Solve each equation.
2³x=4x⁺¹

A solution that is real in its nature is called the real solution of that equation.

X + 5 = 0

We can solve the above equation as,

X + 5 = 0

X = -5

Now, if we plug the value of X into the equation,

X + 5 = 0

-5 + 5 = 0

0 = 0

The solution is real and is true for this equation.

While the solution of an algebraic expression solution is imaginary if it is the square root of a negative number. It involves the imaginary unit iota.

The complete solution of the expression is given below in the picture.

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Given the radius of each sphere, the volume of sphere A and B are 33.49cm³ and 267.95cm³ respectively.

A sphere is simply a geometrical object with a three-dimensional analogue to a two-dimensional circle.

The volume of a sphere is expressed as;

V = (4/3) × π × r³

Where r is radius and π is constant pi ( π = 3.14 ).

Given the data in the question;

First, we determine the volume of sphere A.

V = (4/3) × π × r³

V = (4/3) × 3.14 × (2cm)³

V = (4/3) × 3.14 × 8cm³

V = 33.49cm³

Hence, the volume of sphere A is approximately 33.49cm³.

Next, we determine the volume of sphere B.

V = (4/3) × π × r³

V = (4/3) × 3.14 × (4cm)³

V = (4/3) × 3.14 × 64cm³

V = 267.95cm³

Hence, the volume of sphere Bis approximately 267.95cm³.

Given the radius of each sphere, the volume of sphere A and B are 33.49cm³ and 267.95cm³ respectively.

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Diagonal FH bisect angles EFG and EHG and the angle FEH is congruent to angle FGH.

Why is the triangle congruent?

We need to check all the statements that are applicable for the diagram.

1) EFGH is a rectangle.

The above statements is not true in general. EFGH is a quadrilateral with opposites sides are equal, it may be rectangle in some cases but it is not always be a rectangle.

2). EFGH has 4 congruent sides.

The above statement is also false. EFGH has two opposite sides are congruent.

3). Diagonal FH bisect angles EFG and EHG.

The above statement is true. Triangle EFH and FGH both are isosceles triangles and hence they both have equal angles for the equal sides. Thus, it is true in all cases.

4). Diagonal FH is perpendicular to side Fe.

The above case is not true in general.

5). Angle FEH is congruent to angle FGH.

Yes, the above statement is the correct one, because both the triangles are congruent to each other.

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The complete question is -

Triangle FGH is the image of isosceles triangle FEH after a reflection across line HF. Select all the statements that are a result of corresponding parts of congruent triangles being congruent

The closest side length of each side of the garden is 7 feet .

It is given that Billy is planting a square garden that is 48 ft.²

Putting A = 48 ft.² in equation (1) , we get

       [tex]48=a^2\\a=\sqrt{48} \\a=6.92[/tex]

which is approximately equal to 7

Hence , closest side will be 7 ft.

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Simplification of given expression is , -1

In, mathematics a logarithmic equation is inverse of exponential equation. That means, we can easily convert the logarithmic equation into exponential equation and vice versa. The basic form of the logarithm function can be written as  .

   [tex]log_ax=N[/tex]

where, a is the base of function or equation

            x define expression

Mathematically, if   [tex]a^N=x[/tex] is an exponential function then, this can be converted to logarithmic function as   [tex]log_ax=N[/tex].

There are many formulas that are given to us for solving several simple and complex problems based on logarithmic functions and logarithmic equations.

Some important formulas of logarithms are :

According to question,

log₃(x+1)-log₃(3 x²-3 x-6)+log₃(x-2)

applying formula of log a - log b , equation can be written as

[tex]= log_3(\frac{x+1}{3x^{2} -3x-6} ) + log_3(x-2)\\\\=log_3(\frac{x+1}{3{(x+1)(x-2)} } ) + log_3(x-2)\\\\= log_3(\frac{1}{3 {(x-2)} } ) + log_3(x-2)[/tex]

now, applying formula of log a+ log b

[tex]=log_3(\frac{1}{3{(x-2)} } (x-2)) \\\\=log_3(\frac{1}{3} )\\[/tex]

apply, [tex]log_a(\frac{1}{x}) = -log_a x[/tex]

[tex]= - log_33[/tex]

= -1

Thus, Simplification of given expression is , -1

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Answer:

c - 12 = 4

Step-by-step explanation:

Hello!

Decreasing a number by something is subtracting that number with something.

c decreased by 12 can be represented as c - 12.

When something is something, we set it equal to each other. We can set that equal to 4 to find our equation.

Equation: c - 12 = 4

Answer:

11/20

Step-by-step explanation:

4/5 - 1/4

We need to get a common denominator of 20

4/5 * 4/4 = 16/20

1/4 *5/5 = 5/20

Replacing the fractions with their equivalent fractions

16/20 - 5/20

11/20

The inverse of the fuction y = 3 x + 2 is y = x/3 - 2/3.

So,

All we have to do to find the inverse of an equation is switch the variables x and y.

Now, solve for y as follows:

Therefore, the inverse of the fuction y = 3 x + 2 is y = x/3 - 2/3.

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The statement that shows Sam's purpose for making the investment,  is; Sam wants an investment that offers a higher return than stocks.

We know that bonds refers to a type of investment that could be held for a while the hope of making returns. Both the bonds and the stocks are common types of investments that are traded in the equities market. They are held by buyers with the sole hope that they could yield returns.

However, the bonds tend to be more secure than the stocks. The stocks are actually an overly volatile kind of investment. The bonds when mature tends to provide a high return on investment for the investor compared to the stocks.

Hence, the statement that shows  Sam's purpose for making the investment,  is; Sam wants an investment that offers a higher return than stocks.

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Answer:

5x = 15

2x = 6

10x = 30

8x = 24

21x = 63

Step-by-step explanation:

The equations that can give you an answer of 3 are the following

[tex]3x=9[/tex]

[tex]6x=18[/tex]

[tex]11x=33[/tex]

[tex]10x=30[/tex]

[tex]7x=21[/tex]

The equations above are one step linear equations

and many many more

Answer:6x-24y

Step-by-step explanation:You basically distribute the 6 with the x and the you do the same with the 4y. and thats how u get ur answer

Answer:

6x - 24y

Step-by-step explanation:

I'm guessing that they want you to distribute the 6 to each term inside the parentheses...

Answer:

4

Step-by-step explanation:

because if it has two Pairs that's 2 on each side of the side of the shape

so, when you have a figure of 2 parallel sides id would be a parallelogram

Answer:

[tex]\dfrac{20}{9}[/tex]

Step-by-step explanation:

Answer
[tex]x = \dfrac{20}{9}[/tex]. So the original number = [tex]\dfrac{20}{9}[/tex]

The original number is 20/9.

What is mean by Fraction?

A number expressed as a quotient in which a numerator is divided by denominator is called a fraction.

Given that;

The numbers are 1 5/6 and 3 2/3.

Now, Let us assume original number 'x'.

Change mixed fraction into improper fraction,

1 5/6 = 11/6

3 2/3 = 11/3

Then by the given condition,

Subtracting 7/18 from x and add 11/6, we get;

x - 7/18 + 11/6

And this is equal to = 11/3

So, we get;

x - 7/18 + 11/6 = 11/3

Add 7/18 both side,

x - 7/18 + 11/6 + 7/18 = 11/3 + 7/18

x + 11/6 = 11/3 + 7/18

Subtract 11/6 both side,

x + 11/6 - 11/6 = 11/3 + 7/18 - 11/6

x = 11/3 + 7/18 - 11/6

x = 66/18 + 7/18 - 33/18

x = 40/18

x = 20/9

Thus, The original number is 20/9.

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The value of x in the absolute value inequality is given as5 - 10|3x + 3| < 95 is all set of real numbers

The absolute value inequality is given as

5 - 10|3x + 3| < 95

Subtract 5 from all sides of the absolute value inequality

So, we have

- 10|3x + 3| < 90

Multiply -1/10 to all sides of the absolute value inequality

So, we have

|3x + 3| > -9

This means that the value of x is all set of real numbers

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Answer:

4

Step-by-step explanation:

Well, 2*2 = 4, so the answer is 4

Answer:

the gfc is 1 because of the unknowns which are a and b

Answer: A, C and F

Step-by-step explanation:

Answer:

[tex]A = 4x^2 - 50x + 150[/tex]

Step-by-step explanation:

Since we are given that the path is `x` meters wide, e can express both the length and the width of the pool by subtracting `2x` from the length and width of the outside edge of the path.

[tex]w = 15 - 2x\\l = 10 - 2x[/tex]

Therefore, we can express the area of the pool using [tex]x[/tex] by substituting the values of [tex]w[/tex] and [tex]l[/tex] into the formula for the area of a rectangle.

[tex]A = w \times l\\A = (15-2x)(10 - 2x)\\\to A = 15 \times 10 - 15 \times 2x - 2x \times 10 - 2x \times (-2x)\\\to A = 150 - 30x - 20x + 4x^2\\\to A = 4x^2 - 50x + 150[/tex]

The profit function for the concert is presented as follows;

y = -100•x² + 2,000•x - 1,000

The maximum profit is $9,000

A profit function is one that gives the profit of a business which is the difference between the business's revenue and the costs of the business.

Profit = Revenue - Cost

Revenue = Quantity × Ticket price

Cost = Quantity × unit cost

Which gives;

y = Q•x - Q•c = Q•(x - c)

Given that the quantity, Q, varies, and Q•(x - c) is the product of two variables, a quadratic function is used as follows;

y = a•x² + b•x + c

When x = 0, y = -1000

Which gives;

y = a×0² + b×0 + c = -1000

c = -1000

When x = 3, y = 4,100

Which gives;

4,100 = a•3² + b•3 - 1000 = 9•a + 3•b - 1000

4,100 = 9•a + 3•b - 1000

5,100 = 9•a + 3•b

When x = 6, y = 7,400

Which gives;

7,400 = a•6² + b•6 - 1000 = 36•a + 6•b - 1000

7,400 = 36•a + 6•b - 1000

8,400 = 36•a + 6•b

Solving gives;

a = -100, b = 2,000

Which gives;

y = -100•x² + 2,000•x - 1,000

The profit function is therefore;

The maximum profit is therefore at point, x = -2000/(2×(-100)) = 10

Which gives;

Max profit = -100×10² + 2,000×10 - 1,000 = 9,000

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The formula for r will be C/2π.

The radius of the yo-yo will be 6.02 feet.

It should be noted that circumference simply means the distance round the circle.

In this case, the circumference is given as 37.8 feet.

The formula is illustrated as:

C = 2πr

r = C/2π

The radius will be:

r = C / 2π

r = 37.8 / (2 × 3.14)

r = 37.8 / 6.28

r = 6.02

Therefore, the radius of the yo-yo will be 6.02 feet.

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x=2 y+5 is a tangent line to the circle x^2+y^2=5 at the point (1,-2).

This is further explained below.

Generally, the equation for is  mathematically given as

Differentiate both sides of the equation of the circle with respect to x, treating y=y(x) as a function of x:

[tex]x^2+y^2=5 \Longrightarrow 2 x+2 y \frac{\mathrm{d} y}{\mathrm{~d} x}=0 \\\\ \frac{\mathrm{d} y}{\mathrm{~d} x}=-\frac{x}{y}[/tex]

This gives the slope of any line tangent to the circle at the point (x, y). Rewriting the given line in slope-intercept form tells us its slope is

[tex]x=2 y+5 \\\\ y=\frac{1}{2} x-\frac{5}{2} \\\\ slope =\frac{1}{2}[/tex]

In order for this line to be tangent to the circle, it must intersect the circle at the point (x, y) such that

[tex]-\frac{x}{y}=\frac{1}{2} \\ \\\\ y=-2 x[/tex]

In the equation of the circle, we have

[tex]x^2+(-2 x)^2=5 x^2=5 \\\\ x=\pm 1 \\\\ y=\mp 2[/tex]

If x=-1, then

[tex]-1=2 y+5\\\\ y=-3 \neq 2[/tex]

If x=1, then

[tex]1=2 y+5 \\\\ y=-2$[/tex]

In conclusion, as expected. Therefore x=2 y+5 is a tangent line to the circle x^2+y^2=5 at the point (1,-2).

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Answer to this question is:

Domain of given function is (0,∞),{x|x>0}

Range of given function is (0,∞),{y|y>0}

Inverse of given function is 9/x²

The domain is all values of x that make the expression defined.The entire range of independent variable values is the domain of a function.

This definition can be stated simply as follows: The domain is the collection of all x-values that can be used to make the function "work" and produce actual y-values.

When choosing the domain, keep in mind that a fraction's denominator (bottom) cannot be 0.

Interval Notation:(0,∞)

Set-Builder Notation:{x|x>0}

The set of all valid y values constitutes the range.Graph may be used to find the values.

Interval Notation:(0,∞)

Set-Builder Notation:{y|y>0}

For calculation of inverse:

∴Domain of given function is (0,∞),{x|x>0}

Range of given function is (0,∞),{y|y>0}

Inverse of given function is 9/x²

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The equation of the line y = mx + b can be used to calculate the slope of a line m = ( y₂ - y₁ ) / ( x₂ - x₁ ) that passes through two points.

A line is defined as a curve showing the shortest distance between 2 points.

Here,

The slope of the line passing through two points ( x₁, y₁ ) and ( x₂, y₂ ) is given as :

m = ( y₂ - y₁ ) / ( x₂ - x₁ )

The line that connects two locations has the equation y = mx + b, where m is the slope of the line and b is a constant.

We can find b, using the x, y, and m values in the equation of a line to obtain the y-intercept.

Therefore, the Slope of the line can be given as m = ( y₂ - y₁ ) / ( x₂ - x₁ ) and equation of line y = mx +b.

Hence, option A is correct.

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Answer:

A or Find the slope using the formula m = StartFraction y 2 minus y 1 Over x 2 minus x 1 EndFraction, and then substitute one point and the slope into the equation y = m x + b to find the y-intercept.

Step-by-step explanation:

The process of ordering objects or numbers is known as permutation. Combinations are a method of choosing items or numbers from a collection or group of items so that the order of the items is irrelevant.

Combinations are used to group objects or to determine how many subgroups can be formed from the given collection of objects. Additionally, we use permutations to determine the amount of potential combinations of unrelated objects.

Fir example, if we want to select 2 players from 6. This will be:

= 6! / 2! (6 - 2)!

= 15 ways

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After direct variation y = 6/5 when x = 4.

So,

The initial statement is y = kx

Then to find y when x = 4.

When x = -3:

y = 3/10 × 4 ⇒ y = 6/5

Therefore, after direct variation y = 6/5 when x = 4.

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Aircraft a has 371 seats and Aircraft b has 235 seats.

A linear equation is an algebraic equation that has highest degree of 1.

The total number of seats in both aircraft is 606.

Assuming aircraft b has x number of sheets.

∴ Aircraft a has (x + 136) number of sheets.

From the question itself we can form a numerical expression which is

x + (x + 136) = 606.

2x + 136 = 606.

2x = 606 - 136.

2x = 470.

x = 470/2.

x = 235.

So, Aircraft b has 235 seats and Aircraft a has ( 235 + 136) = 371 seats.

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Answer: $75 difference per shirt

Step-by-step explanation:

The value of x for the equation 4 log₃2-2 log₃x=1 is 2.30946

The logarithm is exponentiation's opposite function in mathematics. This indicates that the exponent to which a set number, base b, must be increased in order to obtain a specific number x, is represented by the logarithm of that number.

Given,

4 log₃2-2 log₃x=1

log₃ 2⁴ - log₃ x² = 1

using logₐ(m - n) = logₐ(m/n)

log₃(2⁴ / x²) = 1

2⁴ / x² = 3¹

x = 4/√3

x = 2.30946

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Answer:

y = - 3x - 12

Step-by-step explanation:

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

calculate m using the slope formula

m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]

with (x₁, y₁ ) = (- 7, 9 ) and (x₂, y₂ ) = (- 5, 3 )

m = [tex]\frac{3-9}{-5-(-7)}[/tex] = [tex]\frac{-6}{-5+7}[/tex] = [tex]\frac{-6}{2}[/tex] = - 3 , then

y = - 3x + c ← is the partial equation

to find c substitute either of the 2 points into the partial equation

using (- 5, 3 )

3 = 15 + c ⇒ c = - 3 - 15 = - 12

y = - 3x - 12 ← equation of line