The algebraic property used in steps 2 and 3 are respectively Addition Property of Equality and Division Property of Equality.
What is the algebraic property used?
We are given the inequality;
3x – 2 > –4
Step 2;
Using Addition Property of Equality, add 2 to both sides;
3x – 2 + 2 > –4 + 2
3x > –2
Step 3;
Using Division Property of Equality, divide both sides by 3 to get;
3x/3 > –2/3
x > –2/3
Thus, we conclude that the algebraic property used in steps 2 and 3 are respectively Addition Property of Equality and Division Property of Equality.
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[tex] \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: [/tex]
[tex]\omega[/tex] is a 2020th root of unity, so [tex]\omega^{2020}=1[/tex] and we have a kind of reflection identity of [tex]\omega^{2020-n} = \omega^{-n}[/tex] for [tex]n\in\Bbb Z[/tex].
Let's evaluate the product first. Denote it by [tex]P_k[/tex]. We split the product where the [tex]j=k[/tex] factor would belong, and pull out powers of [tex]\omega^k[/tex].
[tex]\displaystyle P_k = \prod_{j=1, j\neq k} (\omega^k - \omega^j) \\\\ ~~~~ = \prod_{a=1}^{k-1} (\omega^k - \omega^a) \prod_{b=k+1}^{2019} (\omega^k - \omega^b) \\\\ ~~~~ = (\omega^k)^{k-1} \prod_{a=1}^{k-1} (1-\omega^{a-k}) \cdot (\omega^k)^{2019-k} \prod_{b=k+1}^{2019} (1 - \omega^{b-k}) \\\\ ~~~~ = (\omega^k)^{2018} \prod_{a=1}^{k-1} (1 - \omega^{-a}) \prod_{b=1}^{2019-k} (1 - \omega^b) \\\\ ~~~~ = \omega^{-2k} \prod_{a=1}^{k-1} (1-\omega^{-a}) \prod_{b=1}^{2019-k} (1-\omega^b)[/tex]
Now introduce some factors to "complete" the [tex]b[/tex]-product and have it contain 2019 factors.
[tex]\displaystyle P_k = \omega^{-2k} \prod_{a=1}^{k-1} (1-\omega^{-a}) \frac{\displaystyle \prod_{b=1}^{2019} (1 - \omega^b)}{\displaystyle \prod_{b=2020-k}^{2019} (1 - \omega^b)}[/tex]
It's relatively straightforward to show that if [tex]\zeta[/tex] is an [tex]n[/tex]-th root of unity, then
[tex]\displaystyle \sum_{m=1}^{n-1} (1-\zeta^m) = n[/tex]
which gives
[tex]\displaystyle P_k = 2020 \omega^{-2k} \frac{\displaystyle \prod_{a=1}^{k-1} (1-\omega^{-a})}{\displaystyle \prod_{b=2020-k}^{2019} (1 - \omega^b)}[/tex]
Shifting the index in the denominator and again using the reflection property eliminates all but one factor.
[tex]\displaystyle P_k = 2020 \omega^{-2k} \frac{\displaystyle \prod_{a=1}^{k-1} (1-\omega^{-a})}{\displaystyle \prod_{b=1}^k (1 - \omega^{-b})} \\\\ ~~~~ = \frac{2020 \omega^{-2k}}{1 - \omega^{-k}}[/tex]
Now evaluate the sum. We can exploit symmetry. Split the sum at the 1010th term, so that
[tex]\displaystyle - \sum_{k=1}^{2019} P_k = -2020 \left(\sum_{k=1}^{1009} \frac{\omega^{-2k}}{1 - \omega^{-k}} + \frac{\omega^{-2020}}{1-\omega^{-1010}} + \sum_{k=1011}^{2019} \frac{\omega^{-2k}}{1-\omega^{-k}}\right)[/tex]
The middle terms reduces to 1/2. Shifting the index in the second sum, we can condense it to
[tex]\displaystyle -\sum_{k=1}^{2019} P_k = -2020 \left(\frac12 + \sum_{k=1}^{1009} \left(\frac{\omega^{-2k}}{1-\omega^{-k}} + \frac{\omega^k}{1-\omega^k}\right)\right)[/tex]
Join the fractions.
[tex]\displaystyle \frac{\omega^{-2k}}{1-\omega^{-k}} + \frac{\omega^k}{1-\omega^k} = 1 - \frac{2-\omega^{2k}-\omega^{-2k}}{2-\omega^k-\omega^{-k}} = -(1+\omega^k + \omega^{-k})[/tex]
The remaining sums are trivial.
[tex]-\displaystyle \sum_{k=1}^{2019} P_k = -2020 \left(\frac12 - 1009 - \frac{1-\omega^{1010}}{1-\omega} - \frac{1-(-\omega)^{1010}}{1+\omega}\right) \\\\ ~~~~ = 2020\cdot1009 - 1010 \\\\ ~~~~ = (2000+20)(1000+9) - 1000 - 10 \\\\ ~~~~ = 2\cdot1000^2 + 37\cdot1000 + 170[/tex]
Taking this last result mod 1000, we find the last 3 digits to be 170.
In which interval is the radical function f of x is equal to the square root of the quantity x squared minus 2 times x minus 24 end quantity increasing? HELP!!! PLEASE!!! ASAP!!!
Answer:
Step-by-step explanation: C
Just need help on one answer I’ll give Brainly!
Answer:
Q1 = 8
Hope this helps with your question :)
The line at the start of the "box" is Q1, the middle line in the "box" is the median also called Q2, and the last line of the "box" is Q3.
Select the correct answer.
Function bis nonlinear, and b(9) = 4. Which equation could represent function b?
The nonlinear- function which represents b(9)=4 is given by b(x)=72/x - 4 .
In mathematics and physics, a nonlinear system is one in which the change in the output is not proportional to the change in the input. Engineers, biologists, physicists, mathematicians, and many other scientists are interested in nonlinear problems since the majority of systems are inherently nonlinear.
Nonlinear function graphs don't resemble lines at all. It contains the formula f(x) = ax + b. Its equation can take any form, with the exception of f(x) = ax + b. The slope of the curve is the same between any two points. The point pairs on the graph do not all have equal slopes.
Of all the given options: the only equations which imply that b(9)=4 are :
i) b(x)=72/x - 4 and (iv) b(x)=4
Of this two options only the first option is non-linear.
Hence the required option is b(x)=72/x - 4 .
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Disclaimer:
The missing options are:
i) b(x) = 72/x - 4
ii) b(x) = √x + 7
iii)b(x) = 2x+1
iv) b(x) = 4
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Question 4
Use the given sets to find An B.
Answer
AnB =
Correct
A = {1, 2, 3, 4, 5, 6, 7, 8, 9}
B = {2, 4, 6, 8, 10, 12}
Ta
A n B = {2,4,6,8}
The intersection of Sets :
The intersection of sets for two given sets is the set that contains all the elements that are common to both sets. The symbol for the intersection of sets is "∩''. For any two sets A and B, the intersection, A ∩ B (read as A intersection B) lists all the elements that are present in both sets, and are the common elements of A and B.
Intersections of sets have properties similar to the properties of numbers. The properties of the intersection of sets include the commutative law, associative law, the law of null set and universal set, and the idempotent law. The following table lists the properties of the intersection of sets.
Name of Property/Law Rule
Commutative Law A ∩ B = B ∩ A
Associative Law (A ∩ B) ∩ C = A ∩ (B ∩ C)
Law of ϕ and U ϕ ∩ A = ϕ , U ∩ A= A
Idempotent Law (A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
(A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
Important Notes:
(A ∩ B) is the set of all the elements that are common to both sets A and B.
If A ∩ B = ϕ, then A and B are called disjoint sets.
n(A ∩ B) = n(A) + n(B) - n(A ∪ B)
Given :
A = {1, 2, 3, 4, 5, 6, 7, 8, 9}
B = {2, 4, 6, 8, 10, 12}
Find :
A n B = {2,4,6,8}
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The last factor of 30 in the first multiple of 30 is the same number what is it
Answer:30
Step-by-step explanation:because 30 is a multiple and factor of 30.
2. 50 vehicles belong to an enterprise that has three departments, A, B and C. 6 vehicles are available for the use of all three departments, while 2 vehicles give service to neither of the departments. 24 vehicles are administrated by department B, although 16 of them are also used by the others departments. There are twice as many vehicles in department A as there are in department C. If equal numbers of vehicles are allocated when used by any two departments, find the number of vehicles administrated by each department.
The number of vehicle administered by department A only = 18
The number of vehicle administered by department B only = 8
The number of vehicle administered by department C only = 1
Given data
50 vehicles belong to an enterprise that has three departments, A, B and C
6 vehicles are available for the use of all three departments
2 vehicles give service to neither of the departments
24 vehicles are administrated by department B
16 of the 24 used by B are also used by the others departments
There are twice as many vehicles in department A as there are in department C.
equal numbers of vehicles are allocated when used by any two departments
How to find the number of vehicles administrated by each departmentnumber of vehicles on department A = 2 * number of vehicles in department C.
say A = 2C
B = 24
let x represent the number of vehicles used by any two departments
16 of the 24 used by B are also used by the others departments and equal numbers of vehicles are allocated when used by any two departments hence
16 = 2x + 6
2x = 16 - 6
2x = 10
x = 5
B only = 24 - 2x - 6 = 18 - 2x = 18 - 10 = 8
A only = A - 2x - 6 = 2C - 10 - 6 = 2C - 16
C only = C - 2x - 6 = C - 10 - 6 = C - 16
adding up all the vehicles as allocated to get to the sum of 50
24 - 2x - 6 + A - 2x - 6 + C - 2x - 6 + x + x + x + 6 + 2 = 50
24 - 6 - 6 - 6 + 6 + 2 - 2x - 2x - 2x + x + x + x + A + C = 50
24 - 10 - 6x + 3x + 2C + C = 50
14 - 3x + 3C = 50
3C - 3X = 50 - 14
3C - 3(5) = 36
3C - 15 = 36
3C = 36 + 15 = 51
C = 17
C only
C only = C - 16
C only = 17 - 16
C only = 1
A only
A only = 2C - 16
A only = 2 * 17 - 16 = 34 - 16 = 18
The number of vehicle administered by department A only = 18
The number of vehicle administered by department B only = 8
The number of vehicle administered by department C only = 1
The number of vehicle administered by department A = 2C = 34
The number of vehicle administered by department B = 24
The number of vehicle administered by department C = 17
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Find the values of x, y, and z. The diagram is not to scale.
(Click document to see image.)
The values of x, y, and z. The diagram is not to scale.
x=67z =113y=55This is further explained below.
What are the values of x, y, and z. The diagram is not to scale.?Generally, Angle in a triangle is 180^{\circ}
Therefore
45+68+x=180
x=180-(45+68)
x=67
The angle on a line is also 180^{\circ}
x+z =180
67+z =180
z =180-67
z =113
In conclusion
180=113+12+y
-180-(113+12)=y
y=55
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Simplify the expression where possible. 1/(x^4)^-2
According to the solving the simplified value of the equation 1/(x^4)^-2 is
= x⁸.
What are the simplification rules?PEMDAS stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction and is the main rule for simplifying expressions.
Simplification seems to be the process of substituting a mathematical expression with a simpler, usually shorter counterpart.
What is the simplification method?Designers employ simplification to make designs more memorable and instantly recognizable in logos, emblems, posters, computer icons, as well as other design applications.
According to the given data:Given :[tex]\frac{1}{\left(x^4\right)^{-2}}[/tex]
On simplifying we get:
Use (aⁿ)ˣ = aⁿˣ
so we get:
[tex]\frac{1}{X^{(4)(-2)}}[/tex]
=[tex]\frac{1}{x^{-8}}[/tex]
Use a⁻ⁿ = 1/aⁿ
Now:
= [tex]\frac{1}{\frac{1}{x^8}}\\=x^8[/tex]
According to the solving the simplified value of the equation 1/(x^4)^-2 is
= x⁸
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The sum of two numbers is 15. Using x to represent the smaller of the two numbers, translate "the sum of twice the smaller number and two more than the larger num ber" into a variable expression. Then simplify.
Answer:
Step-by-step explanation: Let x be the smaller and y be the largest number.
Since x+y=13, we deduce y=13-x
Now, for the translation: "two more than the larger number" is y+2 , while "twice the smaller" is 2x
Their sum is y+2+2x
And since we know that y=13-x, we have y+2+2x=13-x+2+2=15+x
Write 4,000,000 in scientific notation.
Answer:
4 × 10^6
Because 10^6 = 100,000
Answer:
4×10^6
Step-by-step explanation:
you write a non zero digit which is 4 placing your decimal after the first non zero digit ..then you count the no. of digits you need to move the beginning decimal to get to where your decimal is ...or to use a scientific calculator to make it simple
-9y > 9. Solve for the inequality
Answer:
-1
Step-by-step explanation:
make it equal to zero
-9y - 9 = 0
add 9 to both sides
-9y = 9
divide both sides by -9 to get y alone
y = 9/-9
simplify
y = -1
Community Gym charges a $60 membership fee and a $65 monthly fee. Workout Gym charges a $190 membership fee and a $55 monthly fee. After how many months will the total amount of money paid to both gyms be the same? What will the amount
Answer:
13 months$905Step-by-step explanation:
You want the cost and the number of months of membership such that the cost of both gyms is the same.
Total costThe total cost of x months of membership at each gym will be the sum of its initial fee and the product of the monthly fee and the number of months.
SetupThe cost at Community Gym is ...
c = 60 + 65x
The cost at Workout Gym is ...
w = 190 +55x
These costs will be the same when ...
c = w
60 +65x = 190 +55x
SolutionSubtracting 60+55x from both sides of the equation leaves ...
10x = 130
x = 13 . . . . . . divide by 10
The cost will be the same after 13 months.
That cost will be ...
60 +65(13) = 60 +845 = 905
The amount paid at each gym after 13 months will be $905.
The cost of gas in England is 96.4 pound sterling/liter. How much is this in U.S. dollars/gallon? (3.875 litres = 1 gallon and 1.47 US$= 1 pound sterling)
The cost of gas in England is 96.4 pound sterling/liter is 549.1185 U.S. dollars/gallon
The cost of gas in England is 96.4 pound sterling/liter.
3.875 liters = 1 gallon
1 liter = 1/3.875 gallon
1.47 US$= 1 pound sterling
pound sterling / liter = 1.47/(1/3.875)) U.S. dollars/gallon
= (1.47)(3.875) U.S. dollars/gallon
96.4 pound sterling / liter = (96.4)(1.47)(3.875) U.S. dollars/gallon
= 549.1185 U.S. dollars/gallon
Sterling is the arena's oldest forex that is still in use and that has been in continuous use due to the fact that its inception.
it is presently the fourth maximum-traded currency in the forex market, after the USA dollar, the euro, and the Javanese yen.
collectively with the ones 3 currencies and Renminbi, it paperwork the basket of currencies which calculate the price of IMF unique drawing rights. As of mid-2021, sterling is likewise the fourth most-held reserve forex in international reserves.
these kinds of currencies are government-issued fiat currencies.
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The length of a rectangle is four less than twice its width. If the perimeter of the rectangle is 76 units, what is its width?
a
13
b
14
c
20
d
26
e
28
In triangle ABC, the size of angle B is 3 times the size of
angle A, and the size of angle C is 16° less than 6 times
the size of angle A.
Find the size of the angles.
Step-by-step explanation:
Here,
angle B = 3 x angle A
angle C = 6 x angle A - 16
now, for a triangle,
angle A + angle B + angle C = 180
or, angle A + 3 x angle A + 6 x angle A - 16 = 180
or, 10 x angle A = 196
so, angle A = 19.6 degrees
angle B = 3 x 19.6 = 58.8 degrees
angle C = 6 x 19.6 - 16 = 101.6 degrees
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Janine is analyzing spending habits of students in her high school. She finds a mean monthly discretionary expense of $70 spent on having meals
out with friends. The standard deviation of a distribution is 5. What is the variance?
Answer:
Variance = 25
Step-by-step explanation:
I'm smart
There are 5 people from Arizona (A), 7 people from California (C), and 3 people from Nevada (N)
equally qualified for a job. There are three job openings for three different people. What is the probability that not all three people are from Arizona
Answer
.9780
Step-by-step explanation:
Probability, not all 3 people are from Arizona
Arizona Equals 1- 2/91 = 89/91 = .9780
Probability is .9780
when you get on the bus to go to school there are x people on the bus. at the next stop 6 people get off the bus. write an expression to represent the remaining number of people on the bus after the first stop.
Answer:
x-6
Step-by-step explanation:
1. Which value is equal to -8
Answer:
Are there options you forgot to put on the question?
Step-by-step explanation:
please help soon!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Answer:
its 63 you got it right
Step-by-step explanation:
Devan borrowed $27. They then earned $33. After paying back what they owed, how much money does Devan have?(1 point)
Answer: Devon would have -6 dollars left
Step-by-step explanation:
Find the measure of RUS.
Answer:
Step-by-step explanation:
the answer is 23.
Answer:
C
Step-by-step explanation:
solve y-3x=13 for y
Answer:
y=3x+13
Step-by-step explanation:
Simply add the 3x to the other side.
Answer: y = 13 + 3x
Step-by-step explanation:
you just add 3x to both sides of the equation !! sorry if it doesn’t help but you basically put it the other way around and put 3x on both sides
Which of the following equations is equivalent to
2x + y = 10
Ay = -2x - 10
By = 2x + 10
y=-2x + 10
y = -10x - 2
Answer:
y = -2x + 10
Step-by-step explanation:
Solve for the value of y.
Based on the given parameters, the value of y in the figure is 13
How to determine the value of y?The measures of the angles are given as:
Angle 1 = 8y
Angle 2 = 6y - 2
The angles are supplementary angles
So. we have
8y + 6y - 2 = 180
Evaluate the like terms
14y =182
Divide both sides by 14
y = 13
Hence, the value of y in the figure is 13
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Cary is 9 years older then Dan. In 7 years the sum of their ages will be 93. Find the age of each man now
Cary's and Dan's present ages are 44 and 35 years respectively.
Let the Dan's present age be x years.
According to the given question.
Cary is 9 years older then Dan.
And, after seven years the sum of their ages will be 93.
So, from the given conditon we can say that
cary's age = 9 + x
And after seven years, sum of the ages of Cary and Dan
9 + x + 7 + x + 7 = 93
⇒ 23 + 2x = 93
⇒ 2x = 93 - 23
⇒ 2x = 70
⇒ x = 70/2
⇒ x = 35
⇒ Dan's present age is 35 years.
Therefore, Cary's age = 35 + 9 = 44 years.
Hence, Cary's and Dan's present ages are 44 and 35 years respectively.
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A train traveled at 100 mph for 2 hours,then wen an additional 2 hours at 85mph. Which expression gives the number of miles the train traveled?
An addition equation to find the volume of this
An addition equation for calculating volume is = 12 units cube.
What is addition equation?The fundamental addition formula, sometimes known as the mathematical equation of addition, is as follows. The addends are 5 and 3, and the sum is 8. It should be noted that an addition fact can have many addends. For instance, 5 + 7 + 9 + 3 = 24.
What is a shape's volume?Volume refers to the amount of space occupied by a 3D shape. The volume of a shape can be calculated by multiplying its height, breadth, and depth. If somehow the shape is made using cubic cm blocks, the volume can be calculated by counting the cubes.
According to the given Data:Total no. of cubes = 12
length of the one cube = 1 unit
So,
volume of the given prism = l x b x h
volume of one prism = 1 unit^3
There are 12 cube are present on adding all the units we get
volume of total cubes are = 12 unit cube.
So,
The volume is = 12 unit cube.
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