Given: Neil bought a house for £235 000
In the first year the value of the house depreciated by 4%
In each of years 2 and 3, the value of the house increased by 6%.
Required: To find out the value of the house at the end of year 3.
Explanation: Since in first year the value of house depreciated by 4%,
[tex]\text{ New price}=235000-\frac{235000\times4\times1}{100}[/tex][tex]undefined[/tex]Are these events independent or dependent? A. Rolling a 6-sided dice twice and recording whether the outcome was even or odd. B. Picking a number from a hat, recording the number and keeping the slip out of the hat, then picking a second number from the same hatA. dependent ; independentB. independent; dependentC. independent ; independentD. dependent ; dependent
Divide the polynomials using long division or synthetic division.
(x³ − 4x²+x+6) ÷ (x − 2)
(Type your answer in the box WITHOUT any spaces.)
I
A/
Answer:To divide a polynomial by a binomial of the form x - c using synthetic division.
Use the Remainder Theorem in conjunction with synthetic division to find a functional value.
Use the Factor Theorem in conjunction with synthetic division to find factors and zeros of a polynomial function.
Step-by-step explanation:
if you have the lessons this is what id do to solve this!
starting with the graph of f(x)=8^x write the equation of the graph that results from Shifting f(x) 5 units upward y=____shifting f(x) 9 units to the left y=____reflecting f(x) about the x axis and the y axis y=
We have the following:
[tex]f(x)=8^x[/tex](a)
for there to be an upward displacement, we must add the function the value that we want it to rise, like this
[tex]f(x)=8^x+5[/tex](b)
for there to be a shift to the left, we must add the exponent from the value we want it to rise, like this
[tex]f(x)=8^{x+9}[/tex](c)
for there to be a shift to the left, we must subtract the exponent from the value we want it to rise, like this
The inverse is:
[tex]\begin{gathered} y=8^x \\ x=8^y \\ \ln x=y\cdot\ln 8 \\ y=\frac{\ln x}{\ln 8} \end{gathered}[/tex]The answer is
[tex]f(x)=\frac{\ln x}{\ln 8}[/tex]Select the correct choice below and fill in any answer boxes in your choice.
Answer;
[tex]x=40[/tex]Explanation;
To get the correct choice, we have to simplify the given equation as follows;
[tex]\begin{gathered} 4x-(2x+6)=3x-46 \\ 4x-2x-6=3x-46 \\ 2x-6=3x-46 \\ 3x-2x=46-6 \\ x\text{ = 40} \end{gathered}[/tex]So, the correct option is A and we fill the value 40 in the box
which equation represents the graph shown below?A. y=4sin(pi/80x)+5B. y=5cos(pi/80x)+4C. y=4cos(pi/80x)+5D. y=5sin(pi/80x)+4
Since the amplitud of the function is 5 and it starts on (0,9) we can say that the function is:
y=5cos(pi/80x)+4
Factor the given polynomial completely and match your result to the correct answer below.18m³ +24m²-24mSelect one:O a. 6m(m-4)(3m + 1)O b. 6m(3m2 +6m-4)O c.6m(m+2)(3m-2)O d. The polynomial is prime.
Given:
[tex]18m^^3+24m^2-24m[/tex]Required:
We need to factorize the given polynomial completely.
Explanation:
Take out the common multiple 6m.
[tex]18m^3+24m^2-24m=6m(3m^2+4m-4)[/tex][tex]Use\text{ 4m=6m-2m.}[/tex][tex]18m^3+24m^2-24m=6m(3m^2+6m-2m-4)[/tex]Take out the common multiple.
[tex]18m^3+24m^2-24m=6m(3m(m+2)-2(m+2))[/tex][tex]18m^3+24m^2-24m=6m(m+2)(3m-2)[/tex]Final answer:
[tex]6m(m+2)(3m-2)[/tex]what is represented by 2 in the ordered pair (2,7)
In (2, 7), 2 is the input
I need help with 10 please it says to find the area of each shaded sector. And round to the hundredth place
Given:
SR = 26 m.
To find:
The area of shaded region.
Solution:
Here, QR ~ PS. So, angle PTS = angle QTR.
So, angle PTS = 73 degrees.
To find the area of the shaded region, we have to subtract the area of unshaded region from the area of the circle.
Here, SR is the diameter and SR = 26. So, the radius of the circle is 13 m.
Since, the unshaded regions are similar to each other. So, the total area of the unshaded region is:
[tex]\begin{gathered} A=2\times\frac{73}{360}\times\frac{22}{7}\times(13)^2 \\ =\frac{542828}{2520} \\ =215.41m^2 \end{gathered}[/tex]The area of the circle is:
[tex]\begin{gathered} A=\pi r^2 \\ =\frac{22}{7}\times(13)^2 \\ =\frac{3718}{7} \\ =531.14 \end{gathered}[/tex]So, the area of shaded region is:
[tex]531.14-215.41=315.73m^2[/tex]Thus, the area of the shaded region is 315.73 m^2.
Find the derivatives of the following using the different rules.1. y = 3x + 29
To find the derivative of the given function, we can use the power rule.
[tex]ax^n\Rightarrow nax^{n-1}[/tex]In this rule, we multiply the exponent of the variable by its numerical coefficient and then subtract 1 from the exponent.
For this function y = 3x + 29, we have two terms. These are 3x and 29. We need to apply the power rule for each term.
Let's start with 3x.
[tex]3x^1\Rightarrow1(3)(x^{1-1})\Rightarrow3x^0\Rightarrow3[/tex]The first derivative for 3x is 3.
For the term 29, since there is no variable, the derivative for 29 is 0.
So, the first derivative of y = 3x + 29 is y' = 3 + 0 or just y' = 3.
[tex]y^{\prime}=3[/tex]After completing the fraction division 5 divided by 5/3, Miko used the multiplication shown to check her work. 3x5/3=3/1x5/3=15/3 or 5
Answer:
Miko found the correct quotient and checked her work using multiplication correctly
Explanation:
When we divided 5 by 5/3, we get:
[tex]5\div\frac{5}{3}=5\times\frac{3}{5}=\frac{5\times3}{5}=\frac{15}{5}=3[/tex]Therefore, the quotient is 3.
Then, to check the division, we need to multiply the quotient 3 by the divisor 5/3. If we get the dividend 5, the division was correct, so
[tex]3\times\frac{5}{3}=\frac{3}{1}\times\frac{5}{3}=\frac{15}{3}=5[/tex]Therefore, Miko found the correct quotient and checked her work using multiplication correctly.
2 + 2 what is it I need to know please :/
2+2=4
The answer is 4
Answer:
4!
Step-by-step explanation:
when you add them together you get 4
I need help with this practice problem I need to know if I’m correct.
Let's draw a picture of our problem:
The trigonometric value function of the angle beta will have the same value of the trigonometric value function of theta. Then, in order to find the cosine of beta, we can use the following right triangle
Therefore,
[tex]\cos \beta=\cos \theta=\frac{-4}{\sqrt[]{61}}[/tex]or equivalently
[tex]\cos \beta=\frac{-4}{61}\sqrt[]{61}[/tex]Therefore, the answer is correct.
the speed of a stream is 4 mph. A boat travels 6 miles upstream in the same time it takes to travel 14 miles downstream. what is the speed of the boat in still water?
The speed of the boat in still water is 10 mph.
What is a expression? What is a mathematical equation?A mathematical expression is made up of terms (constants and variables) separated by mathematical operators. A mathematical equation is used to equate two expressions. Equation modelling is the process of writing a mathematical verbal expression in the form of a mathematical expression for correct analysis, observations and results of the given problem.
We have the speed of a stream as 4 mph. A boat travels 6 miles upstream in the same time it takes to travel 14 miles downstream.
Assume the speed of the boat in still water be [x] mph.
We know that -
time [t] = distance[x]/speed[s]
Then, we can write -
6/(x - 4) = 14/(x + 4)
6(x + 4) = 14(x - 4)
6x + 24 = 14x - 56
56 + 24 = 8x
80 = 8x
x = 10 mph
Therefore, the speed of the boat in still water is 10 mph.
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What is the product of 2 1/2 and 1 1/4
First, we transform the mixed fractions into improper fractions:
[tex]\begin{gathered} 2\frac{1}{2}=\frac{2\cdot2+1}{2}=\frac{5}{2}, \\ 1\frac{1}{4}=\frac{1\cdot4+1}{4}=\frac{5}{4}. \end{gathered}[/tex]Then, we multiply the improper fractions:
[tex]\frac{5}{2}\cdot\frac{5}{4}=\frac{5\cdot5}{2\cdot4}=\frac{25}{8}\text{.}[/tex]Finally, we transform the improper fraction into a mixed fraction:
[tex]\frac{25}{8}=\frac{24+1}{8}=\frac{3\cdot8+1}{8}=3\frac{1}{8}\text{.}[/tex]Answer:
[tex]3\frac{1}{8}\text{.}[/tex]When a tow truck is called, the cost of the service is $150 plus $5 per mile that the car must be towed.
Write and graph a linear equation to represent the total cost of the towing service, which is dependent on the number of miles the car is towed.
Find and interpret the slope and y-intercept of the linear equation
The linear equation to represent the total cost of the towing service is;
C = 150 + 5x.
What is defined as the term linear equation?A linear equation is one in which the variable's highest power is always 1. A one-degree equation is another name for it. A linear equation inside one variable has the standard form Ax + B = 0.For the given question,
The fixed cost of towing service = $150.
The variable cost per mile = $5.
Let the number of miles be 'x'.
The total cost be 'C'.
Thus, the equation becomes;
C = 150 + 5x.
C = 5x + 150 (linear equation to represent the total cost of the towing service)
Find the slope and y-intercept of the linear equation;
Comparing the equation with the slope intercept form of line;
y = mx + c
slope m = 5
y-intercept c = 150.
Thus, the linear equation to represent the total cost of the towing service is; C = 150 + 5x.
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What makes 3 + 7 + 2 = 0 + 2 true?
Assuming that the question for this case is:
[tex]3+7+2=x+0+2[/tex]We can subtract in both sides of the equation 2 and we got:
[tex]x=3+7+2-2=10[/tex]And the solution for this case would be 10
2 3/5 •3= ?A 13/15B 1 2/13C 6 3/5D 7 4/5
Given:
[tex]2\frac{3}{5}\times3=?[/tex]First of all,we multiply 5 and 2 i.e.
[tex]\begin{gathered} 5\times2=10 \\ \end{gathered}[/tex]Then,add 3 to this value.
It becomes 10+3=13
Hence,
[tex]2\frac{3}{5}\times3=\frac{13}{5}\times3[/tex][tex]=\frac{39}{5}[/tex]Now, we convert the obtained value in mixed fraction.
First we divide 39 by 5 and then write the remainder in numerator and dividend in the side as shown.
[tex]\frac{39}{5}=7\frac{4}{5}[/tex]Here, when we divide we get 6 as divident and 4 as remainder, so we express it like this.
Hence, option (4) is correct.
Mark says that an angle showing a turn through of a circle that is 10 inches across
The question involves comparing the measure of two angles different circles
For the first circle
The angle turn through 1/4 of a circle, which means that the circle turns through 90 degrees
For the second circle that is 10 inches across
The angle also turned through 1/4 of the circle. This also implies that the circle turns through 90 degrees
Thus,
The angle turned by the two circles is the same. Hence, Mark is not correct
Find the absolute maximum and absolute minimum values of f on the given interval.
f(x) = xe^(−x^2/98), [−3, 14]
absolute minimum value?
absolute maximum value?
Absolute minimum value and maximum value at f(-3) = -2.7 and
f(14) = 12.14 respectively.
Define function.An association between a number of inputs and outputs is called a function. A function is, to put it simply, an association of inputs where each input is connected to exactly one output. For each function, there is a corresponding range, codomain, and domain.
Given function is -
f(x) = x*e^(−x^2/98)
By differentiating the function, we will get
f'(x) = (1)([tex]e^{-x^{2} /98}[/tex])+ x([tex]\frac{-2x}{98}[/tex] * [tex]e^{-x^{2} /98}[/tex])
f'(x) = ([tex]e^{-x^{2} /98}[/tex] ) - ([tex]\frac{x^{2} }{49}[/tex] * [tex]e^{-x^{2} /98}[/tex])
f'(x) = ([tex]e^{-x^{2} /98}[/tex]) (1 - [tex]\frac{x^{2} }{49}[/tex])
To calculate the maximum and minimum value, (1 - [tex]\frac{x^{2} }{49}[/tex]) must be zero or ([tex]e^{-x^{2} /98}[/tex]) must be zero.
=> (1 - [tex]\frac{x^{2} }{49}[/tex]) =0
=> [tex]\frac{x^{2} }{49}[/tex] = 1
=> [tex]x^{2}[/tex] = 49
=> x = 7 or x= -7
However, -7 is not within our given interval and does not need to be tested. Therefore, put the x = -3,7,14 in given function.
f(-3) = -3 [tex]e^{-9/98}[/tex] = -2.7
f(7) = 7 [tex]e^{-1/2}[/tex] = 4.24
f(14) = 14 [tex]e^{-1/7}[/tex] = 12.14
Absolute minimum value at f(-3) = -2.7 and
Absolute maximum value at f(14) = 12.14
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Which statements describe one of the transformations performed on f(x) = x²to create g(x) = 3(x+5)² -2? Choose all that apply.A. A vertical stretch with a scale factor of 3B. A translation of 2 units to the leftC. A translation of 5 units to the left□ D. A vertical stretch with a scale factor of eP
The transformations performed on the function f(x) to create g(x) include : a vertical stretch with a scale factor of 3 and a translation of 5 units to the left.
We are given a function f(x). The function f(x) is defined as x². We also have another function, g(x). The function g(x) is defined as g(x) = 3(x + 5)² - 2. The function g(x) is formed by performing several transformations on the function f(x). The first transformation is translating the function to the left by 5 units. The next transformation is stretching the function vertically by a scale factor of 3. The last transformation is translating the function downward by 2 units.
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A survey was done on the drink preferences of shoppers at the mall. The results are shown in the table. What is the probability that a shopper, chosen at random, will prefer neither Drink D nor Drink C?
SOLUTION:
Case: probability
Method:
From the table,
The probability that a shopper will prefer neither D nor C is:
[tex]\begin{gathered} Pr(NotDorC) \\ =\frac{n(AorBorE)}{Total} \\ =\frac{9+11+5}{46} \\ =\frac{25}{46} \end{gathered}[/tex]Final answer:
The probability that a shopper will prefer neither Drink D nor C:
25/46
fifty four percent of the items in a refrigerator are dairy products what percent of the items are non dairy products
Given :
54% of the items are dairy products.
A large survey of countries including the USA, China, Russia, France, and others indicated that most people prefer the color blue. In fact, about 24% of the population claim blue as their favorite color. Suppose a random sample of 86 college students were surveyed and 30 of them said that blue is their favorite color. Does this information imply that the color preference of all college students is different (either way) from that of the general population? Use α = 0.05.1. State your null and alternate hypothesis. What is the level of significance? Will you use a left tail, right tail or two-tail test?2. What is the value of the test statistic. 3.lFind the P-value. Sketch the sampling distribution z = to show the area corresponding to the P-value. 4. Based on 1-3, will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α ? State your conclusion in the context of the application.
We have to perform an hypothesis test of a proportion.
The claim is that the sample has a different proportion than the population.
Then, the null and alternative hypothesis are:
[tex]\begin{gathered} H_0\colon\pi=0.24 \\ H_a\colon\pi\neq0.24 \end{gathered}[/tex]The significance level is 0.05.
The sample has a size n=86.
The sample proportion is p=0.349.
[tex]p=X/n=30/86=0.349[/tex]The standard error of the proportion is:
[tex]\begin{gathered} \sigma_p=\sqrt{\dfrac{\pi(1-\pi)}{n}}=\sqrt[]{\dfrac{0.24\cdot0.76}{86}} \\ \sigma_p=\sqrt{0.002121}=0.046 \end{gathered}[/tex]Then, we can calculate the z-statistic as:
[tex]z=\dfrac{p-\pi-0.5/n}{\sigma_p}=\dfrac{0.349-0.24-0.5/86}{0.046}=\dfrac{0.103}{0.046}=2.241[/tex]This test is a two-tailed test*, so the P-value for this test is calculated as:
[tex]\text{P-value}=2\cdot P(z>2.241)=0.025[/tex]* We use a two-tailed test because we are looking for difference above or below the population proportion.
As the P-value (0.025) is smaller than the significance level (0.05), the effect is significant.
The null hypothesis is rejected.
At a significance level of 0.05, there is enough evidence to support the claim that the sample has a different proportion than the population.
Answer:
1) The null and alternative hypothesis are:
[tex]\begin{gathered} H_0\colon\pi=0.24 \\ H_a\colon\pi\neq0.24 \end{gathered}[/tex]2) The test statistic is z=2.241.
3) The P-value is 0.025. The value in the standard normal distribution is:
4) As the effect is significant (the P-value is less than the significance level), there is evidence to reject the null hypothesis.
The conclusion is that this sample has a proportion that is significantly different from that from the population.
The length of an arc of a circle measures 0.3km. The radius of the circle measures 0.7km. What is the degree measure of the central angle of a circle associated with this arc? Use 3.14 for Π. Round your answer to the nearest tenth.
SOLUTION:
Step 1:
In this question, we are given the following:
The length of an arc of a circle measures 0.3km.
The radius of the circle measures 0.7km.
What is the degree measure of the central angle of a circle associated with this arc? Use 3.14 for Π.
Round your answer to the nearest tenth.
Step 2:
The details of the solution are as follows:
[tex]\begin{gathered} \text{Length of an arc of a circle = 0. 3 }km \\ \text{Radius of the circle = 0. 7 }km \\ \text{Degr}ee\text{measure of the central angle of a circle = }\theta \\ \pi\text{ = 3. 14} \end{gathered}[/tex][tex]\begin{gathered} \text{Length of Arc , l = }\frac{\theta}{360^0\text{ }}\text{ x 2}\pi r \\ 0.\text{ 3 = }\frac{\theta}{360^0}\text{ x 2 x 3. 14 x 0.7} \end{gathered}[/tex][tex]\begin{gathered} 0.3\text{ = }\frac{\theta\text{ x 4.396}}{360^0} \\ \end{gathered}[/tex]cross-multiply, we have that:
[tex]\begin{gathered} 360\text{ x 0. 3 = 4.396}\theta \\ \text{Divide both sides by 4.396, we have that:} \end{gathered}[/tex][tex]\begin{gathered} \theta\text{ = }\frac{360\text{ X 0. 3}}{4.396} \\ \end{gathered}[/tex][tex]\begin{gathered} \theta=\text{ }\frac{108}{4.396} \\ \end{gathered}[/tex][tex]\begin{gathered} \theta\text{ = 24.5677889} \\ \theta\approx24.6^{0\text{ }}(\text{ to the nearest tenth)} \end{gathered}[/tex]2. What is the value of 'm' if 9m ÷ 9-3 = 95
Answer:
m = 98
Step-by-step explanation:
9m/9 - 3 = 95
9m/9 = 95 + 3
9m/9 = 98
cross-multiply
9m = 98 × 9
9m = 882
Divide both sides by 9
m = 98
Can someone please help me with these, please? I’ve tried them myself already, but I got confused enough I didn’t end up with an answer
Given:
The cost for a day food, entertainment and hotes is $250.
The cost for round trip air fair is $198.
Explanation:
Let x represents the number of full days that individual can stay at the beach.
The total money available to individual is $1400. So inequality is,
[tex]\begin{gathered} 250\cdot x+198\leq1400 \\ 250x+198\leq1400 \end{gathered}[/tex]Thus inequality for number of days is,
[tex]250x+198\leq1400[/tex]The variable x represent the number of full days that individual can spend at beach trip.
(b)
Solve the inequality for x.
[tex]\begin{gathered} 250x+198-198\leq1400-198 \\ \frac{250x}{250}\leq\frac{1202}{250} \\ x\leq4.808 \end{gathered}[/tex]The maximum whole value of x is 4.
Thus individual can spend 4 complete (full) days at the beach trip.
Use this diagram to answer the questions
4b. 3
Part A
Which expression represents the area of the rectangle?
B. 6+ (40 - 3)
A6(4-3)
D. 2 x 6 x (40 - 3)
C
6+ (40 - 3) + 6 + (40 - 3)
Part B
Which expression is equivalent to the expression you chose in Part A?
B. 246 - 18
A
240-3
D859
C. 80+ 6
Find the area of the rectangle if = 4 Enter your answer in the box
square units
We will answer the question given in the picture.
We can see from the question a part of a linear function, and we can see an open circle at the point (4, -2). We can also see that the arrow of the linear function indicates that the function continues infinitely.
To find the domain and the range of the function we need to remember that:
• The domain of a function is, roughly speaking, all of the values for which the function is defined. In general, is represented by all of the values of x for which this function is defined.
,• The range of a function is, roughly speaking, all the values that the variable y, the dependent variable, takes for each of the values of the independent variable, x.
Therefore, if we check the graph, we have:
The domain of the function1. The values for x are not defined for x = 4 (we can see a small open circle at the point (4, -2). However, the values for x continue infinitely after that. Therefore, the domain of the part shown is as follows:
[tex]\text{ Domain=}x>4[/tex]And we can say that the domain of the function is for all of the values greater (not equal to x = 4) to positive infinity. We can write this in interval notation as follows:
[tex]\text{ Domain=}(4,\infty)[/tex]The range of the functionWe can check from the graph that the values for y start from y = -2. However, y = -2 is not included since we have a small open circle that indicates that (see above).
Therefore, the range of the function is given by:
[tex]y<-2[/tex]And we can say that the values of the range are less than y = -2 (not equal), and they are all smaller than y = -2 (for instance, -3, -4, -5.001, -10.222, and so on). The latter values are less than y = -2. We can write this in interval notation as follows:
[tex]\text{ Range=}(-2,-\infty)_[/tex]Therefore, in summary, we can say that:
1. The inequality to represent the domain of the part shown is x > 4. It means that the domain is those values of the independent variable greater than x = 4 (not equal to 4), and these values extend to positive infinity.
2. The inequality to represent the range of the part shown is y < -2. It means that the range is those values of the dependent variable less than y = -2 (not equal to y = -2), and these values extend to negative infinity.
In the Itty Bitty High School, there are 85 students. There are 27 students whotake French, 51 who take Geometry and 38 who take History. There are 10 thattake Geometry and French, 7 that take History and French and 15 that takeGeometry and History. There are 2 students who take all 3 and 5 that take noneof these subjects.What it the probability that you randomly select a person who is a female giventhey are brown headed? *Fair ColorBrom Blonde Red87682SaleFemale34861271666Your answer
ANSWER:
2.2%
STEP-BY-STEP EXPLANATION:
The probability would be the quotient between the number of people with these characteristics (female, brown hair) and the total number of people
[tex]\begin{gathered} p(\text{female and brown) }=\frac{66}{548+876+82+612+716+66}=\frac{66}{1450}=0.022 \\ p(\text{female and brown) }=2.2\text{\%} \end{gathered}[/tex]Seventh grade > Y.12 Area of compound figures with triangles MRGWhat is the area of this figure?3 mm4 mm8 mm2 mm5 mm6 mm3 mmWrite your answer using decimals, if necessary.square millimeters6 mm
Area of the figure = 72mm²
Explanation:Given:
An irregular figure
To find:
the area of the figure
To determine the area, we will divide the figure into shapes with known areas
We have 1 triangle, 1 rectangle, and 1 square. We will find the area of each of the shapes
Area of triangle = 1/2 × base × height
height = 3mm
base = 4 + 3 + 1 = 8mm
[tex]\begin{gathered} Area\text{ of the triangle = }\frac{1}{2}\times8\times3 \\ \\ Area\text{ of the traingle = 12 mm}^2 \end{gathered}[/tex]Area of rectangle = length × width
length = 8mm, width = 3mm
[tex]\begin{gathered} Area\text{ of the rectangle = 8 }\times\text{ 3} \\ \\ Area\text{ of the rectangle = 24 mm}^2 \end{gathered}[/tex]Area of square = length²
length = 6 mm
[tex]\begin{gathered} Area\text{ of the square = 6}^2 \\ \\ Area\text{ of the square = 36 mm}^2 \end{gathered}[/tex]Area of the figure = Area of triangle + Area of the rectangle + Area of the square
[tex]\begin{gathered} Area\text{ of the figure = 12 + 24 + 36} \\ \\ Area\text{ of the figure = 72 mm}^2 \end{gathered}[/tex]can you help me with this question
Since B is the midpoint of AC, we can conclude:
[tex]\begin{gathered} AC=AB+BC \\ also \\ AB=BC \\ so\colon \\ 6x+1=2x+9 \\ solve_{\text{ }}for_{\text{ }}x\colon \\ 6x-2x=9-1 \\ 4x=8 \\ x=\frac{8}{4} \\ x=2 \end{gathered}[/tex]