The Sum Of A Number And Six

Answer:

Zero real number solutions

Step-by-step explanation:

Doesn't intercept the x-axis -> No real solutions

The set of inequalities y ≤ x² - 3 and y > -x² + 2 do not have a solution

From the graph, we have:

f(x) = x² - 3

g(x) = -x² + 2

Next, we change the equations to inequalities as follows:

y ≤ x² - 3

y > -x² + 2

To modify the graph, we then perform the following transformations:

After the modifications in (a), we have:

y ≤ x² - 3 and y > -x² + 2

Substitute y > -x² + 2 in y ≤ x² - 3

-x² - 2 ≤ x² - 3

This gives

2x² ≤ - 1

Divide by 2

x² ≤ - 0.5

The square root of numbers less than 0 is a complex number

Hence, the set of inequalities do not have a solution

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Step-by-step explanation:

[tex]f(x) = ln( \frac{1}{x} ) [/tex]

To find the derivative, notice we have a function 1/x inside of another function, ln(x)

We use what we call the chain rule,

It states that

derivative of a '

[tex] \frac{d}{dx} f(g(x)) = f '(g(x)) \times \: g '(x)[/tex]

Here f is ln(x)

f is 1/x

So first, we know that

[tex] \frac{d}{dx} ( ln(x) = \frac{1}{x} [/tex]

so

[tex]f'(g(x)) = \frac{1}{ \frac{1}{x} } [/tex]

We know that

[tex]g'(x) = - \frac{1}{ {x}^{2} } [/tex]

So we have

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

The better way to calculate the amount of substance is Volume.

According to the statement

we have to find the all information about the soup cans.

So we know that if we put a liquid in the container then in that case surface area is not important but a volume is more important for this purpose.

So, That's why The weight in the case of soup is not important when evaluating a package to be able to sell the product. A container is usually specified according to its volume, especially those containing liquids. Because the weight can vary concerning the concentration of the product, the best way to calculate the amount of substance to be packaged is the volume.

So, The better way to calculate the amount of substance is Volume.

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Answer:

5445

Step-by-step explanation:

We can use distributive property.  a*(b - c) = (a*b) - (a *c)

  99 = 100 - 1

55 * 99 = 55 * (100 - 1)

            = 55*100 - 55*1

            = 5500 - 55

            = 5445

The equation representing the given situation is, 250 = 70x + 35, where x is the number of cars washed by the company.

In the question, we are given that Scarlett made a profit of $250.00 with her mobile car wash company. She charged $75.00 per car wash and received $35.00 in tips, but also had to pay $5.00 in cleaning supplies per car.

We are asked to represent the situation with an equation.

We assume the number of cars washed by the company to be x.

Charge for washing 1 car = $75.00.

Cost for cleaning supplies for 1 car = $5.00.

Thus the profit on the wash of 1 car = $75.00 - $5.00 = $70.00.

The total profit received by the company, on washing x number of cars = $70x.

The tips received = $35.00.

Thus, the total profit of the company = $(70x + 35).

But, the total profit is given to be $250.00.

Thus, we get an equation, 250 = 70x + 35.

Thus, the equation representing the given situation is, 250 = 70x + 35, where x is the number of cars washed by the company.

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Answer: c

Step-by-step explanation:

The equation of the line is

[tex]y+2=\frac{1}{3}(x-1)\\\\y+2=\frac{1}{3}x-\frac{1}{3}\\\\y=\frac{1}{3}x-\frac{7}{3}[/tex]

To determine which point lies on the line, we can substitute in the x coordinate and see if the equation gives the same y-coordinate.

Therefore, the answer is c.

Answer:

f(g(2)) = 50

Step-by-step explanation:

evaluate g(2) then substitute the result obtained into f(x)

g(2) = 3(2) - 2 = 6 - 2 = 4 , then

f(4) = 2(4 + 1)² = 2(5)² = 2(25) = 50

Answer:

f[g(2)] = 50

Step-by-step explanation:

Given functions:

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

[tex]g(x)=3x-2[/tex]

We are asked to determine f[g(2)], which is known as a composite function. When solving composite functions, you always work inside out.

Step 1: Find the value of g(2) by substituting 2 for x in function g(x).

[tex]\implies g(2)=3(2)-2[/tex]

[tex]\\\implies g(2)=6-2\Rightarrow g(2)=\boxed{4}[/tex]

Step 2: Substitute the found value into the composite function.

[tex]\\\implies f[g(2)]= 2(4+1)^2[/tex]

[tex]\implies f[g(2)] = 2(5)^2 \Rightarrow 2(25) \Rightarrow \boxed{50}[/tex]

Hence, the value of the composite function f[g(2)] is 50.

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Answers:

i) [tex]\sf (x + 2)(x+ 3)[/tex]

ii) [tex]\sf \left(3x+1\right)\left(3x+2\right)\left(9x^2+9x-16\right)[/tex]

Factorize expression's:

i.

[tex]\sf (x + 1)^2 + 3(x + 1) + 2[/tex]

apply perfect square and distributive method

[tex]\sf (x^2 + 2(x)(1) + 1^2) + 3x + 3 + 2[/tex]

expand

[tex]\sf x^2 + 2x + 1 + 3x + 3 + 2[/tex]

collect like terms

[tex]\sf x^2 + 2x + 3x + 3 + 2 + 1[/tex]

add/subtract like terms

[tex]\sf x^2 + 5x + 6[/tex]

breakdown

[tex]\sf x^2 + 3x + 2x+ 6[/tex]

factor common term

[tex]\sf x(x + 3) + 2(x+ 3)[/tex]

collect into groups

[tex]\sf (x + 2)(x+ 3)[/tex]

ii.

[tex]\sf (9x^2 + 9x - 4)(9x^2 + 9x - 10) - 72[/tex]

breakdown

[tex]\sf (9x^2 + 12x - 3x - 4)(9x^2 + 15x - 6x - 10) - 72[/tex]

factor common term

[tex]\sf (3x(3x + 4) -1( 3x + 4)) ( (3x(3x + 5)- 2(3x +5) )- 72[/tex]

collect like terms

[tex]\sf (3x -1)( 3x + 4) (3x- 2)(3x +5) - 72[/tex]

expand

[tex]\sf 81x^4+162x^3-45x^2-126x-32[/tex]

factor

[tex]\sf \left(3x+1\right)\left(3x+2\right)\left(9x^2+9x-16\right)[/tex]

Answer:

[tex]\textsf{1.} \quad (x+3)(x+2)[/tex]

[tex]\textsf{2.} \quad (3x+1)(3x+2)(9x^2+9x-16)[/tex]

Step-by-step explanation:

Question 1

[tex]\textsf{Given expression}: \quad(x+1)^2+3(x+1)+2[/tex]

[tex]\textsf{Let }u=(x+1) \implies u^2+3u+2[/tex]

[tex]\textsf{To factor }\:\:u^2+3u+2:[/tex]

Rewrite the middle term as u + 2u:

[tex]\implies u^2+u+2u+2[/tex]

Factorize the first two terms and the last two terms separately:

[tex]\implies u(u+1)+2(u+1)[/tex]

Factor out the common term (u+1):

[tex]\implies (u+2)(u+1)[/tex]

Replace [tex]u[/tex] with [tex](x+1)[/tex] :

[tex]\implies (x+1+2)(x+1+1)[/tex]

Simplify:

[tex]\implies (x+3)(x+2)[/tex]

Question 2

[tex]\textsf{Given expression}: \quad (9x^2+9x-4)(9x^2+9x-10)-72[/tex]

Expand:

[tex]\implies 9x^2(9x^2+9x-10)+9x(9x^2+9x-10)-4(9x^2+9x-10)-72[/tex]

[tex]\implies 81x^4+81x^3-90x^2+81x^3+81x^2-90x-36x^2-36x+40-72[/tex]

Collect like terms:

[tex]\implies 81x^4+81x^3+81x^3-90x^2+81x^2-36x^2-90x-36x+40-72[/tex]

Combine like terms:

[tex]\implies 81x^4+162x^3-45x^2-126x-32[/tex]

Use the Factor Theorem:

If f(x) is a polynomial, and f(a) = 0, then (x – a)  is a factor of f(x).

[tex]\begin{aligned}\implies f \left(-\dfrac{1}{3}\right) & =81\left(-\dfrac{1}{3}\right)^4+162\left(-\dfrac{1}{3}\right)^3-45\left(-\dfrac{1}{3}\right)^2-126\left(-\dfrac{1}{3}\right)-32\\ & = 1-6-5+42-32\\ & = 0\end{alilgned}[/tex]

Therefore (3x + 1) is a factor.

[tex]\begin{aligned}\implies f \left(-\dfrac{2}{3}\right) & =81\left(-\dfrac{2}{3}\right)^4+162\left(-\dfrac{2}{3}\right)^3-45\left(-\dfrac{2}{3}\right)^2-126\left(-\dfrac{2}{3}\right)-32\\ & = 16-48-20+84-32\\ & = 0\end{alilgned}[/tex]

Therefore (3x + 2) is a factor.

Therefore:

[tex]\implies f(x)=(3x+1)(3x+2)(ax^2+bx+c)[/tex]

Compare the coefficient of x⁴ and the constant to find a and c:

[tex]\implies 3 \cdot 3 \cdot a=81 \implies a=9[/tex]

[tex]\implies 2c=-32 \implies c=-16[/tex]

Therefore:

[tex]\implies f(x)=(3x+1)(3x+2)(9x^2+bx-16)[/tex]

Expand:

[tex]\implies f(x)=81x^4+(81+9b)x^3-(126-9b)x^2-(144-2b)x-32[/tex]

Compare the coefficient of x³ to find b:

[tex]\implies 81+9b=162 \implies b=9[/tex]

Therefore, the fully factorized expression is:

[tex]\implies (3x+1)(3x+2)(9x^2+9x-16)[/tex]

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The surface area of the sphere is 324π square units

From the question, we are to calculate the surface area of the sphere

The surface area of a sphere can be calculated by using the formula

[tex]Surface \ area = 4 \pi r^{2}[/tex]

Where r is the radius of the sphere

In the given diagram,

r = 9

Thus,

Surface area = 4π × 9²

Surface area = 4π × 81

Surface area = 324π square units

Hence, the surface area of the sphere is 324π square units

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The measure of angle GFD of the circumscribed circle is; A: 40°

From the figure, we can apply the arc angles summation formula to get;

Major angle ∠FED + Minor angle ∠FED = 360°

We are given that Major arc FD measures 280°. Thus;

280° + Minor angle ∠FED = 360°

Minor angle ∠FED = 360° - 280°

Minor angle ∠FED = 80°

Also, we know that;

∠FED + ∠FGD = 180°

Thus, putting ∠FED = 80° gives us;

80° + ∠FGD = 180°

Subtract 80° from both sides using subtraction property of equality to get;

∠FGD = 180° - 80°

∠FGD = 100°

Now, GF and GD are the tangents to the circle from the same point G. Thus, we can say that;

GD = GF

Therefore,

∠FDG = ∠GFD = x

(This is because ∠FDG and ∠GFD are the angles opposite to equal sides.

In triangle FGD, we have sum of interior angles = 180°

Therefore, we have the expression;

∠FDG + ∠FGD + ∠GFD = 180° (due to the fact that sum of angles in a triangle is equal to 180°)

x + 100° + x = 180°

2x = 180° - 100°

2x = 80°

x = 40°

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Answer:

40

Step-by-step explanation:

I did a thing and it worked

The values of the x in the given equation are x = 6.46 and x = -0.46

From the question, we are to solve the equation for x

The given equation is

(x+3)² -10 = 2

First, add 10 to both sides of the equation

(x+3)² -10 +10 = 2 +10

(x+3)²  = 12

Now, take the square root of both sides to get

x + 3 = ±√12

∴ x = 3 ± √12

x = 3+√12 OR 3-√12

x = 3 + 3.46 OR 3 - 3.46

x = 6.46 OR -0.46

Hence, the values of the x in the given equation are x = 6.46 and x = -0.46

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The measure of angle D in the convex pentagon ABCDE is 132°

An equation is an expression that shows the relationship between two or more numbers and variables.

Let x represent the measure of angle D, hence:

angle A = x - 40.

∠A + ∠B + ∠C + ∠D + ∠E = 540° (sum of angle in a pentagon)

3(x - 40) + 2x = 540

x = 132°

The measure of angle D in the convex pentagon ABCDE is 132°

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Answer:

Step-by-step explanation:

To find the inverse of a function [tex]f(x)[/tex], we let [tex]f(y)=x[/tex] and then solve again for y.

For example, using [tex]g(x)=-0.5x[/tex], we will let [tex]g(y)=x[/tex], giving [tex]x=-0.5y[/tex]. This means that [tex]y=-2x[/tex], and thus [tex]g^{-1}(x)=-2x[/tex].

Using similar logic, we see that:

Answer:

False

Step-by-step explanation:

They both have 1 solution

15x + 4 = 3(6x +5)

15x +4 = 18x + 15  Multiple everything in the parentheses by 3.

4 = 3x + 15  Subtract 15 x from both sides

-11 = 3x   Subtract 15 from both sides

-11/3 = x  Divide both sides by 3

There is only one solution for this equation.

2x -8 = 8x + 36

-8 = 6x + 36  Subtract 2x from both sides

--44 = 6x  Subtract 36 from both sides

-44/6 = x Divide both sides by 6.

There is only one solution for this equation.

Answer:

false

Step-by-step explanation:

They both have solutions

Answer:

ans is 42635 when expressed in standard form

Step-by-step explanation:

Answer:

35.5

Step-by-step explanation:

→ Write down the formula

2 × π × r = 222.94

→ Simplify

6.28 × r = 222.94

→ Divide both sides by 6.28

r = 35.5

Answer:

35.5 = r

Step-by-step explanation:

The circumference of a circle is given by

C = 2 * pi *r  where r is the radius

222.94 = 2 * 3.14 * r

222.94 = 6.28 r

Divide each side by 6.28

222.94/6.28 = r

35.5 = r

The total number of children can be calculated using linear equation in one variable. The marbles are shared among 7children.

Solution

Total number of marbles = 47

The number of marbles received by each child = 6

Let the number of children be x

Then according to the question'

Total number of marbles received by all children + 5 = Total number of marbles

6x + 5 = 47

6x = 47- 5

6x = 42

x = 7

now, the number of children is 7.

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Find y intercept

Point(0,-7)

Find vertex

x coordinate

y coordinate

Vertex at (-2,-27)

Slope

Answer:

10

Step-by-step explanation:

The x-coordinate of the vertex of a quadratic equation in the form

[tex]f(x)=ax^2+bx+c\quad \textsf{is} \quad -\dfrac{b}{2a}[/tex]

Given function:

[tex]f(x)=5x^2+20x-7[/tex]

[tex]\implies a=5, \quad b=20, \quad c=-7[/tex]

x-coordinate of the vertex

[tex]\implies -\dfrac{b}{2a}=-\dfrac{20}{2(5)}=-2[/tex]

To find the y-coordinate of the vertex, substitute the found value of x into the function:

[tex]\begin{aligned}\implies f(-2) & =5(-2)^2+20(-2)-7\\& = 5(4)-40-7\\& = 20-47\\& = -27\end{aligned}[/tex]

Therefore, the coordinates of the vertex are (-2, -27).

The y-intercept is when the curve crosses the y-axis, so when x = 0.

To find the y-coordinate of the y-intercept, substitute x = 0 into the function:

[tex]\begin{aligned}\implies f(0) & =5(0)^2+20(0)-7\\& = 0 + 0-7\\& = -7\end{aligned}[/tex]

Therefore, the coordinates of the y-intercept are (0, -7).

To find the slope of the line passing through the vertex and the y-intercept, simply substitute the found points into the slope formula:

[tex]\implies \sf slope=\dfrac{change\:in\:y}{change\:in\:x}=\dfrac{-27-(-7)}{-2-0}=\dfrac{-20}{-2}=10[/tex]

Therefore, the slope of the line passing through the vertex and the y-intercept of the given quadratic function is 10.

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Answer:

90 is the right answer

Step-by-step explanation:

because it give the value pf o is tan o

Answer:

4 : 9

Step-by-step explanation:

ratio of corresponding sides and ratio of perimeter are equal

then ratio of perimeter = 4 : 9

Answer:

5.2 Inch to cm = 13.208 cm

5.2 feet to cm = 158.496 cm

Step-by-step explanation:

Inch: 1 inch = 2.54 cm, so multiply 2.54 * 5.2. We get the answer: 13.208 cm

Feet: 1 foot = 30.48 cm, so multiply 30.48 * 5.2. We get the answer: 158.496 cm

The range is -6.

To find the answer, calculate as follows:

Range = Maximum - Minimum

Range = 8 and Maximum = 2

Let, the minimum be x.

Now, substitute values in the formula.

8 = 2 - x

x = 2 - 8

x = -6

Therefore, the answer is -6.

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The volume of the prism would be enlarged by a factor of 8

The scale factor is given as:

k = 2

Let the initial volume be v and the final volume be V.

The relationship between both volumes is

V = k^3 * v

This gives

V = 2^3 * v

Evaluate

V = 8v

Hence, the volume of the prism would be enlarged by a factor of 8

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