Answer:
the answer is C 0.75 to 1.0
Over the intervals 0.25 to 0.5, 0.5 to 0.75, and 0.75 to 1.0, the exponential and linear functions are approximately equal.
To determine over which interval the exponential and linear functions are approximately the same, we first need to define the functions. Let's denote the exponential function as [tex]\( f(x) = e^x \)[/tex] and the linear function as ( g(x) = mx + c ), where ( m ) is the slope and ( c ) is the y-intercept.
Given the intervals, we'll first need to calculate the values of the exponential function at the endpoints of each interval and then find the linear function that best approximates those values. We'll choose the linear function with the same value at the starting point of each interval and approximate the slope ( m ) based on the difference in the exponential function values at the endpoints of the interval.
Let's start with each interval:
1. Interval from 0.25 to 0.5:
- Endpoint 1:[tex]\( f(0.25) = e^{0.25} \approx 1.284 \)[/tex]
- Endpoint 2: [tex]\( f(0.5) = e^{0.5} \approx 1.649 \)[/tex]
- Approximating a linear function starting at ( f(0.25) ):
[tex]- \( m = \frac{f(0.5) - f(0.25)}{0.5 - 0.25} = \frac{1.649 - 1.284}{0.5 - 0.25} \approx 0.73 \)[/tex]
[tex]- \( c = f(0.25) \approx 1.284 \)[/tex]
- So, the linear function is [tex]\( g(x) \approx 0.73x + 1.284 \)[/tex]
2. Interval from 0.5 to 0.75:
- Endpoint 1:[tex]\( f(0.5) = e^{0.5} \approx 1.649 \)[/tex]
- Endpoint 2: [tex]\( f(0.75) = e^{0.75} \approx 2.117 \)[/tex]
- Approximating a linear function starting at ( f(0.5) ):
[tex]- \( m = \frac{f(0.75) - f(0.5)}{0.75 - 0.5} = \frac{2.117 - 1.649}{0.75 - 0.5} \approx 0.934 \)[/tex]
[tex]- \( c = f(0.5) \approx 1.649 \)[/tex]
- So, the linear function is [tex]\( g(x) \approx 0.934x + 1.649 \)[/tex]
3. Interval from 0.75 to 1.0:
- Endpoint 1: [tex]\( f(0.75) = e^{0.75} \approx 2.117 \)[/tex]
- Endpoint 2: [tex]\( f(1.0) = e^{1.0} \approx 2.718 \)[/tex]
- Approximating a linear function starting at ( f(0.75) ):
[tex]- \( m = \frac{f(1.0) - f(0.75)}{1.0 - 0.75} = \frac{2.718 - 2.117}{1.0 - 0.75} \approx 1.202 \)[/tex]
[tex]- \( c = f(0.75) \approx 2.117 \)[/tex]
- So, the linear function is [tex]\( g(x) \approx 1.202x + 2.117 \)[/tex]
4. Interval from 1.25 to 1.5:
- Endpoint 1:[tex]\( f(1.25) = e^{1.25} \approx 3.490 \)[/tex]
- Endpoint 2: [tex]\( f(1.5) = e^{1.5} \approx 4.482 \)[/tex]
- Approximating a linear function starting at ( f(1.25) ):
[tex]- \( m = \frac{f(1.5) - f(1.25)}{1.5 - 1.25} = \frac{4.482 - 3.490}{1.5 - 1.25} \approx 3.946 \)[/tex]
- [tex]\( c = f(1.25) \approx 3.490 \)[/tex]
- So, the linear function is[tex]\( g(x) \approx 3.946x + 3.490 \)[/tex]
Now, we can compare each linear approximation to the exponential function within its respective interval:
1. For the interval from 0.25 to 0.5, the linear function [tex]( g(x) \approx 0.73x + 1.284 \)[/tex] is approximately equal to the exponential function [tex]\( f(x) = e^x \).[/tex]
2. For the interval from 0.5 to 0.75, the linear function[tex]\( g(x) \approx 0.934x + 1.649 \)[/tex] is approximately equal to the exponential function [tex]\( f(x) = e^x \).[/tex]
3. For the interval from 0.75 to 1.0, the linear function [tex]\( g(x) \approx 1.202x + 2.117 \)[/tex] is approximately equal to the exponential function [tex]\( f(x) = e^x \).[/tex]
4. For the interval from 1.25 to 1.5, the linear function [tex]\( g(x) \approx 3.946x + 3.490 \)[/tex] is not approximately equal to the exponential function [tex]\( f(x) = e^x \).[/tex]
So, the exponential and linear functions are approximately the same over the intervals from 0.25 to 0.5, 0.5 to 0.75, and 0.75 to 1.0.
How do you evaluate the limit?
Answer:
Step-by-step explanation:
First of all the answer.
[tex]\lim_{n \to -\infty}2^x= 1/ \lim_{n \to \infty}1/2^x = 0[/tex]
[tex]\lim_{n \to -\infty} 1-2^x = 1[/tex]
This approaches 0/1 = 0
The graph is included to show that this is the answer I get.
Solve 8 sin ( 5 x ) = 3 for the two smallest positive solutions a and b, with a < b
The two smallest positive solutions, with a < b, are approximately
x ≈ 0.384 and x ≈ 1.64.
We have,
Isolate the sine function by dividing both sides by 8.
sin(5x) = 3/8
The inverse sine function (also called arcsin) is on both sides of the equation.
5x = arcsin(3/8).
Divide both sides by 5.
x = (1/5) arcsin(3/8)
Now,
arcsin(3/8) is approximately 0.4115.
So,
x = (1/5) * 0.4115
x = 0.384.
Since the sine function repeats itself every 2π radians, we add multiples of 2π/5 to the solution to find other valid values of x.
The second smallest positive solution
x = 0.384 + (2π/5) = 1.64.
Therefore,
The two smallest positive solutions, with a < b, are approximately
x ≈ 0.384 and x ≈ 1.64.
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The equation 8 sin(5x) = 3 is solved by isolating the variable term, using inverse sine to get an angle, and considering the period of the sine function when finding additional solutions.
Explanation:To solve the equation 8 sin (5x) = 3, we first isolate the term involving the variable 'x'. Start by dividing both sides by 8. We get sin(5x) = 3/8.
The inverse sine (also known as arcsin or sin^-1) is used to get the angle whose sine is 3/8. So we have 5x = sin^-1(3/8).
To solve for 'x', we then divide both sides by 5: x = sin^-1(3/8) / 5.
This value of 'x' is a solution, but sine function repeats every 2π intervals, so we add multiples of 2π/5 to 'x' to get additional solutions.
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What questions can you ask yourself as you decide on a career field?
What do I love to do?
What brings me joy?
What am I naturally good at?
All of the above
Answer:
What do I love to do?
What brings me joy?
What am I naturally good at?
Answer is all of the above - last choice
the surface area of the figure below is 50.80m^2 True or False
For this case we have that by definition, the surface area of a cone is given by:
[tex]SA = \pi * r * s + \pi * r ^ 2[/tex]
Where:
A: It's the radio
s: It's the slant
Substituting according to the data we have:
[tex]SA = \pi * 2.1 * 5.6 + \pi * (2.1) ^ 2\\SA = 36.9264 + 13.8474\\SA = 50.7738 \ m ^ 2[/tex]
If we round:
[tex]SA = 50.8 \ m ^ 2[/tex]
Answer:
True
HELP ME OUT PLEASE!! A circular plot with a 250 ft diameter is watered by a spray irrigation system. To the nearest square foot, what is the area that is watered as the sprinkler rotates through an angle of 58∘?
≈ 41,179 ft^2
≈ 31,634 ft^2
≈ 2,517 ft^2
≈ 7,909 ft^2
Answer:
7,909 ft²
Step-by-step explanation:
If the diameter is 250 ft, that must mean the radius is 125 ft. For this problem, use the formula for area of a sector.
[tex]\frac{58}{360} \pi 125^{2}[/tex]
Answer:
≈ 7,909 ft^2
Step-by-step explanation:
Find the next term of the arithmetic sequence shown. 23, 28, 33, 38, . . .
A) 42
B) 43
C) 45
D) 48
ANSWER
B) 43
EXPLANATION
The given arithmetic sequence is 23, 28, 33, 38, . . .
We can observe the following pattern,
23+5=28
28+5=33
33+5=38
To obtain the subsequent terms, we add 5.
Therefore the next term is
38+5=43
6.
What is the value of x?
24
30
40
38
You must do pythagorean theorem:
a^2 + b^2 = c^2
a and b are the legs and c is the hypotenuse
In this triangle...
a = 16
b = x
c = 34
so...
16^2 + x^2 = 34^2
256 + x ^2 = 1156
x^2 = 900
Then to isolate x square root both sides. This will get rid of the square on the x value.
This will give you...
30
x is 30
Hope this helped!
Answer: second option
Step-by-step explanation:
You need to apply the Pythagorean theorem:
[tex]a^2=b^2+c^2[/tex]
Where "a" is the hypotenuse and "b" and "c" are the legs of the right triangle.
In the figure you can observe the value the hypotenuse of the right triangle and the value of one of this leg. Then, you need to solve for the other leg from [tex]a^2=b^2+c^2[/tex]:
[tex]b^2=a^2-c^2[/tex]
[tex]b=\sqrt{a^2-c^2}[/tex]
Substituting values, you get that the value of "x" is:
[tex]b=\sqrt{(34)^2-(16)^2}[/tex]
[tex]b=30[/tex]
It’s simple but my teacher has “33 ft” as an answer and Idk why...
Answer:
33 ft
Step-by-step explanation:
The distance of interest is apparently not the straight-line distance from the top of the vertical ladder to the ground (20 ft), but the length of the arc the top of the ladder makes as it rotates. That arc length (s) is given by ...
s = r·θ . . . . . where θ is the angle in radians subtended by the arc
Here, the radius is 20 ft, and the angle is the supplement of 85°, or 95° = (95/180)π radians.
s ≈ (20 ft)·(1.65806 radians) ≈ 33.16 ft ≈ 33 ft
A wall map is 45 cm high and 27 cm wide. Ashley wants to proportionately shrink it so its height is 12 cm. How wide would it be then?
Answer:
20 centimeters
Step-by-step explanation:
27 / x = 12
x = 2.25 times
width would be 45 / 2.25 = 20 centimeters
Evaluate the expression.
9! - 4!(5!)
Answer:
360,000
Step-by-step explanation:
The exclamation point is factorial:
9! = 9×8×7×6×5×4×3×2×1 = 362880
5! = 5×4×3×2×1 = 120
4! = 4×3×2×1 = 24
362880 - 120× 24 = 360000
Construct a scatterplot and identify the mathematical model that best fits the data. Assume that the model is to be used only for the scope of the given data and consider only linear, quadratic, logarithmic, exponential, and power models. Use a calculator or computer to obtain the regression equation of the model that best fits the data. You may need to fit several models and compare the values of R2.
y = 7.82x0.844
y = 6.81 e0.316x
y = 7.19 + 12.8 ln x
y = 4.40 + 5.00x
Answer:
i)
Find the attached
ii)
The mathematical model that best fits the data is;
y = 7.19 + 12.8 ln x
Step-by-step explanation:
i)
A scatter-plot can easily be constructed using applications such as Ms. Excel and Stat-Crunch.
In Ms. Excel we first enter the data in any two adjacent columns. Next, highlight the data, then click the insert ribbon and select the scatter-plot option.
Excel returns a scatter-plot chart as shown in the attachment below.
ii)
After obtaining the scatter-plot, we shall need to add a trend line in order to determine the mathematical model that best fits the data given.
Click anywhere inside the chart, then select the design tab under chart tools. Click on the Add Chart element in the upper left corner of the excel workbook and select more trend-line options. This feature will enable us to fit any trend-line to our data.
Select any trend line option ensuring you check the boxes; Display Equation on chart and Display R-squared value on chart.
Find the attached for the various trend-lines.
The mathematical model that best fits the data is;
y = 7.19 + 12.8 ln x
Since it has the largest R-squared value of 0.9905
[tex]\rm y = 7.19 + 12.8\ ln\ x[/tex], in this function, the scatter points follow this function. Then the correct option is C.
What is a function?The function is an expression, rule, or law that defines the relationship between one variable to another variable. Functions are ubiquitous in mathematics and are essential for formulating physical relationships.
Construct a scatterplot and identify the mathematical model that best fits the data.
Assume that the model is to be used only for the scope of the given data and consider only linear, quadratic, logarithmic, exponential, and power models.
Use a calculator or computer to obtain the regression equation of the model that best fits the data.
You may need to fit several models and compare the values of R2.
A. [tex]\rm y = 7.82x + 0.844[/tex], in this function, the scatter points do not follow this function.
B. [tex]\rm y = 6.81 e^{0.316x}[/tex], in this function, the scatter points do not follow this function.
C. [tex]\rm y = 7.19 + 12.8\ ln\ x[/tex], in this function, the scatter points follow this function. Because the best fits.
D. [tex]\rm y = 4.40 + 5.00x[/tex], in this function, the scatter points do not follow this function.
The graph is shown.
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quadrilateral ABCD is located at A(-2,2) B (-2,4) C(2,4) D(2,2). The quadrilateral is then transformed using the rule (x-2, y+8) to form A’B’C’D’. Describe what characteristics you would find if the corresponding vertices were connected with line segments !!! HELP WILL MARK AS BRAINLIEST PLS HELP!!
Answer:
Those line segments are parallel and all the same length.
Step-by-step explanation:
Translation modifies the coordinates of each point the same way, moving the point some distance in some direction. Hence the line segments connecting original to image points will all have the same length and direction.
_____
Comment on the question
The question is pretty open-ended. If it is multiple choice, perhaps there is a choice that more or less expresses the idea above. Otherwise, it is hard to tell what the question is driving at.
The population of California is approximately 3.8 × 107, and the population of Nebraska is approximately 1.9 × 106. Which state has the larger population, and by how many times is its population larger than the other state’s?
Answer:
california by 205.2
Step-by-step explanation:
3.8 times 107=406.6-California
1.9 times 106=201.4-Nebraska
205.2
Answer: The Population of California is larger than the population of Nebraska by 20 times
Step-by-step explanation:
Given
Population of California , [tex]P_C=3.8\times 10^{7}[/tex]
and population of Nebraska, [tex]P_N=1.9\times 10^{6}[/tex]
From above data it is clear that the population of California is larger than the population of Nebraska
And Ratio, [tex]\frac{P_C}{P_N}=\frac{3.8\times 10^{7}}{1.9\times 10^{6}}=\frac{38}{1.9}=20[/tex]
Thus the population of California is larger than the population of Nebraska by 20 times
After a group of 24 parts were tested. 5 were found defective. About what percent of the parts were defective
Answer:
20.8%
Step-by-step explanation:
5/24= defective
^divide, and multiply by 100 to get percent
:)
The required percentage of defective parts is 20.8%.
What is the percentage?The percentage is the ratio of the composition of matter to the overall composition of matter multiplied by 100.
Here,
After a group of 24 parts was tested. 5 were found defective.
The required percentage of the defective parts is given as,
= 5 / 24 × 100%
= 0.208 × 100%
= 20.8%
Thus, the required percentage of defective parts is 20.8%.
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Please answer this multiple choice question CORRECTLY for 30 points and brainliest!
Answer:
B
Step-by-step explanation:
If we imagine this is on a clock, then point P is at the 9-o'clock position. When we rotate it 90 degrees clockwise, that's a quarter rotation, so P' will be at the 12-o'clock position. So the coordinates of P' will be (0, 5).
What is the smallest fractional portion of an inch that you can measure with the ruler shown in Exam Figure A1?
A. 1/32"
B. 1/16"
C. 1/8"
D. 1/64"
Answer:
The minimun portion of inches that can be measured with the ruler is 1/16''.
Step-by-step explanation:
The precision , what is the minimun measurement that can be aprecciated by the ruler can be determined counting the number of disivion within 2 units. To this ruler we can count 16 division bwtween one unit and asecutive one.
Which shows reflection
Is it c?
Will give BRAINLIEST. Can't see these very well
Answer:
B
Step-by-step explanation:
You could take these images (if you can visualize it)
And paste it into SM paint, and then reflect it.
But, if you look closely, A is just the notes higher, but not reflected, or changed in any other way.
C the second note is lower in the reflection. The notes shouldn't change in a reflection. Just the order.
If Ruth contributed $10,000 toward the total group investment of $200,000 for the purchase of a seafood restaurant, what percentage of the company does she own?
Answer:
20 %
Step-by-step explanation:
10000/200000
The percentage of the company she owns is equivalent to 5%.
What is percentage?In mathematics →
a percentage is a number or ratio expressed as a fraction of 100.it is often denoted using the percent sign "%".Given is that if Ruth contributed $10,000 toward the total group investment of $200,000 for the purchase of a seafood restaurant.
Assume that the percentage of the company she owns is equivalent to [x] percent. So, we can write -
10000 = [x]% 0f 200000
10000 = [x]/100 × 200000
10000 = [x] × 2000
[x] = (10000/2000)
[x] = 5%
Therefore, the percentage of the company she owns is equivalent to 5%.
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You select a letter randomly from a bag containing the letters s, p, i, n, n, e, and r. find the probability of selecting an s. description
Answer:
1/7
Step-by-step explanation:
Answer:
i think it would be 1/7 also
Step-by-step explanation:
(25 points)
Mr. Price earns $350 per week working in an appliance store. In addition, he earns 3% commission on all of his sales.
Last week, he sold $3, 800 worth of appliances.
What was Mr. Price's total income for the week?
A) $114
B) $236
C) $464
D) $556
Answer:
c
Step-by-step explanation:
commission is basically 3 percent of what he sells he gets, sooooo 3 percent of 3800 is 114 (3800 × .03 because change the percent to decimal) and add it to his pay :)
Answer:
D
Step-by-step explanation:
A science teacher needs to choose 12 students out of 16 to serve as peer tutors how many different ways can the teacher choose the12 students
He can ask the student to raise their hands and pick them
the teacher can ask them to pick the closest number. the teacher can choose the students with the highest grades.
The slope of a speed-time graph indicates
Answer:
acceleration
Step-by-step explanation:
In a speed-time graph, the speed data is on the Y-axis and the time is represented on the X-axis.
A slope is the variation of Y-values over the variation of the X-values between to points.
So, the slope of a speed-time graph would be the variation of the speed over the variation of time.
Let's say your units are m/s for the speed and s for the time.
The slope units would then be calculated by dividing m/s by s... (m/s) / s....
That would give you m/s², which is an acceleration unit.
(3^2)^6 = _____
A) 3^-4
B) 3^4
C) 3^8
D) 3^12
The answer is:
The correct option is:
D) [tex](3^{2})^{6}=3^{12}[/tex]
Why?To solve the problem, we need to remember the power of a power property, it's defined by the following way:
[tex](a^{m})^{n}=a^{m*n}[/tex]
When we have a power of a power, we must keep the base and then, the new exponent will be the product between the two original exponents.
So, we are given the expression:
[tex](3^{2})^{6}[/tex]
Then, calculating we have:
[tex](3^{2})^{6}=3^{2*6}=3^{12}[/tex]
Hence, we have that the correct option is:
D) [tex](3^{2})^{6}=3^{12}[/tex]
Answer:
The correct answer is option D). 3^12
Step-by-step explanation:
Points to remember
Identities
(xᵃ)ᵇ = xᵃᵇ
xᵃ * xᵇ = x⁽ᵃ ⁺ ᵇ⁾
xᵃ/xᵇ = x⁽ᵃ ⁻ ᵇ⁾
It is given that (3^2)^6
To find the correct option
(3^2)^6 can be written as, (3²)⁶
By using above identities,
(3²)⁶ = 3⁽² ˣ ⁶⁾
= 3¹²
Therefore the correct answer is option D). 3^12
A jewelry box contains two gold hoop earrings and two silver hoop earrings. You randomly choose two earrings. What is the probability that both are silver hoop earrings? Write your answer as a fraction or percent rounded to the nearest tenth. The probability that both are silver hoop earrings is .
Answer:
fraction is 2/4
Step-by-step explanation:
Gavin and Jack are practicing shots against their goalie. On their last 15 attempts, Gavin made 6 and Jack made 7. Based on this performance, what is the probability that they both make their next shot?
Answer:
[tex]\dfrac{14}{75}[/tex]
Step-by-step explanation:
Gavin made 6 out of 15 shots, so the probability that Gavin's next shot will be successful is
[tex]\dfrac{6}{15}=\dfrac{2}{5}[/tex]
Jack made 7 out of 15 shots, so the probability that Jack's next shot will be successful is
[tex]\dfrac{7}{15}[/tex]
The probability that they both make their next shot successfully is
[tex]\dfrac{2}{5}\cdot \dfrac{7}{15}=\dfrac{14}{75}[/tex]
Divide and write in simplest form
3 1/3 divided by 2 3/5
Answer:
Step-by-step explanation:
If we change those mixed fractions to improper we will have a much easier time with them:
[tex]\frac{\frac{10}{3} }{\frac{13}{5} }[/tex]
Now that looks horrible. The rule for dividing fractions by fractions is a simple one, thankfully. Bring the bottom fraction up next to the top fraction, then flip it upside down and change the sign to multiplication:
[tex]\frac{10}{3}[/tex]×[tex]\frac{5}{13}[/tex]
Multiply straight across the top and straight across the bottom to get
[tex]\frac{50}{39}[/tex]
Depending upon what your teacher calls "simplest form", this form may be the simplest, or the mixed fraction may be the simplest. The mixed fraction equivalent to this improper is
[tex]1\frac{11}{39}[/tex]
At the movie theatre, child admission is $5.70 and adult admission is $9.40. On Wednesday, 173tickets were sold for a total sales of $1345.00. How many child tickets were sold that day?
Answer: 76 child tickets were sold that day.
Step-by-step explanation:
You need to set up a system of equations.
Let be "c" the number of child tickets sold that Wednesday and "a" the number of adult tickets sold that Wednesday.
Then:
[tex]\left \{ {{a+c=173} \atop {9.40a+5.70c=1345.0}} \right.[/tex]
Using the Method of Elimination, you can multiply the first equation by -9.40, then add both equations and finally solve for "c":
[tex]\left \{ {{-9.40a-9.40c=-1626.2} \atop {9.40a+5.70=1345.0}} \right.\\..................................\\-3.7c=-281.2\\\\c=\frac{-281.2}{-3.7}\\\\c=76[/tex]
A spinner has 4 equal sections. after 12 spins, the spinner landed on section a 4 times, section b 5 times, section c 2 times, and section d 1 time. what is the experimental probability of the spinner stopping on section a?
Answer:
The Experimental Probability of the spinner stopping on Section A is 1/3.
Step-by-step explanation:
Number of time spinner landed on section A = 4
Number of time spinner landed on section B = 5
Number of time spinner landed on section C = 2
Number of time spinner landed on section D = 1
Total Number of spins = 12
[tex]Probability=\frac{Number\:of\;Favorable\:outcome}{Number\:of\:total\;outcome}=\frac{4}{12}=\frac{1}{3}[/tex]
Therefore, The Experimental Probability of the spinner stopping on Section A is 1/3.
Which numbers are necessary to solve this problem? Franklin brought 4 jump ropes to the park. Each jump rope can be used by 3 people at a time. Fifteen friends came to the park to jump rope. How many friends could play jump rope if all of Franklin's ropes were being used? A. 4 jump ropes, 3 people B. 3 people, 15 friends C. 4 jump ropes, 15 friends D. 4 jump ropes, 3 people, 15 friends
Answer:
D. 4 jump ropes, 3 people, 15 friends
Step-by-step explanation:
In order to answer the question how many friends could play, you need to be able to determine the smaller of ...
(number of ropes) × (friends per rope)number of friendsYou can find the first of these numbers using the values of answer A, but if the number of Franklin's friends is smaller than 12, then you need to know that in order to properly answer the question. Hence, we believe you need to know ...
the number of jump ropesthe number of friends per ropethe number of friendsThere are 16 girls in a school club. The number of girls is four more than twice the number of boys. Find the number of boys
The problem describes a situation with 16 girls who are four more than twice the number of boys. By setting up and solving the simple algebra equation 2x + 4 = 16, we find there are 6 boys in the club.
Explanation:This is a word problem which can be solved using simple algebra. Given that there are 16 girls in the club and the number of girls is four more than twice the number of boys, we can set up the equation:
2x + 4 = 16
Where x stands for the number of boys. Solving this equation, we subtract 4 from both sides to get: 2x = 12. Dividing both sides by 2, we find that there are x = 6 boys in the club.
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