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We are given the equation:

log₂(x-3) + log₂x - log₂(x+2) = 2

When is it defined:

in this equation, log₂(x-3) and log₂(x+2) can only be defined when

x-3 >0 and x+2 > 0

solving for the values of x, we get:

x > 3    and   x > -2

which basically means x > 3

Because we are looking for an inequality which is true for both x>-2 and x>3

Hence, x will have a value greater than 3

Solving for x:

using the product rule [logₐb + logₐc = logₐ(bc)]

log₂[(x-3)(x)] - log₂(x+2) = 2  

using the quotient rule [logₐb - logₐc = logₐ(b/c)]

log₂[(x-3)(x) / (x+2)] = 2

from the property [ aˣ = b  ⇒  logₐb = x]

(x² - 3x) / (x+2) = 2²

x² - 3x = 4x + 8

x² - 7x - 8 = 0

x² + x - 8x - 8 = 0

x(x+1) - 8(x+1) = 0

(x-8)(x+1) = 0

(x-8) = 0    OR    (x+1) = 0

x = 8      OR      x = -1

We know that the equation is defined only for x > 3

We can see that x = 8 satisfies that inequality

Therefore, x = 8

Answer: In order: 25, 15, 67.

Step-by-step explanation:

In picture

Answer:

The answer is below

Step-by-step explanation:

The question is not complete, the complete question is:

A rectangle is constructed with its base on the x-axis and two of its vertices on the parabola y 81-x², what are the dimensions of the rectangle with the maximum area? what is the area?

Solution:

Given the parabola:

y = 81 - x²

Let the points (a,0) and (-a, 0) be points on the axis, this points touch the parabola at (a, 81 - a²) and (-a, 81 - a²).

Points (a,0), (-a, 0), (a, 81 - a²), (-a, 81 - a²) form the rectangle.

The length of rectangle = distance between (a,0) and (-a, 0) or between (a, 81 - a²) and (-a, 81 - a²) = 2a

The breadth of rectangle = distance between (a,0) and (a, 81 - a²) or between (-a, 0) and (-a, 81 - a²) = 81 - a²

Area of rectangle (A) = length × breadth = 2a × (81 - a²) = 162a - 2a³

A = 162a - 2a³

The maximum area is at dA / da = 0

dA / da = 162 - 8a² = 0

162 - 6a² = 0

6a² = 162

a² = 27

a = 5.2 units

Length = 2a = 2(5.2) =  10.4 units

Breadth = 81 - a² = 81 - 5.2² = 54 unit

Area = length × breadth = 54 × 10.4 = 561 unit²

Answer:

108 cm^2

Step-by-step explanation:

A= 1/2 b×h

so

1/2 12×18

Answer: 8

Step-by-step explanation:

Answer:

[tex]( - 4)( - 2) = \boxed{8}[/tex]

Answer:

The answer is $0.07 cents per inch

Step-by-step explanation:

If you take 6 feet into inches (72)

and divide the total cost by the amount of inches (5.04 / 72)

You get the answer

Answer:You will need to buy 1/2 a pound of cashews and 1/2 a pound of peanuts

Answer:

C. 15

Step-by-step explanation:

It's 13.888 about 14, so 15 is the closest answer.

b

Step-by-step explanation:

taking 250÷18=13.88~

Answer:

okay i will answer but i dont see the question

Step-by-step explanation:

The expenses with the given profit and revenue is $47,000. Therefore, option C is the correct answer.

Given that, the profit is $3,000 and the revenue is $50,000.

The profit or gain is equal to the selling price minus the cost price. Loss is equal to the cost price minus the selling price.

Here, Profit = Revenue - Expenses

3000 = 50000 - Expenses

⇒ Expenses = 50000-3000

= 47000

The expenses with the given profit and revenue is $47,000. Therefore, option C is the correct answer.

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Answer:

Step-by-step explanation:

y-7= -4/3 (x+6)

y-7= -4/3x-8

add 7 to both sides

y= -4/3x-1

The equation in slope-intercept form for the given line is 3y = -4x -3

The definition of an equation is a mathematical statement that shows that two mathematical expressions are equal.

Given that a line passes through the point (-6,7) and has a slope of -4/3.

Standard equation of a line = y = mx + c where m is the slope,

Therefore, putting values of x and y to get the value of c

7 = -6*-4/3 + c

c = 7 - 8

c = -1

Equation = y = -4x/3 -1

Or, 3y = -4x -1

Hence, the equation of a line passing through the point (-6,7) and has a slope of -4/3 is 3y = -4x -1

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Answer:

A deduction can only lower your taxable income and the tax rate that is used to calculate your tax. This can result in a larger refund of your withholding. A credit reduces your tax giving you a larger refund of your withholding, but certain tax credits can give you a refund even if you have no withholding.

Step-by-step explanation:

Answer: There are  62 tricycles in the shop.

Step-by-step explanation:

Given: Number of wheels in a bicycle = 2

Number of wheels in a tricycle = 3

Let x = Number of bicycles , y = Number of tricycles.

As per given , we have

x+y =120       (i)

2x+3y=302       (ii)

Multiplying 2 on both sides of (i)

2x+2y=240      (iii)

Subtract (iii) from (ii) , we get

y= 62

Put value of y in (i) , we get

[tex]x+62=120\\\\\Rightarrow\ x=120-62\\\\\Rightarrow\ x= 58[/tex]

Hence, there are 58 bicycles and 62 tricycles in the shop.

Answer:

They are both 3 dimensional shapes. A cylinder is essentially a prism for a circle.

Step-by-step explanation:

Step-by-step explanation:

- They are both 3 dimensional

-They both have 2 bases

Answer:

y = 3

General Formulas and Concepts:

Step-by-step explanation:

The question is asking us what the y-value is when x = -2.

According to the graph, when x = -2, y = 3.

Therefore, y = 3.

Answer:

C and D

multiply the 6 and -3 to the numbers inside the parenthesis

Answer:

When  Item Price * (%) is < or > to $ Off

Step-by-step explanation:

Answer:

Step-by-step explanation:

tan x° = [tex]\frac{3}{4}[/tex] ⇒ x° ≈ 36.87° ≈ 36.9°

Answer:

Step-by-step explanation:

hi

[tex]\dfrac{6}{\sqrt{12}}=\dfrac{2\times 3}{\sqrt{2^2\times 3}}=\dfrac{3}{\sqrt{3}}=\sqrt{3}[/tex]

thanks

Answer:

9a-65

Step-by-step explanation:

simply write 9a+(add the remaining numbers all together)

Answer:

An equation in standard form for the line is:

[tex]\frac{5}{2}x-y=-4[/tex]

Step-by-step explanation:

Given the points

The slope between two points

[tex]\mathrm{Slope\:between\:two\:points}:\quad \mathrm{Slope}=\frac{y_2-y_1}{x_2-x_1}[/tex]

[tex]\left(x_1,\:y_1\right)=\left(-2,\:-1\right),\:\left(x_2,\:y_2\right)=\left(0,\:4\right)[/tex]

[tex]m=\frac{4-\left(-1\right)}{0-\left(-2\right)}[/tex]

[tex]m=\frac{5}{2}[/tex]

Writing the equation in point-slope form

As the point-slope form of the line equation is defined by

[tex]y-y_1=m\left(x-x_1\right)[/tex]

Putting the point (-2, -1) and the slope m=1 in the line equation

[tex]y-\left(-1\right)=\frac{5}{2}\left(x-\left(-2\right)\right)[/tex]

[tex]y+1=\frac{5}{2}\left(x+2\right)[/tex]

[tex]y=\frac{5}{2}x+4[/tex]

Writing the equation in the standard form form

As we know that the equation in the standard form is

[tex]Ax+By=C[/tex]

where x and y are variables and A, B and C are constants

so

[tex]y=\frac{5}{2}x+4[/tex]

[tex]\frac{5}{2}x-y=-4[/tex]

Therefore, an equation in standard form for the line is:

[tex]\frac{5}{2}x-y=-4[/tex]

Answer:

42

Step-by-step explanation:

start with the first 5x2 then add

Answer:

42

Step-by-step explanation:

5×2+32

10+32

42

Answer:  8

Step-by-step explanation:

Answer:

8

Step-by-step explanation:

Answer:

The width is 22.5 feet and the length 45 feet.

Step-by-step explanation:

Let l represent the length of the rectangle and  w represent the width of the rectangle. The cost of the fence (C) is:

C = 2l + 2w + 2l + 6w

C = 4l + 8w

360 = 4l + 8w

360 = 4(l + 2w)

90 = l + 2w

l = 90 - 2w

Area (A) of rectangle = length * width

A = l * w

Put l = 90 - 2w

A = (90 - 2w)w

A = 90w - 2w²

The maximum area is at A' = 0, hence:

Differentiating gives:

A' = 90 - 4w

Put A' = 0

0 = 90 - 4w

4w = 90

w = 22.5 feet

l = 90 - 2w = 90 - 2(22.5) = 45 feet

Hence the width is 22.5 feet and the length 45 feet.