Use the distributive property to find the equivalent expression.
3(2 + x) =__+__x

It is not possible to construct a triangle whose sides are 8 cm 7 cm and 15 cm because the sum of two sides(8 cm and 7 cm) are equal to third side(15 cm).

In the given question, we have to check that it is possible to construct a triangle whose sides are 8 cm 7 cm and 15 cm.

The sum of the two sides must always be greater than the third side in order to determine whether or not the given sides may form a triangle.

To check this we firstly add the 8 and 7

8 + 7 = 15

15 = 15

Now we add 8 and 15

8 + 15 > 7

23 > 7

Now we add 7 and 15

7 + 15 > 8

22 > 8

It is not possible to construct a triangle whose sides are 8 cm 7 cm and 15 cm because the sum of two sides(8 cm and 7 cm) are equal to third side(15 cm).

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The solution of the given equation 3x² + 13x + 14 = 0 is given by :

option b . x = -2.

As given in the question,

Given equation is equal to :

3x² + 13x + 14 = 0

Solve the given equation by method of splitting middle term we get,

3x² + 13x + 14 = 0

⇒ 3x² + 6x + 7x + 14 = 0

⇒ 3x ( x + 2 ) + 7 ( x + 2 ) = 0

⇒ ( 3x + 7 ) ( x + 2 ) = 0

⇒ ( 3x + 7 ) = 0 or ( x + 2 ) = 0

⇒ x = -7/3 or x = -2

From the given options:

Option b . x = -2 is the solution of the equation 3x² + 13x + 14 = 0.

Therefore, the solution of the given equation 3x² + 13x + 14 = 0 is equal to : option b. x = -2.

The above question is incomplete, the complete question is:

Which of the following is one of the solution of equation 3x² + 13x + 14 = 0?

a. x = 2       b. x = -2         c. x = 3          d. x = -3

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a. Drive Co.'s revenue function R(X) = (-x^2/5)-247x.

b. The price per drive when revenue is maximized is $370.6.

c. The maximum profit made from the sale of these drives if Drive Co. incurs production costs of $179 per drive and has fixed costs of $250 is $6030.

What is a function?

The characteristic that every input is associated to exactly one output defines a function as a relationship between a set of inputs and a set of allowable outputs.

Drive Co. sells portable 905 hard drives when the price is $66/drive and can sell 800 hard drives when the price is $87/drive.

Let p be the price of hard drive and x be the number of hard drives.

When x = 905, then p = 66.

So, (x1,p1) = (905,66)

Similarly, when x = 800, then p = 87.

So, (x2,p2) = (800,87)

So, the equation of line between x and p is given as -

p-p1=[(p2-p1)/(x2-x1)}(x-x1)

Plugging in the values -

p-66=[(87-66)/(800-905)](x-905)

p-66=(21/-105)(x-905)

-105p+6930=21x-19005

-105p-21x=-19005-6930

-105p-21x=-25935

p=(-21x/105)-247

p=(-x/5)-247

So, the price function p(x) = (-x/5)-247.

Now, derive the revenue function -

Revenue = Number of drives × price per drive

Revenue = x · p(x)

Revenue = x · (-x/5)-247

Revenue = (-x^2/5)-247x

Therefore, the revenue function R(x) = (-x^2/5)-247x.

For maximum revenue -

dR/dx = R'(x) = 0 and R''(x)<0

dR/dx = -2x/5-247 = 0

-2x/5 = 247

x = -1235/2

x = -617.5

Therefore, the number of drives at maximum revenue is 618 drives.

The price per drive is -

p(618) = {-618/5}-247

p(618) = -370.6

Therefore, the price per drive is $370.6.

Now, the profit S(x) = Revenue R(x) - Total Cost C(x)

C(x) = fixed cost + variable cost

Variable cost = number of drives × price per drive

C(x) = 250 + 179x

Substituting the values -

S(x) = [(-x^2/5)-247x] - [250+179x]

S(x) = (-x^2/5)-68x-250

In order to get maximum profit differentiate the equation and equate it to 0.

dS/dx = 0

dS/dx = -2x/5-68 = 0

-2x/5 = 68

x = 340/-2

x = -170

d^2S/dx^2 = -2/5<0

So, maximum profit will be at x = 170.

S max = S(170) = [(170)^2 /5] - 68 (170) - 250

= 5780 - 11560 - 250

= -6030

Therefore, the maximum profit earned is $6030.

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By linear equation, the cost of playing skee ball is $0.25 and that of air hockey is $0.50.

Elimination and substitution are the two most frequently used techniques for simultaneous linear equations. Some questions have a more obvious solution, frequently because it makes solving the equations easier, while others leave the method selection up to the individual.

Loss by Jake=$4

Loss by Adam=$6.50

let,

hockey played=y

skee-ball played=x

for Jake,

8x+4y=4

for Adam,

10x+8y=6.50

solving both equation simultaneously,

we will get

x=1.5/6=0.25

y=0.50

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Option A will be the solution set of given linear inequalities.

What exactly are linear inequalities?

In mathematics, inequality denotes a mathematical expression in which neither side is equal. In Math, an inequality occurs when a connection produces a non-equal comparison between two expressions or two integers. The equal sign "=" in the phrase is substituted by any of the inequality symbols, such as greater than symbol (>), less than symbol (<), greater than or equal to symbol (≥), less than or equal to symbol (≤), or not equal to symbol ((≠). In mathematics, there are three types of inequalities: polynomial inequalities, rational inequalities, and absolute value inequalities.

Hence,

For given inequalities

y>4x+7

8x-2y>10

there will be no solution as the graph of these inequalities does not coincide anywhere.

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(-4,0) is the solution of the set to the system of inequalities

What is Inequality?

A  relationship between two expressions or values that are not equal to each other is called 'inequality.

The given two inequalities are

y<-3x-6 and 4x-2y<-8

Let us check for (-4, 0)

0<-3(-4)-6

0<6

4x-2y<-8

4(-4)-2(0)<-8

-16<-8

So (-4, 0) is the solution of the inequalities.

Now let us check for (0, 2)

replace y value as 2 and x as 0.

2<-3(0)-6

2<-6

Which is not true.

Hence, (-4,0) is the solution of the set to the system of inequalities.

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64 marks must he score on her fourth exam to get exactly 80% in the course.

From the question, we are informed that Lisa is currently taking physics as one of her electives in school and that her grade at the end of the year is determined by the average of four exams, each worth 100 points.

First test = 84,

Second test = 92

Third test = 80

Forth test = y

And,

The fourth test is denoted by y.

Since we are told that she must get exactly 80% in the course and she took 4 exams, that means her total score will be: = 80 × 4 = 320

Therefore,

⇒ 84 + 92 + 80 + y = 320

⇒ 256 + y = 320

⇒ y = 320 - 256

⇒ y = 64

The fourth exam's score is 64.

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Answer:  ∠M = 98°

Step-by-step explanation:

∠M = (3x-16)°

∠N = x°

∠O = (x+6)°

Sum of all angles in a triangle is 180°

Therefore, 180° = ∠M +∠N +∠O, substitute the given values

180° = (3x-16)° + x° + (x+6)° (add/combine like terms)

180° = (3x+x+x)° + (-16+6)°

180° = (5x)° + (-10°)

180° + 10° = (5x)° ( divide both sides by 5)

190°/5 = (5x)°/5

38° = x°

IF ∠M = (3x-16)°, substitute x°=38°,

Therefore, ∠M = 3(38)-16

                   ∠M = 114 - 16 = 98°

The outcome of translating y = x^2 downward by 3 units and to the left by 4 units is y = (x+4)^2-3.

A graph is a mathematical depiction of a network that describes the connection between lines and points.

We must determine which function results from translating y = x^2 downward by 3 units and to the left by 4 units.

A translation is a movement of the graph that is either horizontally or vertically parallel to the axis.

Replace x with x+4 y = (x+4)^2-3 to shift the graph four units to the left.

As a result, y = (x+4)^2-3 is the result of translating y = x^2 by 3 units downward and 4 units to the left.

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In given word problem Andre has more money than Elena after four weeks.

What is word problem?

A math word problem is a question that is written as one or more sentences and asks students to use their mathematical understanding to solve an issue from "real world." In order to understand the word problem, they must be familiar with the terminology that goes along with the mathematical symbols that they are used to.

Here,

Initial investment of Andre= $50

She add $5 per week for next 4 weeks , then

=> 4×5 = $20

Then total savings of Andre = $50+$20=$70.

Now Initial investment of Elena = $5

She adds $10 for each 4 weeks then

=> 10×4=$40.

Now total savings of Elena = $40+$5=$45.

Here $70>$45.

Hence after 4 weeks Andre has more money than Elena.

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Answer:

  4f-7

Step-by-step explanation:

You want the perimeter of a triangle whose side lengths are f, 2f, and f-7.

The perimeter is the sum of the side lengths:

  P = f +2f +(f -7)

  P = 4f -7

The perimeter of the triangle is 4f-7.

__

Additional comment

No such triangle can exist. The sum of the two shortest sides is f+(f-7) = 2f-7, which is not greater than the longest side, 2f. The triangle inequality is not satisfied.

Answer:

Step-by-step explanation:

To find the y-intercept on a graph, you must look at where the line intercepts through the y-axis.

For this problem it would be 16

Answer:

The Slope would be 3

The approximate angle between the base and a triangular face is; 79.43°

The dimensions of the pyramid are;

Side Length; a = 22 inches

Edge length; b = 61 inches

Using Pythagoras theorem:

h = height of pyramid

h = √(b² - 1/2a²)

h = √(61² - 22²/2)

h = √(3479)

h = 58.98

Slant height :

L = √(58.983² + 22²/4)

L =√(3479 + 121)

L = 60

Sinα = h/L

α = sin⁻¹(58.983/60)

α = 79.43°

That is the approximate angle found.

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Answer: C = P$3.89

Step-by-step explanation:

Answer:

[tex]\frac{12}{7}[/tex]

Step-by-step explanation:

[tex]3 /\frac{7}{4} = \frac{12}{7}[/tex]

Here is a trick to divide fractions
Take the 2nd fraction in this case [tex]\frac{7}{4}[/tex] and flip it [tex]\frac{4}{7}[/tex]
Change the division sign to a multiplication sign and multiply

[tex]\frac{3}{1} * \frac{4}{7} = \frac{12}{7}[/tex]

The correct options regarding given function  are (b), (c) and (f).

What is a function?

An association between each element of a non-empty set A and at least one element of a different non-empty set B is called a function.

The set of values that we are permitted to enter into our function is known as the domain of a function.

A function's range is the collection of values it gives as outputs.

The options given are:

a)The function f(x) = 9,000([tex]1.05^{x}[/tex]) represents the situation.

b)The function f(x) = 9,000([tex]0.95^{x}[/tex]) represents the situation.

c)After 2 years, the farmer can estimate that there will be about 8,120 bees remaining.

d)After 4 years, the farmer can estimate that there will be about 1,800 bees remaining.

e)The domain values, in the context of the situation, are limited to whole numbers.

f)The range values, in the context of the situation, are limited to whole numbers.

Now given,

Number of bees initially = 9000

Rate of decay = 5%

f(x) = Number of bee population after x years

So, f(x) = 9000([tex]0.95^{x}[/tex])

After 2 years, number of bees is

  f(2) = 9000(0.95²) = 8122.5 ≈ 8120

After 4 years, number of bees is

 f(4) = 9000(0.95⁴) = 7330.56≈7330

The domain of the function is number of years(x) which can be whole as well as decimal numbers.

The range of the function is number of bees after x years, which should be a positive integer number. So whole numbers are the domain.

Therefore the correct options regarding given function  are (b), (c) and (f).

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Answer: 25

Step-by-step explanation:

The sum of all exterior angles = 360

Using this, we can get the equation

2x+20+x+40+3x+6x=360

12x+60=360

12x=300

Simplify, and you get

x = 25

Therefore , as a result, the x, y, and z axes are the axis that the supplied coordinate plane problem is solved with.

The phrase "coordinate plane" describes a two-dimensional area made up of two rational numbers. It appears at the point where the X and Y axes connect, called the origin. A coordinate grid's numbers can be used to find points.

Here,

The x, y, and z axes of the Cartesian coordinate system,

which is used to represent three-dimensional space, are three axes that are mutually perpendicular to one another.

The origin is the location where the three axes converge.

Therefore , as a result, the x, y, and z axes are the axis that the supplied coordinate plane problem is solved with.

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The four ways to depict an inequality: solution graph, interval notation, set notation, and equation notation.

In mathematics, an inequality displays the connection between two values in an unbalanced algebraic statement. One of the two variables on the two sides of the inequality may be greater than, greater than or equal to, less than, or less than or equal to another value, according to inequality signals.

There are four ways to depict an inequality: solution graph, interval notation, set notation, and equation notation.

A comparison of any two values, known as an inequality, demonstrates whether one value is less than, larger than, or equal to the value on the opposite side of the equation. A relationship in a real-world scenario can be demonstrated by writing an inequality.

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The difference between X₁ and X₁ is "X₁ is a random variable, and X₁ is a specific numerical value".

Any variable whose value is uncertain or any function that values each result of an experiment is referred to as a random variable. Random variables are frequently identified by letters and fall into one of two categories: discrete variables, which have definite values, or continuous variables, which can take on any value within a continuous range.

A parameter's estimated value in numbers, specifically. The sample mean is the most accurate estimate of the population mean. A range or interval of values used to gauge the parameter. The value of the parameter under consideration may or may not be included in this estimation.

So we can say the difference between X₁ and X₁ is "X₁ is a random variable, and X₁ is a specific numerical value".

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No. of pages read on day of planning are represented by vertical intercept.

The difference between vertical intercept of original plan and line drawn by Lin is 10 unit.

The slope intercept form equation for a straight line with a slope, 'm', and 'c' as the y-intercept can be given as:

y = mx + c.

Given,

No. of pages Lin read on day of planning = 30

No. of pages she plans to read everyday = 40

If total pages are y for x days then,

y = 40x + 30

which is the original plan of Lin.

comparing it with slope-intercept form of line

y = mx + c

slope m = 40

vertical intercept of original plan c = 30 unit

slope m represents the no. of pages read everyday.

vertical intercept represents the no. of pages read on day of planning.

vertical intercept c' of line drawn by Lin = 40 unit

Comparing intercept of original plan to line drawn by Lin

= |c - c'|

= |30 - 40|

= 10 unit

Hence, the vertical intercept represents the no. of pages read on day of planning.

There is 10 unit difference between vertical intercept of original plan and line drawn by Lin.

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Argument A is strong based on the logic.

We know that logic is the process whereby we can be able to decode if an argument is true or false when we look at the premises and the conclusion. It is normal that the premise of the argument should be able to give rise to the conclusion.

In this case, we must note that the professor is an authority figure in chemistry and what he says must be taken as trust worthy as shown in the deductive argument that led to this conclusion as shown above.

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The value of x is 27

From the information given, we have that the triangle is an isosceles triangle.

The properties of an isosceles triangle are;

We have that;

First side = x + 2

Second side = 2x - 25

Equate the two sides, we get;

x+ 2 = 2x - 25

collect like terms

x - 2x = - 25 - 2

subtract like terms

-x= -27

make 'x' the subject

x = 27

Hence, the value is 27

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The answer is 56 crickets.

The quantitative relation between two amounts shows the number of times one value contains or is contained within the other. for example-"the ratio of computers to students is now 2 to 1"

Given here: Greta has two pet lizards, Draco and Lizzy. For every 11 crickets Greta feeds Draco, she always feeds Lizzy 8 crickets. Thus the ratio is 11:8 Therefore if Draco got 77 crickets ten lizzy should get 8×7=56 crickets.

Hence, The answer is 56 crickets.

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x<9

5(x+3)<60

x+3< 12

x<9

Which simply means x will always be less than 9 in this case.

These are the Simple inequation in one variable which can be simply solved by operating the same operations on both sides to get a simplified answer.

We follow the same algorithm even for equations where the left side and right sides are equal so we simply transfer the left digits to the right side simply by changing their operational signs like + will be -, * will be /.

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As an example, if the centre of dilation were in the centre of a circle with radius 5 and was going through a scale factor 2 dilation, the radius would be 10.

The process of dilation involves growing an object's size without changing its shape. The size of the item may increase or decrease depending on the scale factor. Using dilation math, a square with side 5 units may be made wider to have side 15 units, but the shape of the square remains the same.

Enlargements that generate a bigger picture are called dilations. Reduction describes the dilatation that shrinks the picture. The starting figure is stretched or contracted during a dilatation. The scale factor (or ratio) and the centre of a dilatation are both described. Occasionally, the words "dilate" and words like "magnify," "disptend," "expand," "inflate," and "swell" are used interchangeably. All of these words suggest "expanding in size or volume," but the word "dilate" specifically refers to increased diameter.

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The trinomials in factor form are listed below:

Case A: (x + 2)²

Case B: (x - 3) · (x - 9)

Case C: (x - 7) · (x + 6)

Case D: (x - 9) · (x + 1)

Herein we find four case of trinomials of the form x² + b · x + c that must be factorised, this can be done by using the following algebraic property:

x² - (r₁ + r₂) · x + r₁ · r₂ = (x - r₁) · (x - r₂)

Where r₁, r₂ are real roots of the quadratic equation.

Now we proceed to factorise each trinomial (quadratic equation), that is, their factor form:

Case A

x² + 4 · x + 4

x² + (2 + 2) · x + 2 · 2

(x + 2) · (x + 2)

(x + 2)²

Case B

x² - 12 · x + 27

x² - (3 + 9) · x + (- 3) · (- 9)

(x - 3) · (x - 9)

Case C

x² - x - 42

x² - (7 - 6) + (- 7) · 6

(x - 7) · (x + 6)

Case D

x² + 8 · x - 9

x² - (- 9 + 1) · x + (- 9) · 1

(x - 9) · (x + 1)

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The average rate of change for the quadratic function in the graph is 1

The proportion of change along the y-axis to change along the x-axis represents the average rate of change between the two points.

Since the graph of a quadratic function is curved and lacks a constant rate of change between any two distinct locations, it is referred to as the average rate of change rather than the slope.

The average rate of change between points (2, 0) and (3, 1) is

= (0 - 1) / (2 - 3)

=-1 / -1

= 1

The average rate of change within the interval 1 to 3 is 1

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