What is the slope of the line represented by the equation y = x-3?
O-3
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value of n could be 252 degree. Option 4

It is important to note that a decagon is a ten-sided polygon.

Since, a decagon has 10 sides

And we know that the angle made in one complete rotation is 360 degree.

Now find angle made in each rotation by the decagon is 360,

= 360/ 10

= 36 degrees

Now, we have to find possible values of n, and 'n' must be a multiple of 36

Apply hit and trial from given options.

We observe that from given options, only 252 degree is multiple of 36.

Therefore, value of n could be 252 degree. Option 4

The complete question is ;

A regular decagon is rotated n degrees about its center, carrying the decagon onto itself. The value of n could be:

1. 10 degrees

2. 150 degrees

3. 225 degrees

4. 252 degrees

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Answer:

[tex]\dfrac{1}{2} \left(25 \arcsin \left(\dfrac{x}{5}\right) -x\sqrt{25-x^2}\right) + \text{C}[/tex]

Step-by-step explanation:

Fundamental Theorem of Calculus

[tex]\displaystyle \int \text{f}(x)\:\text{d}x=\text{F}(x)+\text{C} \iff \text{f}(x)=\dfrac{\text{d}}{\text{d}x}(\text{F}(x))[/tex]

If differentiating takes you from one function to another, then integrating the second function will take you back to the first with a constant of integration.

Given indefinite integral:

[tex]\displaystyle \int \dfrac{x^2}{\sqrt{25-x^2}}\:\:\text{d}x[/tex]

Rewrite 25 as 5²:

[tex]\implies \displaystyle \int \dfrac{x^2}{\sqrt{5^2-x^2}}\:\:\text{d}x[/tex]

Integration by substitution

[tex]\boxed{\textsf{For }\sqrt{a^2-x^2} \textsf{ use the substitution }x=a \sin \theta}[/tex]

[tex]\textsf{Let }x=5 \sin \theta[/tex]

[tex]\begin{aligned}\implies \sqrt{5^2-x^2} & =\sqrt{5^2-(5 \sin \theta)^2}\\ & = \sqrt{25-25 \sin^2 \theta}\\ & = \sqrt{25(1-\sin^2 \theta)}\\ & = \sqrt{25 \cos^2 \theta}\\ & = 5 \cos \theta\end{aligned}[/tex]

Find the derivative of x and rewrite it so that dx is on its own:

[tex]\implies \dfrac{\text{d}x}{\text{d}\theta}=5 \cos \theta[/tex]

[tex]\implies \text{d}x=5 \cos \theta\:\:\text{d}\theta[/tex]

Substitute everything into the original integral:

[tex]\begin{aligned}\displaystyle \int \dfrac{x^2}{\sqrt{5^2-x^2}}\:\:\text{d}x & = \int \dfrac{25 \sin^2 \theta}{5 \cos \theta}\:\:5 \cos \theta\:\:\text{d}\theta \\\\ & = \int 25 \sin^2 \theta\end{aligned}[/tex]

Take out the constant:

[tex]\implies \displaystyle 25 \int \sin^2 \theta\:\:\text{d}\theta[/tex]

[tex]\textsf{Use the trigonometric identity}: \quad \cos (2 \theta)=1 - 2 \sin^2 \theta[/tex]

[tex]\implies \displaystyle 25 \int \dfrac{1}{2}(1-\cos 2 \theta)\:\:\text{d}\theta[/tex]

[tex]\implies \displaystyle \dfrac{25}{2} \int (1-\cos 2 \theta)\:\:\text{d}\theta[/tex]

[tex]\boxed{\begin{minipage}{5 cm}\underline{Integrating $\cos kx$}\\\\$\displaystyle \int \cos kx\:\text{d}x=\dfrac{1}{k} \sin kx\:\:(+\text{C})$\end{minipage}}[/tex]

[tex]\begin{aligned} \implies \displaystyle \dfrac{25}{2} \int (1-\cos 2 \theta)\:\:\text{d}\theta & =\dfrac{25}{2}\left[\theta-\dfrac{1}{2} \sin 2\theta \right]\:+\text{C}\\\\ & = \dfrac{25}{2} \theta-\dfrac{25}{4}\sin 2\theta + \text{C}\end{aligned}[/tex]

[tex]\textsf{Use the trigonometric identity}: \quad \sin (2 \theta)= 2 \sin \theta \cos \theta[/tex]

[tex]\implies \dfrac{25}{2} \theta-\dfrac{25}{4}(2 \sin \theta \cos \theta) + \text{C}[/tex]

[tex]\implies \dfrac{25}{2} \theta-\dfrac{25}{2}\sin \theta \cos \theta + \text{C}[/tex]

[tex]\implies \dfrac{25}{2} \theta-\dfrac{5}{2}\sin \theta \cdot 5 \cos \theta + \text{C}[/tex]

[tex]\textsf{Substitute back in } \sin \theta=\dfrac{x}{5} \textsf{ and }5 \cos \theta = \sqrt{25-x^2}:[/tex]

[tex]\implies \dfrac{25}{2} \theta-\dfrac{5}{2}\cdot \dfrac{x}{5} \cdot \sqrt{25-x^2} + \text{C}[/tex]

[tex]\implies \dfrac{25}{2} \theta-\dfrac{1}{2}x\sqrt{25-x^2} + \text{C}[/tex]

[tex]\textsf{Substitute back in } \theta=\arcsin \left(\dfrac{x}{5}\right) :[/tex]

[tex]\implies \dfrac{25}{2} \arcsin \left(\dfrac{x}{5}\right) -\dfrac{1}{2}x\sqrt{25-x^2} + \text{C}[/tex]

Take out the common factor 1/2:

[tex]\implies \dfrac{1}{2} \left(25 \arcsin \left(\dfrac{x}{5}\right) -x\sqrt{25-x^2}\right) + \text{C}[/tex]

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Answer:

10830.87 mm³

Step-by-step explanation:

Hello!

Volume of a cylinder: [tex]V = \pi r^2(h)[/tex]

The radius for this cylinder is 9.525, after dividing 19.05 by 2.

Plug it into the volume formula to solve for the volume.

The volume is approximately 10830.87 cubic millimeters.

Answer:

0 < x ≤ 10 and 0 < y ≤ 20.

Step-by-step explanation:

I did the test and got it right ma bois.

Answer: Its A

Step-by-step explanation:

Answer:

11

Step-by-step explanation:

25 than you divided 5 + 2 than you multiply 3 and you get

Answer:

11

Step-by-step explanation:

Answer:

16482 this is the answer

The image of the given polygon under a dilation with a scale factor of 1/3 about the center of dilation ​(0, 0) ​is A'(0, 0), B'(1, 2) and C'(-1, 1)

Transformation is the movement of a point from its initial location to a new location. Types of transformation are reflection, translation, rotation and dilation.

Rigid transformation is a transformation that preserves the shape and size of a figure such as translation, reflection and rotation.

Let us assume that the polygon has vertex at A(0, 0), B(3, 6) and C(-3, 3)

The image of the given polygon under a dilation with a scale factor of 1/3 about the center of dilation ​(0, 0) ​is A'(0, 0), B'(1, 2) and C'(-1, 1)

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the number of ways to line up the 12 people if the bride must be next to the maid of honor is 132 ways.

The formula for finding the number of arrangement is given as;

Permutation = [tex]\frac{n!}{n - r!}[/tex]

Where;

From the question given, we can deduce that

n = 12 people

r = 2, because they must come next to each other and is taken as 2

We then have,

Permutation = [tex]\frac{12!}{12 - 2!}[/tex]

Permutation = [tex]\frac{12!}{10!}[/tex]

Permutation = 132 ways

This is to tell us that the number of ways to arrange the people is 132 ways

Thus, the number of ways to line up the 12 people if the bride must be next to the maid of honor is 132 ways.

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Answer:

522.1

Step-by-step explanation:

Calculated based on 2 given angles and 1 given side.

Side b = 204

Angle ∠A = 90°

Angle ∠B = 23°

Angle ∠C = 180-90-23 = 67°

a = b·sin(A)/sin(B) = 522.09815

Answer:

[tex]x =\frac{5}{3} \pm \frac{\sqrt{10}}{3} \\\\x=2.72076\\x=0.612574\\[/tex]

Step-by-step explanation:

The quadratic equation is:

[tex]3x^2 - 10x + 5 = 0[/tex]

The roots (solutions) of a quadratic equation of the form
[tex]a^2 + bx + c = 0\\[/tex]

are

[tex]x = \frac{ -b \pm \sqrt{b^2 - 4ac}}{ 2a }[/tex]

in this case we have a = 3, b = -10, and c = 5

So, substituting for a, b and c we get

[tex]x = \frac{ -(-10) \pm \sqrt{(-10)^2 - 4(3)(5)}}{ 2(3) }[/tex]

[tex]x = \frac{ 10 \pm \sqrt{100 - 60}}{ 6 }\\[/tex]

[tex]x = \frac{ 10 \pm \sqrt{40}}{ 6 }[/tex]

Simplifying we get

[tex]x = \frac{ 10 \pm 2\sqrt{10}\, }{ 6 }\\\\x = \frac{ 10 }{ 6 } \pm \frac{2\sqrt{10}\, }{ 6 }\\\\x = \frac{ 5}{ 3 } \pm \frac{ \sqrt{10}\, }{ 3 }\\\\\frac{ 5}{ 3 } + \frac{ \sqrt{10}\, }{ 3 } = 2.72076\\\\\\[/tex]  (First root/solution)

[tex]\frac{ 5}{ 3 } - \frac{ \sqrt{10}\, }{ 3 } = 0.612574[/tex]  (Second root/solution)

Answer: 260 m

Step-by-step explanation:

Let's say the length of NR is x m, using the similarity rule, we get the equation

175/130 = (175+175)/x

x = 260

So NR is 260 m

Using proportions, considering the ratio given in the double number line, it is found that there are 7.04 yards in 4 miles.

A proportion is a fraction of a total amount, and the measures are related using a rule of three. Due to this, relations between variables, either direct or inverse proportional, can be built to find the desired measures in the problem.

Researching the problem on the internet, it is found that there are 3.52 yards in 2 miles. Hence the following rule of three is used to find the number of yards in 4 miles.

2 miles - 3.52 yards

4 miles - n yards

Applying cross multiplication:

2n = 4 x 3.52

Simplifying by 2:

n = 2 x 3.52

n = 7.04 yards.

There are 7.04 yards in 4 miles.

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The difference of 32 · x² / (x² + 8 · x + 15) - 14 · x² / (x² - 9) is equal to [18 · x³ - 166 · x²] / [(x + 3) · (x + 5) · (x - 3)]. (Correct choice: B)

In this question we have a subtraction between two rational functions, which have to be simplified by algebra properties. The complete procedure is presented below:

32 · x² / (x² + 8 · x + 15) - 14 · x² / (x² - 9)        Given

32 · x² / [(x + 3) · (x + 5)] - 14 · x² / [(x + 3) · (x - 3)]         Factorization

[x² / (x + 3)] · [32 / (x + 5) - 14 / (x - 3)]      Distributive and associative properties

[x² / (x + 3)] · [32 · (x - 3) - 14 · (x + 5)] / [(x + 5) · (x - 3)]      Subtraction of rational numbers with distinct denominators

[x² / (x + 3)] · [32 · x - 96 - 14 · x - 70] / [(x + 5) · (x - 3)]     Distributive property / (- 1) · a = - a

[x² / (x + 3)] · (18 · x - 166) / [(x + 5) · (x - 3)]      Distributive property / Definitions of addition and subtraction

[18 · x³ - 166 · x²] / [(x + 3) · (x + 5) · (x - 3)]         Mutiplication between rational numbers / Multiplication between powers / Distributive property

The difference of 32 · x² / (x² + 8 · x + 15) - 14 · x² / (x² - 9) is equal to [18 · x³ - 166 · x²] / [(x + 3) · (x + 5) · (x - 3)]. (Correct choice: B)

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Answer: 12x-4, 14 counters, 8 more counters

Step-by-step explanation:  There are 2x yellow, 4x+6 red, and 6x-10 blue counters. We need to add all of this up to get the total number of counters. We first add the x's : 2x+4x+6x = 12x. Then we add the numbers. 6+ (-10) is -4. So, our expression is 12x-4. Next, we need to find out how many blue counters are in the bag. We know that there are 44 total counters and we need to find x because all the counters have x in them. 12x-4 = 44. We first add 4 to both sides to get 12x= 48 and x = 4. blue has 6x-10 counters so, blue has 24-10 = 14 counters. Red has 16+6 = 22 counters. 22-14 = 8 more counters

(-3²)-2(-3-4)-(-1³)

=9-2(-7)+1

=9+14+1

=24

Hope it helps!

Answer: look at the picture

Step-by-step explanation:

Substitute [tex]y=\ln(2x+1)[/tex] and [tex]dy=\frac2{2x+1}\,dx[/tex], so that

[tex]\displaystyle \int \frac2{(2x+1) \ln(2x+1)} \, dx = \int \frac{dy}y = \ln|y| + C = \boxed{\ln|\ln(2x+1)| + C}[/tex]

Answer: x = 9 and x  = 6

Step-by-step explanation:

[tex]x^2-15x+54=0\\(x-9)(x-6)\\x-9=0\\x=9\\x-6=0\\x=6[/tex]

Therefore, x = 9 and x  = 6

Answer:

2

Step-by-step explanation:

→ Find 2 points from the line

( 0 , 6 ) and ( - 3 , 0 )

→ Find the change in y coordinates

0 - 6 = -6

→ Find the change in x coordinates

-3 - 0 = -3

→ Divide the 2 results

-6 ÷ -3 = 2

The volume of the cylinder is [tex]3.14 inch^3[/tex]

Given that

radius = 1 inch

height = inch

we know that volume =[tex]V = \pi \times r^2 \times h[/tex]

now put the value in the above equation we get

[tex]V = 3.14 \times 1 ^2\times 1\\V = 3014inch ^3[/tex]

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The function  which represents the modified design of the roller coaster is Y = 2f(0.15x²-10x+C).

What  define a function?

A technical definition of a function is: a relation from a set of inputs to a set of possible outputs where each input is related to exactly one output.

Here, the function: Y = 2f(0.3x - 1) + 10

Therefore, to transform the function

We have to compare with a general function and integrate.

g(x) = f(bx + c).

We now integrate to transform the function which gives us

Y = (2f) integral {0.3x - 1} + 10

Y = 2f { 0.15x² - x } + 10x + c

Y = 2f(0.15x²-10x+C)

Modified design of roller coaster is

Y = 2f(0.15x²-10x+C)

Thus, the function  which represents the modified design of the roller coaster is Y = 2f(0.15x²-10x+C).

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Answer:

A graph 1

Step-by-step explanation:

The average density of the rod is  0.704 kg/m.

For given question,

We have been given the linear density in a rod 5 m long is 10 / x + 4 kg/m, where x is measured in meters from one end of the rod.

We need to find the

The length of rod is, L = 5 m.

The linear density of rod is, ρ = 10/( x + 4) kg/m

To find the average density we need to integrate the linear density from x₁ = 0 to x₂ = 5,  

The expression for the average density is given as,

⇒ ρ'

[tex]=\int\limits^5_0 {\rho} \, dx\\\\=\int\limits^5_0 {\frac{m}{L} } \, dx\\\\=\int\limits^5_0 {\frac{10}{5(x+4)} }\, dx\\\\=\int\limits^5_0 {\frac{2}{x+4} }\, dx[/tex]                        ......................(1)

Using u = x + 4  

du = dx

u₁ = x₁ + 4

u₁ = 0 + 4

u₁ = 4

and

u₂ = x₂ + 4

u₂ = 5 + 4

u₂ = 9

By entering the values above into (1), we have:

⇒ ρ'

[tex]=2\int\limits^9_4 {\frac{1}{u} } \, du\\\\ = 2[(log~u)]_4^{9}\\\\=2[(log~9-log~4)]\\\\=2\times[0.352][/tex]

= 0.704

Thus, we can conclude that the average density of the rod is  0.704 kg/m.

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Answer:

it's C. 2.26s

but you wrote 2.16s nearly

The soccer ball was in the air for approximately 2.31 seconds. Rounding to two decimal places, the answer is approximately 2.30 seconds,  option D.

To determine how long the soccer ball was in the air, we can use the vertical motion of the ball. When a projectile is launched at an angle, its vertical motion can be analyzed separately from its horizontal motion.

In this case, the initial velocity of the soccer ball can be divided into vertical and horizontal components. The initial velocity in the vertical direction can be calculated using the sine of the launch angle:

Vertical component (Vy) = initial speed (v) * sin(angle)

Vy = 16 m/s * sin(45°)

Vy = 11.31 m/s

Now, we can use the vertical motion equation to find the time the ball spends in the air:

Vertical displacement (y) = Vy * time - (1/2) * gravity * time^2

Since the ball reaches the same vertical position when it lands as when it was launched, the vertical displacement is 0. Therefore, we can set the equation equal to zero:

0 = (11.31 m/s) * time - (1/2) * 9.8 m/s^2 * time^2

Simplifying the equation:

4.9 * time^2 = 11.31 * time

Dividing both sides by time:

4.9 * time = 11.31

time = 11.31 / 4.9

time ≈ 2.31 seconds

Therefore, the soccer ball was in the air for approximately 2.31 seconds. Rounding to two decimal places, the answer is approximately 2.30 seconds, which corresponds to option D.

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(i) The percentage of students who got high scores in both the subjects English and Mathematics is 46%.

(ii) The total number of students who got high scores either in Mathematics or in English if 300 students had attended the exam exists 138.

The probability exists in the analysis of the possibilities of happening of an outcome, which exists acquired by the ratio between favorable cases and possible cases.

The number of students who got high scores in Mathematics was 75%.

The number of students who got high scores in English was 65%.

(i) The percentage of students who got high scores in both the subjects

100% - 6% = 94%

(75% + 65%) - 94%

= 140% - 94%

= 46%

Therefore, the percentage of students who got high scores in both the subjects English and Mathematics is 46%.

(ii) The total number of students who got high scores either in Mathematics or in English if 300 students had attended the exam

= 300 [tex]*[/tex] 46%

= 300 [tex]*[/tex] (46 / 100)

= 300 [tex]*[/tex] 0.46

= 138.

Therefore, the total number of students who got high scores either in Mathematics or in English if 300 students had attended the exam exists 138.

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Answer: -250000

Step-by-step explanation:

Sum of arithmetic terms = n/2[2a + (n - 1)d],

where 'a' is the first term,

'd' is the common difference between two numbers,

and 'n' is the number of terms.

'a' = -1

'd' = -2

'n' = 500

plug and chug

(500)/2[2(-1) + ((500)-1)*(-2)] =

250[-2 + -998] =

250[-1000] =

-250000

Justine only goes when her friend goes [Conditional probability].

Mom going to PROM and Justine going to PROM are separate events. [Independent events].

Mom can't go until she finishes the housework is dependent on events

independent events. [Dependent events]

The probability exists in the analysis of the possibilities of happening of a result, which exists acquired by the ratio between favorable cases and possible cases.

Conditional probability directs to the possibility that some outcome happens given that another possibility contains also occurred.

Justine only goes when her friend goes [Conditional probability]

Independent events exist as those possibilities whose occurrence exists not dependent on any other event.

Mom going to PROM and Justine going to PROM are separate events. [Independent events]

Dependent events exist that depend upon what occurred before.

Mom can't go until she finishes the housework is dependent on events

independent events. [Dependent events]

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Using the Poisson distribution, there is a 0.8335 = 83.35% probability that 2 or fewer will be stolen.

In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by:

[tex]P(X = x) = \frac{e^{-\mu}\mu^{x}}{(x)!}[/tex]

The parameters are:

The probability that a rental car will be stolen is 0.0004, hence, for 3500 cars, the mean is:

[tex]\mu = 3500 \times 0.0004 = 1.4[/tex]

The probability that 2 or fewer cars will be stolen is:

[tex]P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)[/tex]

In which:

[tex]P(X = x) = \frac{e^{-\mu}\mu^{x}}{(x)!}[/tex]

[tex]P(X = 0) = \frac{e^{-1.4}1.4^{0}}{(0)!} = 0.2466[/tex]

[tex]P(X = 1) = \frac{e^{-1.4}1.4^{1}}{(1)!} = 0.3452[/tex]

[tex]P(X = 2) = \frac{e^{-1.4}1.4^{2}}{(2)!} = 0.2417[/tex]

Then:

[tex]P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) = 0.2466 + 0.3452 + 0.2417 = 0.8335[/tex]

0.8335 = 83.35% probability that 2 or fewer will be stolen.

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The value of x is $30.

From the question, it can be deduced that Jeremy's investment account earns a compound interest. Compound interest is when the amount invested and the interest that has already accrued increases in value anytime interest is paid.

The formula that can be used to determine the value of x is:

x = FV / (1+r)^t

X = 480 / (2^4) = $30

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Check the picture below.

Make sure your calculator is in Degree mode.

Let [tex]X(s)[/tex] and [tex]Y(s)[/tex] denote the Laplace transforms of [tex]x(t)[/tex] and [tex]y(t)[/tex].

Taking the Laplace transform of both sides of both equations, we have

[tex]\dfrac{dx}{dt} + 3x + \dfrac{dy}{dt} = 1 \implies \left(sX(s) - x(0)\right) + 3X(s) + \left(sY(s) - y(0)\right) = \dfrac1s \\\\ \implies (s+3) X(s) + s Y(s) = \dfrac1s[/tex]

[tex]\dfrac{dx}{dt} - x + \dfrac{dy}{dt} = e^t \implies \left(sX(s) - x(0)\right) - X(s) + \left(sY(s) - y(0)\right) = \dfrac1{s-1} \\\\ \implies (s-1) X(s) + s Y(s) = \dfrac1{s-1}[/tex]

Eliminating [tex]Y(s)[/tex], we get

[tex]\left((s+3) X(s) + s Y(s)\right) - \left((s-1) X(s) + s Y(s)\right) = \dfrac1s - \dfrac1{s-1} \\\\ \implies X(s) = \dfrac14 \left(\dfrac1s - \dfrac1{s-1}\right)[/tex]

Take the inverse transform of both sides to solve for [tex]x(t)[/tex].

[tex]\boxed{x(t) = \dfrac14 (1 - e^t)}[/tex]

Solve for [tex]Y(s)[/tex].

[tex](s - 1) X(s) + s Y(s) = \dfrac1{s-1} \implies -\dfrac1{4s} + s Y(s) = \dfrac1{s-1} \\\\ \implies s Y(s) = \dfrac1{s-1} + \dfrac1{4s} \\\\ \implies Y(s) = \dfrac1{s(s-1)} + \dfrac1{4s^2} \\\\ \implies Y(s) = \dfrac1{s-1} - \dfrac1s + \dfrac1{4s^2}[/tex]

Taking the inverse transform of both sides, we get

[tex]\boxed{y(t) = e^t - 1 + \dfrac14 t}[/tex]

The solution to 0.5^(x + 2) > 9 is x  > -5.170

The inequality expression is given as:

0.5^(x + 2) > 9

Take the logarithm of both sides of the inequality expression

log(0.5^(x + 2)) > log(9)

Rewrite the inequality expression as

(x + 2)log(0.5) > log(9)

Divide both sides by log(0.5)

x + 2 > -3.170

Subtract 2 from both sides

x  > -5.170

Hence, the solution to 0.5^(x + 2) > 9 is x  > -5.170

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