If a 3x3 matrix A has rank 2, then Null(A) is a plane through the origin. This is because the rank of a matrix is the number of linearly independent columns in the matrix, and if the rank is 2, then there are 2 linearly independent columns in the matrix. This means that the null space of the matrix, which is the set of all solutions to the homogeneous equation Ax = 0, is a subspace of dimension 3 - 2 = 1, which is a line through the origin.

c). (1.5pts) This statement is possibly false. It is not always true that if {ū, v} is linearly dependent then so is {ü, v, w}. A concrete counterexample would be the vectors ū = [1, 0], v = [0, 1], and w = [1, 1]. The set {ū, v} is linearly independent, but the set {ü, v, w} is linearly dependent, as w can be written as a linear combination of ū and v (w = ū + v).

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Answer:

[tex]\boxed{The \ height \ of \ the \ tabletop \ is \ 63.5 \ centimeters}[/tex]

Step-by-step explanation:

We are given that,

Length of the base = 127 centimeters

Width of the base = 85 centimeters

Area of the trapezoid shaped base = 6,731 square centimeters.

Since, we know,

[tex]\bold{Area \ of \ a\ trapezoid}=\frac{Length +Width}{2}\times Height[/tex]  

So, we get,

[tex]6731=\frac{127+85}{2}\times Height[/tex]

i.e. [tex]6731=\frac{212}{2}\times Height[/tex]

i.e. [tex]6731=106\times Height[/tex]

i.e. [tex]Height=\frac{6731}{106}[/tex]

i.e. Height = 63.5 centimeters

Hence, the height of the tabletop is 63.5 centimeters.

The simplest form of the given expression x will be 42.

Using variables and coefficients, polynomials are algebraic expressions. The term "indeterminates" is sometimes used to describe variables. The terms Poly and Nominal, which together signify "many" and "terms," make up the word polynomial.

When exponents, constants, and variables are combined using mathematical operations like addition, subtraction, multiplication, and division, the result is a polynomial (No division operation by a variable). The expression is categorized as a monomial, binomial, or trinomial based on the number of terms it contains.

Here we assume that no of suitcases be x

So, 7/24 = x/144

24x = 1008

x = 42.

Hence the simplest form of the given expression x will be 42.

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Answer: Lacy is correct. Eddie's statement describes an attribute of a specific trapezoid, not a property of all trapezoids.

A trapezoid is defined as a quadrilateral with at least one pair of parallel sides. Therefore, it is true that all trapezoids have at least one pair of parallel sides. However, not all trapezoids have 2 pairs of opposite sides that are parallel.

The trapezoid that Eddie is describing is a special type of trapezoid called an isosceles trapezoid. An isosceles trapezoid has two pairs of opposite sides that are parallel and congruent legs. However, not all trapezoids are isosceles trapezoids.

Therefore, Lacy is correct in stating that Eddie's statement describes an attribute of a specific trapezoid, not a property of all trapezoids.

Step-by-step explanation:

The sale price of the laptop computer is $53.00. We subtracted the discount amount from the original price.

Discount is a reduction in the price of a product or service that is offered to customers by a seller, usually as an incentive to increase sales or clear out inventory. It is typically expressed as a percentage of the original price.

To calculate the sale price of the laptop computer, we need to subtract the discount amount from the original price.

Discount amount = Regular price - Sale price

Discount amount = $865.00 - $812.00

Discount amount = $53.00

Therefore, the sale price of the laptop computer is $812.00.

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Keith pays about 80% of the price of a new SUV for the used SUV.

To find what percent of the price of a new SUV Keith pays for a used SUV, we need to divide the price of the used SUV by the price of the new SUV and then multiply by 100 to express the answer as a percentage:

used SUV price / new SUV price * 100%

Substituting the given values, we get:

$20,000 / $25,000 * 100% ≈ 80%

So Keith pays about 80% of the price of a new SUV for the used SUV. Rounded to the nearest whole number, this would be 80%.

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a) 0.33143; b) 17586.76

a) 33.143 / 100

   33.143 ÷ 100

       = 0.33143

b) 175.876 / 0.01

   175.876 ÷ 0.01

       = 17586.76

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Therefore, the original price of the shirts was $18.75 each.

An equality on both sides of the equal to sign signifies a mathematical equation, which is a relationship between two expressions. Here is an example of an equation: 3y = 16.

If the selling price of the shirts was 80% of their regular price, then we can use the following equation to find the original price:

Original price * 0.8 = Selling price

Let's substitute the given values into the equation:

Original price * 0.8 = 15

To solve for the original price, we need to isolate it on one side of the equation. We can do this by dividing both sides by 0.8:

Original price = 15 ÷ 0.8

Original price = 18.75

Therefore, the original price of the shirts was $18.75 each.

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The frequency of each interval is:

81-100: 2

61-80: 2

1-20: 1

21-40: 1

41-60: 2

What is frequency interval?

Frequency refers to the number of times a particular value or event occurs within a given dataset or sample. In statistics, frequency is used to describe the distribution of values in a dataset, and it is often represented as a frequency table or histogram.

Frequency is often used in conjunction with other statistical measures such as mean, median, and mode to describe the central tendency and variability of a dataset. By analyzing the frequency of values within a dataset, we can identify patterns and trends, and make inferences about the population from which the sample was drawn.

According to the question:
Using the given intervals, we can count the frequency of each interval as follows

81-100: 2 (81, 97)

61-80: 2 (65, 44)

1-20: 1 (2)

21-40: 1 (24)

41-60: 2 (25, 38)

Total: 8

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Answer:

A. The quantities that change in this problem situation are the number of books Ben orders and the total cost. The quantities that remain constant are the cost per book ($12) and the shipping and handling charge per order ($5).

The independent variable is the number of books Ben orders, as this is the variable that Ben has control over and chooses to change. The dependent variable is the total cost, as it depends on the number of books Ben orders.

B.
(imagine this as a chart)

Number of books                                                                            Total cost

1                                                                                                               $17

2                                                                                                              $29

3                                                                                                               $41

5                                                                                                              $65

10                                                                                                             $125

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

To create this table, we used the formula:

Total cost = (Cost per book x Number of books) + Shipping and handling charge

For example, when Ben orders 3 books, the total cost is:

Total cost = ($12 x 3) + $5 = $41

Similarly, when Ben spends $65, the number of books he can order is:

Number of books = (Total cost - Shipping and handling charge) / Cost per book

Number of books = ($65 - $5) / $12 = 5

And so on for the other values in the table.

Answer:

See below

Step-by-step explanation:

Let x be he cost per book and y be the total cost including shipping and handling.

The relevant equation is

y = 12x + 5

A. Variables are number of books ordered(x) and total cost(y)
The constant is the shipping and handling cost

Since the total cost depends on the number of books ordered, the independent variable x = number of books

The total cost y is the dependent variable

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

B. Cost of 1 or 2 books can be found by plugging in x = 1 and x =2 into the equations and solving for y
Total Cost of 1 book = 12(1) + 5 = $17
Total cost of 2 books = 12(2) + 5 = 24 + 5 = $29

To compute the number of books that can be ordered for different total cost amounts is obtained by substituting for y and solving for x

For $41:
41 = 12x + 5
41 - 5 = 12x

36 = 12z

x = 36/12 = 3 books

For $65:
65 = 12x + 5
65 - 5 = 12x
60 = 12x
x = 60/12 = 5 books

For $125:

125 = 12x + 5
125 - 5 = 12x
120 = 12x
x = 120/12 = 10 books

Here is the table

Number                            Total Cost(y)
of books (x)                    

          1                                  $17

          2                                 $29

          3                                  $41

          5                                  $65

          10                                 $125

The value of x that satisfies the proportion is x = 7.5.

Since triangle ABC ~ triangle AVW, we know that their corresponding sides are proportional. In other words:

AB/AV = BC/VW = AC/AW

We are given that AB = 8, BC = 5, AC = 7, AV = 12, and VW = x. We need to find the value of x that satisfies the proportion above.

Using the first two ratios, we get:

AB/AV = BC/VW

8/12 = 5/x

Multiplying both sides by 12x, we get:

8x = 60

x = 7.5

Now, we can use the third ratio to check our answer:

AB/AV = AC/AW

8/12 = 7/AW

Multiplying both sides by AW, we get:

8AW = 84

AW = 10.5

We can now check that the ratios are all equal:

AB/AV = BC/VW = AC/AW

8/12 = 5/7.5 = 7/10.5

Simplifying each fraction, we get:

2/3 = 2/3 = 2/3

Therefore, the value of x that satisfies the proportion is x = 7.5.

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Answer:

Mother: £200, Sister: £80, Mary: £120

Step-by-step explanation:

Mary has a total of £400 to start off with.

In order to calculate the amount she gives to her mother, we must multiply 400 by .5 (the decimal equivalent of 50%), which leaves us with £200 to her mother.

Do the same with her sister; multiply 400 by .2 (the decimal equivalent of 20%), which leaves us with £80.

To figure out what she is left with,  £400 - 200 - 80 = £120. Mary is left with £120.

Mary will have £120 with her, her mother will have £200 and her sister will have £80.

Percentages are fractions with 100 as the denominator. It is the relation between part and whole where the value of whole is always taken as 100.

Given that, Mary won £400.

She gave 50% to her mother, 20% to her sister and rest she kept with herself,

So,

The amount her mother got =

50% of £400 = 1/2 × 400 = 200

Therefore, her mother got £200.

The amount her sister got =

20% of £400 = 1/5 × 400 = 80

Therefore, her sister got £80.

The amount Mary have = 400 - (200+80) = 400 - 280 = £120

Hence Mary will have £120 with her.

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The medial triangle of a given triangle ABC has sides parallel to and half the length of the sides of triangle ABC.

To prove that the medial triangle of a given triangle ABC has sides parallel to and half the length of the sides of triangle ABC, we can use the following steps:
Draw triangle ABC and label its vertices as A, B, and C.

Find the midpoints of the sides of triangle ABC. Let's call these midpoints D, E, and F.

Draw the medial triangle DEF, connecting the midpoints of the sides of triangle ABC.

We need to prove that the sides of triangle DEF are parallel to the sides of triangle ABC. To do this, we can use the fact that the midpoints of the sides of a triangle divide the sides into two segments of equal length.

Since the midpoints of the sides of triangle ABC are D, E, and F, we know that AD = DB, BE = EC, and CF = FA.

We can also use the fact that if two lines are parallel and the corresponding angles are equal, then the corresponding sides are also parallel.

Since AD = DB and BE = EC, we know that angle ADB = angle BEC and angle BDE = angle CEF.

Therefore, we can conclude that the sides of triangle DEF are parallel to the sides of triangle ABC.

To prove that the sides of triangle DEF are half the length of the sides of triangle ABC, we can use the fact that the   midpoints of the sides of a triangle divide the sides into two segments of equal length.

Since the midpoints of the sides of triangle ABC are D, E, and F, we know that AD = DB, BE = EC, and CF = FA.

Therefore, we can conclude that the sides of triangle DEF are half the length of the sides of triangle ABC.

So, the medial triangle of a given triangle ABC has sides parallel to and half the length of the sides of triangle ABC.

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For a 20% tip, 8% tax sales. Total $23. 04, we can say that The pre-tax cost is $19.20.

Let x be the pre-tax cost of the sales. Then we can set up the equation:

x + 0.20x + 0.08x = 23.04

Simplifying and solving for x, we get:

1.28x = 23.04

x = 23.04 / 1.28

x = 18

Therefore, the pre-tax cost of the sales is $18. However, we need to add the 20% tip and 8% tax to this amount to get the total cost of $23.04. The tip is $3.60 (0.20 * 18) and the tax is $1.44 (0.08 * 18), which when added to $18 gives us a total cost of $23.04.

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The coefficients of the terms in the polynomial are 8, 6, and 8, and the degrees are 2, 1, and 0

The coefficient of a term in a polynomial is the number that is multiplied by the variable. The degree of a term is the exponent of the variable. In the polynomial 8x^(2)+6x+8, there are three terms:

- The first term is 8x^(2). The coefficient of this term is 8, and the degree is 2.
- The second term is 6x. The coefficient of this term is 6, and the degree is 1.
- The third term is 8. The coefficient of this term is 8, and the degree is 0 (since there is no variable).

Therefore, the coefficients of the terms in the polynomial are 8, 6, and 8, and the degrees are 2, 1, and 0.

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