Given:
Expression is
[tex]3r^2-2r+4[/tex]
To prove:
If r is any rational number, then [tex]3r^2-2r+4[/tex] is rational.
Step-by-step explanation:
Property 1: Every integer is a rational number. It is Theorem 4.3.1.
Property 2: The sum of any two rational numbers is rational. It is Theorem 4.3.2.
Property 3: The product of any two rational numbers is rational. It is Exercise 15 in Section 4.3.
Let r be any rational number.
We have,
[tex]3r^2-2r+4[/tex]
It can be written as
[tex]3(r\times r)-2r+4[/tex]
Now,
3, -2 and 4 are rational numbers by property 1.
[tex]r^2=r\times r[/tex] is rational by Property 3.
[tex]3r^2\text{ and }-2r[/tex] are rational by Property 3.
[tex]3r^2+(-2r)+4[/tex] is rational by property 2.
So, [tex]3r^2-2r+4[/tex] is rational.
Hence proved.
We want to prove that if r is rational, then:
[tex]3*r^2 - 2r + 4[/tex] Is also rational.
The proof is below:
Remember that a rational number is any number that can be written as a quotient of two integer numbers.
Then we have:
[tex]r = \frac{a}{b}[/tex]
Where a and b are integers.
If we replace that in our equation, we get:
[tex]3*(\frac{a}{b}) ^2 - 2*(\frac{a}{b}) + 4\\\\\frac{3a^2}{b^2} - \frac{2a}{b} + \frac{4}{1}[/tex]
Note that if a is an integer, then 3*a^2 is also an integer.
If b is an integer, then b^2 is also an integer.
Now we can perform the sum of these fractions to get:
[tex]\frac{3a^2}{b^2} - \frac{2a}{b} + \frac{4}{1}\\\\\frac{3a^2 - 2ab + 4b^2}{b^2}[/tex]
And because integers are closed under the addition, then the numerator of that fraction is also an integer, then we have that the given expression can be written as a quotient of two integers, thus, it is a rational number.
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Amy earns $12.50 per hour at her work. She uses the equation shown to calculate the number of hours, she worked last week
12.5h = 312.5 how many hours did Amy work last week?
Answer:
25
I hope this helps.
Answer:
Amy worked 25 hours
Step-by-step explanation:
To first start off we have the equation 12.5h= 312.5
from here, we divide by 12.5 to have h by itself inorder to find how many hours she worked as h stands for hours.
Once you divide 312.5 by 12.5 you get h=25
Once again, the h stands for hours, so this means that Amy worked 25 hours, and that for one hour she earned 12.5 dollars.
If you want to find how much money she made for example in 5 hours, all you have to do is plug in the 5 for h and multiply it by 12.50 to get how much she earned in that certain amount of hours!
Hope this helps! If you have any questions, feel free to ask! :)
Simplify the algebraic expression by combining like (or similar) terms.
-7x - 8y- 5x - 6y
Answer:
-12x - 14y
Step-by-step explanation:
First I would move them side by side so I can see it better so:
-7x - 5x - 8y - 6y
now combine the x values together and the y values together:
-12x - 14y
quarter note + whole note + quarter note
Answer:
1 1/2 notes
Step-by-step explanation:
A rectangular garden is fenced on all sides with 278 feet of fencing. The garden is 9 feet longer than it is
wide. Find the length and width of the garden.
The formulas below are the cost and revenue functions for a company that manufactures and sells small radios.
C(x)= 64,000 + 45x and R(x) = 49x
a. Use the formulas shown to write the company's profit function, P, from producing and selling x radios.
b. Find the company's profit if 24,000 radios are produced and sold.
a. The company's profit function is P(x) =
(Simplify your answer.)
How do i do this problem
Answer:
no clue
Step-by-step explanation:
please help !!
12x + 14 = 6x + 7 (+) 9x - 5 find the value of x
Answer:
x=4
Step-by-step explanation:
A game store sold 206 games last week for $26 each the store owner had a sales goal of $5,000 for the week how much did the store owner actually earn
Answer:
5356$
Step-by-step explanation:
206*26=5356
Find the solution of y=x-3
Answer:
x=3
Step-by-step explanation:
0 = x -3
-x = -3
change the sign to positive so x= 3
What is 7.103 × 105 in standard notation?
Answer:
710300 is your answer if im not mistaken
Work out this problem: 2x + 3x -12x Show you work
Answer:
-7x
Step-by-step explanation:
2x + 3x - 12x
Add 2x + 3x, resulting in 5x
5x - 12x = -7x
It's negative because the -12 is the bigger number.
Hope this helps :D
Find −6 3/8−1/4. Write your answer as a mixed number in simplest for
Answer:
Usually, a mixed number is the simplest way to express an improper fraction – but sometimes, the fraction ... Don't express the answer as a decimal. Instead ... So, add the whole number back in to get a final result of 6 1/2. ... Write out the factors for the numerator of your fraction, then write out the factors for the denominator.
Step-by-step explanation:
Answer:
mixed number is a number consisting of a whole number and a proper fraction.
Below are examples of mixed numbers:
4\dfrac12, 1\dfrac38, 12\dfrac564
2
1
,1
8
3
,12
6
5
Step-by-step explanation:
make me as brain liest
In a group of students, 30 played chess, 19 played volleyball, 25 played
basketball, 14 played both volleyball and chess, 8 played both baskethall
and volleyball, 15 played both basketball and chess and 5 played both
three events. How many played chess only, basketball only and volleyball
only? How many students are there in all?
Answer:
Explained below.
Step-by-step explanation:
Denote the events as follows:
C = chess
V = volleyball
B = basketball
The data provided is as follows:
n (C) = 30
n (V) = 19
n (B) = 25
n (C ∩ V) = 14
n (B ∩ V) = 8
n (B ∩ C) = 15
n (C ∩ V ∩ B) = 5
Consider the Venn diagram below.
The number of students who played only chess is marked in pink:
n (Only C) = 6
The number of students who played only volleyball is marked in blue:
n (Only V) = 2
The number of students who played only basketball is marked in orange:
n (Only B) = 7
The number of students who played all three is marked in grey:
n (C ∩ V ∩ B) = 5
pls help plz pls plz
Answer: c
i’m pretty sure it’s c
Answer:
B
Step-by-step explanation:
15 rows with 7 cookies is just multiplying, 15*7, to get how many cookies that's in the tray. There's also 9 trays, so we multiply (15*7) by 9
I multiplied a number by itself and got 324. What was the
number?
Answer:
18
Step-by-step explanation:
Answer:
18
Step-by-step explanation:
18 times 18 equals 324 and the number 18 is being multiplied by itself
Select the correct answer from each drop-down menu. Identify the investment type and complete the description. _____offer corporations a loan. Investors tend to choose this investment type because it offers______ and _____
1.a bonds 1.b real estate investment 1.c saving account 1.c stocks
2.a risk 2.b safety
3.a dividends base on company profits 3.b income based on an interest rate
Answer: bonds/safety/income based on an interest rate
Step-by-step explanation:
:)
PLEASE HELP!!! ILL GIVE BRANLIEST *EXTRA POINTS* dont skip :((
Answer:
28
Step-by-step explanation:
For the area, since it is 49, each side should be equal, so the sides are 7. To get perimeter, you add all the sides together, or since all four side are equal, take 7 and multiply it by 4.
Community Gym charges a $50 membership fee and a $70 monthly fee. Workout Gym charges a $190
membership fee and a $60 monthly fee. After how many months will the total amount of money paid to
both gyms be the same? What will the amount be?
After
months the amount paid to both gyms will be $
Answer:
14 month
Step-by-step explanation:
Lets create equations for these two gyms. I believe this is an algebra 1 problem.
Community Gym: y=70x+50
Workout Gym: y=60x+190
Because both gyms are equal to y, we can set them together and solve for x.
70x+50=60x+190
10x=140
x=14
You can find the cost by plugging in 14 into both of the equations. You get a cost of $1030 after 14 months
Therefore, after 14 months you will have paid the same amount for both gyms.
Answer:
x = 14 months, $1030
Step-by-step explanation:
The expressions are,
→ 70x + 50
→ 60x + 190
Forming the equation,
→ 70x + 50 = 60x + 190
Now required value of x will be,
→ 70x + 50 = 60x + 190
→ 70x - 60x = 190 - 50
→ 10x = 140
→ x = 140/10
→ [ x = 14 months]
The amount of money will be,
→ 70(14) + 50 = 60(14) + 190
→ 980 + 50 = 840 + 190
→ $1030 = $1030
Hence, it will take 14 months.
Explain why each of the following integrals is improper. (a) 4 x x − 3 dx 3 Since the integral has an infinite interval of integration, it is a Type 1 improper integral. Since the integral has an infinite discontinuity, it is a Type 2 improper integral. The integral is a proper integral. (b) [infinity] 1 1 + x3 dx 0 Since the integral has an infinite interval of integration, it is a Type 1 improper integral. Since the integral has an infinite discontinuity, it is a Type 2 improper integral. The integral is a proper integral. (c) [infinity] x2e−x2 dx −[infinity] Since the integral has an infinite interval of integration, it is a Type 1 improper integral. Since the integral has an infinite discontinuity, it is a Type 2 improper integral. The integral is a proper integral. (d) π/4 cot(x) dx 0 Since the integral has an infinite interval of integration, it is a Type 1 improper integral. Since the integral has an infinite discontinuity, it is a Type 2 improper integral. The integral is a proper integral.
Answer:
a
Since the integral has an infinite discontinuity, it is a Type 2 improper integral
b
Since the integral has an infinite interval of integration, it is a Type 1 improper integral
c
Since the integral has an infinite interval of integration, it is a Type 1 improper integral
d
Since the integral has an infinite discontinuity, it is a Type 2 improper integral
Step-by-step explanation:
Considering a
[tex]\int\limits^4_3 {\frac{x}{x- 3} } \, dx[/tex]
Looking at this we that at x = 3 this integral will be infinitely discontinuous
Considering b
[tex]\int\limits^{\infty}_0 {\frac{1}{1 + x^3} } \, dx[/tex]
Looking at this integral we see that the interval is between [tex]0 \ and \ \infty[/tex] which means that the integral has an infinite interval of integration , hence it is a Type 1 improper integral
Considering c
[tex]\int\limits^{\infty}_{- \infty} {x^2 e^{-x^2}} \, dx[/tex]
Looking at this integral we see that the interval is between [tex]-\infty \ and \ \infty[/tex] which means that the integral has an infinite interval of integration , hence it is a Type 1 improper integral
Considering d
[tex]\int\limits^{\frac{\pi}{4} }_0 {cot (x)} \, dx[/tex]
Looking at the integral we see that at x = 0 cot (0) will be infinity hence the integral has an infinite discontinuity , so it is a Type 2 improper integral
A landscaper needs to estimate the amount of material required for a project building a circular brick patio for a client. The diameter of the patio is 13m and one bundle of brick will cover 116ft^2. How many bundles does he need?
Answer:
13 bricks
Step-by-step explanation:
Given parameter:
Diameter of the patio = 13m
Area of one bundle of brick = 116ft²
Unknown:
The number of bricks that will cover the patio = ?
Solution:
To solve this problem, we must find the area of the patio first.
Area of the patio = π r²
r is the radius of the patio;
Radius = [tex]\frac{13}{2}[/tex]
So area = π x ( [tex]\frac{13}{2}[/tex]) ² = 132.75m²
Now, let us convert the Area of the brick from ft² to m²;
1ft² = 0.093m²
116ft² = 0.093 x 116 = 10.79m²;
The number of bricks required to cover the patio = [tex]\frac{132.75}{10.79}[/tex] = 12.3 bricks,
Realistically, about 13 bricks would be needed.
Water freezes at 0 celcius and 32 fahrenheit
Answer:
true
Step-by-step explanation:
pls help
A coffee shop needs to purchase custom cups. Company C charges a flat fee of $155 plus $15 per cup. Company D does not have a flat fee but charges $30 per cup.
What is the minimum number of cups that must be ordered for company C to be a better deal than company D?
Answer:
i think it would be 11 cups
Add 38 and 29, then multiply by 2.
Answer:
134
Step-by-step explanation:
Luna brought all her savings with her to ieme, in euros,
For the first week, she spent the same amount of money
each day. The table shows how much Luna had left at the
end of each day
What was the initial value of Luna's savinge?
A.) 35 euros
B.) 350 euros
C.) 315 euros
D.) 80 euros
Answer:
The answer is B. 350 euros
Step-by-step explanation:
350 euros was the initial value of Luna's savings. Thus option B is correct.
What is the sum?Merging objects and identifying them since one big bunch is done through addition. In arithmetic, addition is the technique of adding two or more integers together. The product can meet are the quantities that are included, and the outcome of the operation, or the final response, is referred to as the sum.
The data that is presented suggests that there will not be a change in the first week this remains still at 315. They left out the amount that was 35.
Therefore the initial amount of the savings that the person has was the sum of both amounts.
= 315 + 35
= $350
Therefore, option B is the correct option.
Learn more about sum, Here:
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Luna brought all her savings with her to Rome, in euros. For the first week, she spent the same amount of money each day. The table shows how much Luna had left at the end of each day. What was the initial value of Luna's savings? Day Euros ? 315 1 2 280 3 245 4 210 5 175 6 140 7 105 35 euros 315 euros 80 euros 350 euros
what is the constant proportionality of this graph
Answer:
0 to 10
Step-by-step explanation:
Given: x - 6 > -2
Choose the solution set.
{x | x ∈ R, x > 4}
{x | x ∈ R, x > -8}
{x | x ∈ R, x < 4}
{x | x ∈ R, x < -8}
The first one is the answer
solve for brainliest A keem measured the capacity of a glass as
2 7 0 mL and the capacity of a Th e rm o s® a s
1 L. He said that the ratio 270 :1 compare s
the capacity of the glass with the cap a c i t y
of the Th e rm o s®. Is his reasoning corre c t ?
E x p l a i n .
Answer:
[tex]his \: reasoning \: is \: correct.[/tex]
Step-by-step explanation:
[tex]since \: the \: ratio \: of \: the \: glass \: to \:that \\ \: of \: the \: thermos \: is \: in \: the \: fraction : \\ \frac{2 7 0 mL }{1mL} \: which \: is \:270 mL : 1mL \\ then \: the \: keem \: is \: right.[/tex]
Most major ski resorts offer family programs that provide ski and snowboarding in- struction for children. The typical classes provide four to six hours on the snow with a certified instructor. The following data are the location and the daily rate at 6 ski resorts reported on January 20, 2006. Resort Location Daily Rate Beaver Creek Colorado $137 Deer Valley Utah $115 Diamond Peak California $95 Hunter New York $79 Mount Bachelor Oregon $83 Butternut Massachusetts $108 How many variables are in the data set? ______________ Use whole number, example 3
Answer:
2
Step-by-step explanation:
Given the data:
Resort___________Location _____Daily Rate
Beaver Creek_____ Colorado______ $137
Deer Valley________ Utah_________$115
Diamond Peak______California_____ $95
Hunter____________New York_____ $79
Mount Bachelor_____Oregon______ $83
Butternut________ Massachusetts__ $108
The dataset above has two variables wgjich corresponds to the characteristic of each distinct element in the dataset. The variables are usually recognized as columns in a tabular dataset. The variables in the dataset above are :
Location and Daily rate which are attributes of each distinct Ski Resorts which corresponds to elements of the data.
What is the solition of 8 +a = 13?
Answer:5
Step-by-step explanation:
Answer:
a=5
Step-by-step explanation:
1. Subtract 8 from 13 to isolate the variable
a=5
Find the value of x.
its 38 (sorry if i get u wrong)