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The general form of an ellipse is 15x2+4y2+30x−16y−29=0.
What is the standard form of the ellipse?
Answers
The equation of ellipse in standard form is (x + 1)² / 4 + (y - 2)² / 15 = 1.
How to determine the standard form of the ellipseIn this problem we find the equation of an ellipse in general form, whose standard form can be found by algebra properties. The general form of the equation of an ellipse centered at (h, k) is introduced below:
(x - h)² / a² + (y - k)² / b² = 1
Where:
a, b - Lengths of the semiaxes.(h, k) - Coordinates of the center.The complete procedure is now presented:
15 · x² + 4 · y² + 30 · x - 16 · y - 29 = 0
(15 · x² + 30 · x) + (4 · y² - 16 · y) = 29
15 · (x² + 2 · x) + 4 · (y² - 4 · y) = 29
15 · (x² + 2 · x + 1) + 4 · (y² - 4 · y + 4) = 29 + 15 · 1 + 4 · 4
15 · (x + 1)² + 4 · (y - 2)² = 29 + 15 + 16
15 · (x + 1)² + 4 · (y - 2)² = 60
(x + 1)² / 4 + (y - 2)² / 15 = 1
The equation of ellipse is (x + 1)² / 4 + (y - 2)² / 15 = 1.
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Related Questions
10 ptQuestion 10A can of soup has a volume of 80 in and mass of 10 ounces. A can of tuna has a volume of 56 in and mass of 8ounces. About how much less is the density of the soup than the tuna (give your answer in ounces/square inch).Round your answer to the nearest 1000th.SOUPSTUNA CHUNKSBrineLENTIL0.0179 ounces per per square inches less0.1429 ounces per per square inches less0.1250 ounces per per square inches less0.0099 ounces per per square inches less
Answers
We have that the general formula for the density given the volume and the mass is:
[tex]d=\frac{m}{v}[/tex]in this case, the densities for the can of soup and the can of tuna are:
[tex]\begin{gathered} d_{soup}=\frac{10}{80}=\frac{1}{8} \\ d_{tuna}=\frac{8}{56}=\frac{1}{7} \end{gathered}[/tex]the difference between these two densities is:
[tex]\frac{1}{7}-\frac{1}{8}=\frac{1}{56}=0.0179[/tex]therefore, there is 0.0179 less density of the soup than the tuna
I need help with this problem, please help
Answers
Answer:
d.
Step-by-step explanation:
the slope is the factor of x.
a perpendicular slope turns the original slope upside-down and flips the sign.
the original slope is -3/7.
the perpendicular slope is then 7/3.
the only answer option with the correct slope is d.
so, d. must be correct.
let's check that (-2, 2) is on this line :
2 = 7/3 × -2 + 20/3 = -14/3 + 20/3 = 6/3 = 2
2 = 2
correct.
so yes, the point (-2, 2) is on this line, and d. is indeed correct.
Use Part Il of the Fundamental Theorem of Calculus to evaluate the definite integral
Answers
Answer:
[tex]4\ln (2)+\frac{49}{3}\approx19.1059[/tex]Given:
[tex]\int ^{-1}_{-2}\frac{7x^5-4x^2}{x^3}dx[/tex]Simplify:
[tex]\int \frac{7x^3-4}{x}dx[/tex]Expand:
[tex]\int (7x^2-\frac{4}{x})dx[/tex]Apply linearity:
[tex]7\int x^2dx-4\int \frac{1}{x}dx[/tex]Apply power rule and the standard integral ln(x)
[tex]7(\frac{x^3}{3})-4\ln (x)[/tex]Now, applying the Fundamental Theorem of Calculus Part 2
[tex]\int ^{-1}_{-2}\frac{7x^5-4x^2}{x^3}dx=(7(\frac{(-1)^3}{3})-4\ln (-1))-(7(\frac{(-2)^3}{3})-4\ln (-2))[/tex][tex]=4\ln (2)+\frac{49}{3}[/tex]Or approximately
[tex]\approx19.1059[/tex]f(x) = x2 + 1 g(x) = 5 – x
(f + g)(x) =
x to the power of 2 – x + 6
then (f – g)(x) =??
Answers
The function operation ( f - g )( x ) in the functions f(x) = x² + 1 and g(x) = 5 - x is x² + x - 4.
What is the function operation ( f - g )( x ) in the given functions?A function is simply a relationship that maps one input to one output. Each x-value can only have one y-value.
Given the data in the question;
f(x) = x² + 1g(x) = 5 - x( f - g )( x ) = ?To find ( f - g )( x ), replace the function designators in ( f - g ) with the actual functions.
( f - g )( x ) = f( x ) - g( x )
( f - g )( x ) = ( x² + 1 ) - ( 5 - x )
Remove the parenthesis using distributive property
( f - g )( x ) = ( x² + 1 ) - ( 5 - x )
( f - g )( x ) = x² + 1 - 5 + x
Collect and add like terms
( f - g )( x ) = x² + x + 1 - 5
( f - g )( x ) = x² + x - 4
Therefore, the function operation ( f - g )( x ) is x² + x - 4.
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The required function would be (f – g)(x) = x² + x - 4.
What is the function?A mathematical expression that defines the connection between two variables is considered a function.
The given functions following as
f(x) = x² + 1 and g(x) = 5 - x
We have to determine the function (f – g)(x).
(f – g)(x) = f(x) - g(x)
Substitute the values of functions f(x) = x² + 1 and g(x) = 5 - x in the function (f - g).
(f – g)(x) = (x² + 1) - (5 - x)
Open the parenthesis and apply the arithmetic operation,
(f – g)(x) = x² + 1 - 5 + x
Rearrange the terms likewise and combine them,
(f – g)(x) = x² + x + 1 - 5
Apply the subtraction operation to get
(f – g)(x) = x² + x - 4
Therefore, the required function would be (f – g)(x) = x² + x - 4.
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The maintenance department at the main campus of a large state university receives daily requests to replace fluorescent lightbulbs. The distribution of the number of daily requests is bell-shaped and has a mean of 42 and a standard deviation of 10. Using the 68-95-99.7 rule, what is the approximate percentage of lightbulb replacement requests numbering between 42 and 62.
Answers
we are given
mean=42
Std=10
if the mean=42 + std =10 42+10=52
if the mean=42 - std=10 42-10=32
Rule -- 68-95-99.7
68% of the measures are within 1 standard deviation of the mean.
42+10=52
95% are within 2.
42+20=62
99.7% are within 3.
42+30=72
The normal distribution is symmetric, which means that 50% of the measures are above the mean and 50% are below.
we are ask for the porcentage of request between 42-62 (between the mean and 2+std)
62 is two standard deviations above the mean.
Of the 50% of the measures below the mean, 95% are between 42 and 62, so
0.95(50)=47.5
The approximate percentage of light bulb replacement requests numbering between 42 and 62 is of 47.5%
diameter = 10.5in.we are learning something about area of circle.
Answers
Use the formula for the area of a circle:
[tex]A=\pi\cdot r^2[/tex]First, we need to find the radius r. Since the radius is half the diameter, then:
[tex]\begin{gathered} r=\frac{10.5\text{ in}}{2} \\ \therefore r=5.25in \end{gathered}[/tex]Substitute the value for r in the formula for the area of the circle:
[tex]\begin{gathered} A=\pi\cdot(5.25in)^2 \\ \approx86.6in^2 \end{gathered}[/tex]Therefore, the area of a circle of diameter 10.5 in is approximately 86.6 squared inches.
5) Solve the formula r/m = c for m.
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We have the following:
[tex]\frac{r}{m}=c[/tex]solving for m:
[tex]\begin{gathered} r=m\cdot c \\ m=\frac{r}{c} \end{gathered}[/tex]find the missing value in the given equivalent ratios 9:5=__:35
Answers
First, let's express the proportion as an equation, where the blank space is the variable x.
[tex]\frac{9}{5}=\frac{x}{35}[/tex]To find the value of x, we multiply both sides by 35.
[tex]\begin{gathered} \frac{9}{5}\cdot35=\frac{x}{35}\cdot35 \\ x=63 \end{gathered}[/tex]Therefore, the missing value is 63.3. Express the given integral as the limit of a Riemann sum but do not evaluate:
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Expression of the integral [tex]\int\limits^3_0 {(x^{3}-6x) } \, dx[/tex] as the limit of a Riemann sum without any evaluation will be
Lim(n → ∞) ∑(n = 1 → ∞) [{(27i³/n³) - (18i/n)} * (3i/n)]
As per the question statement, we are provided with an integral [tex]\int\limits^3_0 {(x^{3}-6x) } \, dx[/tex] ,
And we are required to determine the expression of the above mentioned integral as the limit of a Riemann sum without any evaluation.
To start with, we need to know the formula [Δx = {(b - a)/n}]
And here, from our given integral [tex]\int\limits^3_0 {(x^{3}-6x) } \, dx[/tex], we get that, (a = 0) and
(b = 3). Therefore substituting the values of "a" and "b" in the formula to calculate Δx, we get,
[Δx = {(3 - 0)/n} = (3/n)]
Also, [(x[tex]_{i}[/tex]) = {a + (Δx)i} = {0 + (3/n)i) = (3i/n)],
Given, [ρ(x) = (x³ - 6x)], and thus, [ρ(x[tex]_{i}[/tex]) = ρ(3i/n)]
Or, [ρ(x[tex]_{i}[/tex]) = {(3i/n)³ - 6(3i/n)}]
Or, [ρ(x[tex]_{i}[/tex]) = {(27i³/n³) - (18i/n)}]
Then, Lim(n → ∞) ∑(n = 1 → ∞) ρ(x[tex]_{i}[/tex])Δx
= Lim(n → ∞) ∑(n = 1 → ∞) [{(27i³/n³) - (18i/n)} * (3i/n)]
Reimann Sum: In Mathematics, a Riemann sum is a certain kind of approximation method for an integral by a finite sum. Named after renowned German mathematician Bernhard Riemann, one very common application of the Reimann Sum is in approximating the area of functions or lines on a graph, and also the length of curve.To learn more about Integrals and Reimann Sum, click on the link below.
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PLEASE HELP!!
Ninas math class is 6 and 4/5 meters long and 1 and 3/8 meters wide. What is the area of the classroom?
Answers
Answer: 374/40 or 9.35 or 9 and 7/20
Step-by-step explanation: see photo for explanation
A pottery factory purchases a continuous belt conveyor kiln for $68,000. A 6.3% APR loan with monthly payments is taken out to purchase the kiln. If the monthly payments are $765.22, over what term is this loan being paid?
Answers
Based om the cost of the continuous belt conveyor kiln and the monthly payments, as well as the APR of the loan, the term this loan will be paid is 120 months or 10 years.
How to find the term of the loan?When given the cost of a loan, the APR, and the monthly payments, you can find out the term of the loan by using the NPER function on a spreadsheet.
The Rate would be:
= 6.3% / 12 months in a year
= 0.525%
The Pmt is the payment of $765.22. This amount should be in negatives.
The Present Value or Pv should be the loan amount of $68,000.
The term on the loan would then be 120 months which is 10 years.
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A sample of size 39 will be drawn from a population with mean 47 and standard deviation 11. Use the TI-84 Plus calculator.
(a) Is it appropriate to use the normal distribution to find probabilities for x?
(b) Find the probability that x will be between 48 and 49. Round the answer to at least four decimal places.
(c) Find the 41st percentile of x. Round the answer to at least two decimal places.
O It is appropriate to use the normal distribution to find probabilities for x.
The probability that x will be between 48 and 49 is
The 41st percentile of x is
O It is not appropriate to use the normal distribution to find probabilities for x.
Answers
Utilizing a calculator First, STAT > EDIT, then enter all the x values in L1 and the probability for each x value in L2 to determine the mean and standard deviation of a probability distribution.
When the mean and standard deviation are known, how can you get the probability?If the data are normally distributed, you may calculate the likelihood of a specific event by knowing the mean and standard deviation. If you have them, you can use the formula z = (x - (mean)) / to determine the z-score (standard deviation).Utilizing a calculator First, STAT > EDIT, then enter all the x values in L1 and the probability for each x value in L2 to determine the mean and standard deviation of a probability distribution. In the second order, STAT > CALC > 1-Var Stats > 1-Var Stats L1, L2 ENTER.A population with a mean of 47 and a standard deviation of 11 will yield a sample size of 39. Calculate with the TI-84 Plus. A) Is the usage of the negative,Therefore,
An answer to the question is:
A) Yes
B) n = 39 = / n = 11 / 39 = 1.7614 P(48 49) = P[(48-4...
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. In a 30°-60-90° triangle, the hypotenuse is 7 yards long.Find the exact lengths of the legs?
Answers
ANSWER
The lengths of the legs of the triangle are 6.06 yards and 3.6 yards.
EXPLANATION
First, let us make a sketch of the problem:
To find the length of the legs, we have to apply trigonometric ratios SOHCAHTOA.
We have that:
[tex]\sin (60)=\frac{\text{opposite}}{\text{hypotenuse}}[/tex]From the diagram:
[tex]\begin{gathered} \sin (60)=\frac{x}{7} \\ \Rightarrow x=7\cdot\sin (60) \\ x\approx6.06\text{ yds} \end{gathered}[/tex]We also have that:
[tex]\sin (30)=\frac{\text{opposite}}{\text{hypotenuse}}[/tex]From the diagram:
[tex]\begin{gathered} \sin (30)=\frac{y}{7} \\ \Rightarrow y=7\cdot\sin (30) \\ y=3.5\text{ yds} \end{gathered}[/tex]The lengths of the legs of the triangle are 6.06 yards and 3.5 yards.
Hello I'd like some help on my practice question I'd prefer if it's quick because I have other questions I need to solve thank you
Answers
f(x) = -2
the answer is the second option
The horizontal line at y = -2 which is parallel to x-axis
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Find (w ∘ w)(−1) for w(x)=3x^2+3x−3.
Answer: (w ∘ w)(−1)=
Answers
Answer:
15
Step-by-step explanation:
wow(-1) means w(w(-1))
so we can find out what w(-1) is
3(-1)^2+3(-1)-3=3-3-3
which is -3
then we can find w(-3)
3(-3)^2+3(-3)-3
which is 15
Please help me with this problem just wanted to be sure that I am correct in order to help my son to under stand the break down of this problem. I believe that the answer is -3 but I am not sure please help?Solve for x.14x−1/2(4x+6)=3(x−4)−18 Enter your answer in the box.x =
Answers
SOLUTION
We want to solve for x in the equation
[tex]14x-\frac{1}{2}\mleft(4x+6\mright)=3\mleft(x-4\mright)-18[/tex]First we expand the brackets in both sides of the equation, this becomes
[tex]\begin{gathered} 14x-\frac{1}{2}(4x+6)=3(x-4)-18 \\ 14x-2x-3=3x-12-18 \end{gathered}[/tex]Note that the minus sign multiplies the items in the brackets too
Now, we collect like terms we have
[tex]\begin{gathered} 14x-2x-3x=-12-18+3 \\ 9x=-27 \\ \text{divide both sides by 9, we have } \\ \frac{9x}{9}=\frac{-27}{9} \\ x=-3 \end{gathered}[/tex]Hence x = -3
Mike made $120 last week working d days. Express the amount he made each day in terms of d.
Answers
Since he made $120 in d days
To find his earn in eac
Fiona was playing a game in which she rolled two number cubes. Cube #1 had the integers 1, 2, 3, 4, 5 and 6 on its faces. Cube # 2 has the integers –1, – 2, – 3, –4, –5 and –6 on its faces. She rolled Cube # 1 and got a 2. After she rolled Cube # 2 the sum of the value on the cubes was 0. What number did she roll on Cube # 2?
Answers
When Fiona rolled Cube #2, she got a number -2 as she got 2 while rolling Cube #1 and the sum of the value on the cubes was 0.
As we know Cube #1 has integers 1, 2, 3, 4, 5, 6 on its faces and Cube #2 has -1, -2, -3, -4, -5, -6 on its faces.
It is also given that Fiona got 2 when she rolled cube #1 and the value of the sum on both the cubes is 0.
So it is clear that on Cube #2, the number must be -2 as in that case only, the sum of the values will be 0.
To prove
2 + (-1) = 1
2 + (-2) = 0
2 + (-3) = -1
Hence, the number she got on Cube #2 is -2.
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I need help with my pre-calculus homework, please show me how to solve them step by step if possible. The image of the problem is attached. The homework was a pdf so the choices can't expand, my teacher told me to just write the transformation of the given function in the format.
Answers
Answer:
• Amplitude: 3
,• Period: π/2
,• Phase Shift: 1/8 (to the left)
,• Vertical Shift: -0.5
Explanation:
Given the trigonometric function:
[tex]y=3\sin (4x+\frac{1}{2})-0.5[/tex]Comparing with the form below:
[tex]\begin{gathered} y=A\sin (Bx-C)+D\text{ where:} \\ \text{Amplitude}=A \\ Period=\frac{2\pi}{B} \\ Phase\text{ Shift}=\frac{C}{B} \\ \text{Vertical Shift}=D \end{gathered}[/tex]Thus, we have that:
[tex]\begin{gathered} \text{Amplitude,}A=3 \\ Period,\frac{2\pi}{B}=\frac{2\pi}{4}=\frac{\pi}{2} \\ Phase\text{ Shift,}\frac{C}{B}=-\frac{1}{2}\div4=-\frac{1}{8} \\ \text{Vertical Shift, }D=-0.5 \end{gathered}[/tex]The phase shift, -1/8 indicates a shift to the left.
The table shows claims and their
probabilities for an insurance
company.
Amount of claim
(to the nearest $20,000)
$0
$20,000
$40,000
$60,000
$80,000
$100,000
Probability
0.70
0.16
0.09
0.03
0.01
0.01
Answers
Answer:
Step-by-step explanation:
This is an equation! Solutions: x=1.
Graphical form: Equation 3%2Ax-x%2B2=4 was fully solved.
Text form: 3*x-x+2=4 simplifies to 0=0
Cartoon (animation) form: simplify_cartoon%28+3%2Ax-x%2B2=4+%29
For tutors: simplify_cartoon( 3*x-x+2=4 )
If you have a website, here's a link to this solution.
What is the surface area of fish tank in the shape of a cube that has a volume of 90 cubic inches.
Answers
You know that the volume of the fish tank in the shape of a cube:
[tex]V=90in^3[/tex]By definition, the formula for calculating the volume of a cube is:
[tex]V=a^3[/tex]Where "a" is the length of each edge of the cube.
If you solve for "a", you get this formula:
[tex]a=\sqrt[3]{V}[/tex]In this case, knowing the volume of the cube, you can substitute it into the second formula and evaluate, in order to find the length of each edge of the cube:
[tex]\begin{gathered} a=\sqrt[3]{90in^3} \\ \\ a\approx4.48in \end{gathered}[/tex]The surface area of a cube can be found using this formula:
[tex]SA=6a^2[/tex]Where "a" is the length of each edge of the cube.
Substituting the value of "a" into the formula and evaluating, you get:
[tex]SA=6(4.48in)^2\approx120in^2[/tex]Hence, the answer is: Second option.
translate the following verbal statement into an algebraic equation and solve. Paid 24,998 for a car which was 1,815 less than sticker price what was the sticker price of the caruse x for your vairableequation_______x=______
Answers
paid price = 24,998
it is the amount that is 1815 less than the sticker price,
so the sticker price or price of the car is x
so x = 24,998 + 1815
x =26,813
so the price of car is x = 26,813.
Determine if the two triangles shown are similar. If so, write the similarity statement.Question options:A) Impossible to determine.B) ΔBCG ∼ ΔEFGC) ΔGCB ∼ ΔGFED) The triangles are not similar.
Answers
ANSWER
Option D: The triangles are not similar
STEP BY STEP EXPLANATION
Now, two (2) triangles are said to be similar if the three (3) angles of triangle A are congruent or equal to the corresponding three (3) angles of triangle B.
If you take a close look at the two (2) triangles, you will notice that the only angle in ∆BCG that is equal to the corresponding angles in ∆EFG is ∆BGC; the two (2) remaining angles in ∆BCG are not congruent with the two (2) corresponding angles in ∆EFG
Hence, it can be concluded that both triangles are not similar.
1)Find the probability of randomly selecting the correct access code on the first try 4 digits (0 through 9)2)find the probability of NOT selecting the correct access code on the first try
Answers
There are 10 digits from 0 to 9.
First digit 10 ways
Second digit 10 ways
Third digit 10 ways
Fourth digit 10 ways
[tex]\text{There are 10}\times10\times10\times10\text{ ways for four digits.}[/tex][tex]\text{There are 10}000\text{ ways for four digits.}[/tex]Hence the total outcomes =10000
Selecting the correct access code on the first try given favorable outcomes =1.
[tex]\text{The probability of randomly selecting the correct access code on the first try=}\frac{favorable\text{ outcome}}{\text{Total outcomes}}[/tex][tex]\text{=}\frac{1}{10000}[/tex][tex]=0.0001[/tex]Hence the probability of randomly selecting the correct access code on the first try is 0.0001.
The probability of not selecting the correct access code on the first try=1-The probability of selecting the correct access code on the first try
The probability of not selecting the correct access code on the first try=1-0.0001
Hence the probability of not selecting the correct access code on the first try=0.9999.
Suppose that $2000 is invested at a rate of 3.9%, compounded monthly. Assuming that no withdrawals are made, find the total amount after six years.Round your answer to the nearest cent.
Answers
Compound interest formula:
[tex]\begin{gathered} A=P(1+\frac{r}{n})^{nt} \\ \\ A\colon\text{Amount} \\ P\colon\text{ Principal} \\ r\colon\text{ interest rate (in decimals)} \\ n\colon\text{ number of times interest is compounded in a year} \\ t\colon\text{ time (in years} \end{gathered}[/tex]Given data:
P= $2,000
r= 3,9% =0.039
n=monthly= 12
t= 6 years
[tex]\begin{gathered} A=2000(1+\frac{0.039}{12})^{12(6)} \\ \\ A=2000(1.00325)^{72} \\ \\ A\approx2526.33 \end{gathered}[/tex]Then, the total amount after six years is $
Solve the following system of equations by graphing. Graph the system below and enter the solution set as an ordered pair in the form (x,y).if there are no solutions, enter none and enter all if there are infinite solutions.X - y = 0X + y = - 4
Answers
EXPLANATION
Since we have the system of equations:
(1) x - y = 0
(2) x + y = -4
Isolating x in (1):
x = y
Plugging in x=y into (2):
y + y = -4
Adding like terms:
2y = -4
Dividing both sides by 2:
y = -4/2
Simplifying:
y = -2
Plugging in y=-2 into (1):
x - (-2) = 0
Removing the parentheses:
x + 2 = 0
Subtracting -2 to both sides:
x = -2
The solution of the system of equations is (-2, -2)
Representing the graph:
Find The Circumference of the following circle in terms of Pi. A. 25piB. 50piC 12 5pi
Answers
Solution
Given the circle with 25 yard diameter
Circumference of a circle = 2πr
[tex]\begin{gathered} diameter\text{ =25yrd} \\ d=2r \end{gathered}[/tex]Circumference = πd
[tex]C=25\pi[/tex]Therefore the correct answer = 25pi
Hence the correct answer is Option A
The start of a quadratic
sequence is
8, 18, 30, 44, 60, …
What is the nth term rule for this sequence?
Answers
Answer:
The correct option is D
98
The general term of the sequence is n(n+7)
8 = 1x8, 18 = 2x9, 30 = 3x10, 44 = 4x11, 60 = 5x12 … so in general T(n) = n(n + 7)
Or because the second differences are constant we know it uses squares
8 = 1^2 + 7, 18 = 2^2 + 14, 30 = 3^2 + 21, 44 = 4^2 + 28, 60 = 5^2 + 35 … so in general T(n) = n^2 + 7n
26/w = 5/6 what is w?
Answers
Here, the expression is 26/w=5/6.
Therefore,
[tex]\begin{gathered} \frac{26}{w}=\frac{5}{6} \\ w=\frac{26\times6}{5} \\ w=\frac{156}{5} \\ w=31.2 \end{gathered}[/tex]So, the value of w is 31.2
Solve each of the following equations. Show its set on a number line. |4x-4(x+1)|=4
Answers
Solving this equation, we have:
[tex]\begin{gathered} |4x-4\mleft(x+1\mright)|=4 \\ |4x-4x-4|=4 \\ |-4|=4 \\ 4=4 \end{gathered}[/tex]Since the final sentence is always true, the solution set is all real numbers.
Showing it in the number line in blue, we have: