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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An interior wall of a building is made up of metal studs
Answer:
4 5/8"
Step-by-step explanation:
quizlet will have all the answers btw
Write the ratio as a unit rate. Simplify. 216 diving student and 12 instructor
Answer:
Ratio students to classroom in school A equals 216:12 which simplifies to 18:1. Ratio students to classroom in school B equals 104:4 which simplifies to 26:1.Total students combining school A and B equals 216+104=320 and Total classroom 12+4=16, if both school combined are assumed to be a single school C, than ratio of of students to the classroom in school C equals 320:16 which simplifies to 20:1 which means 20 students in every classroom, now total students in 4 classroom of School C equals 20x4=80, these 4 classroom are separated as School B and other 12 classroom School A, now ratio still 20:1 (same in both schools). initially students in School B were 104 and now are 80, then total students transferred equals 104-80= 24.
Write a word phrase for each expression. 2(n+1)
Answer:
A number increased by one multiplied by two.
Two distributed into a number and one.
Step-by-step explanation:
First one is way more likely.
HELP ME QUICK PLEASEE!!
4. (03.03 HC)
The table shows the outputs, y, for different inputs, x:
Input (x) 3 4 5 6
Output (y) 3 4 5 6
Part A: Do the data in this table represent a function? Justify your answer. (3 points)
Part B: Compare the data in the table with the relation f(x) - 2x + 16. Which relation has a greater value when x = 6? (2 points)
Part C: Using the relation in Part B, what is the value of x if f(x) - 40? (5 points)
(10 points)
Answer:
A) is a function
B) f(x) is greater
C) x = 12
Step-by-step explanation:
A) Yes, the relation is a function.
A function has a unique input value for each output value. There are no repeated x-values, so this relation is a function.
__
B) The table says y=6 for x=6.
f(6) = 2(6)+16 = 28
f(x) has a greater value for x=6.
__
C) Substituting the given information, we have ...
40 = 2x +16
24 = 2x
12 = x
The value of x is 12 when f(x) is 40.
Consider the following simplified algebraic expression.
10x*y+6x - 7y2-6
Which could be the original expression?
8x*y+7x? - 5y?+3x-2+ 2x*y- x - 2y + 3x-4
Bx+y+ 7x? -5y?+3x-2+ 2x*y-x - 2y2 - 3x-4
8x*y+7x? - 5y2 + 3x+2+2xºy- x? - 2y3 - 3x-4
8x+y+7x2 - 5y2 + 3x-2-2x*y-x-2y? - 3x-4
Answer:
the answer is the second option hope this helped
Step-by-step explanation:
Answer:
b :)
Step-by-step explanation:
I'll mark you as brainliest if you give me the correct answer!!!
Ayana's favorite brownie recipe calls for 1/2 of a cup of flour for every 1/3 of a cup of cocoa powder. For a bake sale, she plans to make a larger batch with 1 cup of cocoa powder. How much flour should she use?
Answer:
1 1/2 cups of flour
Step-by-step explanation:
Her original recipe calls for 1/3 cups of cocoa powder and 1/2 cups of flour. I saw we just added 2 more units to the cocoa powder! So, I decided that we should also add 2 more units to the flour and that is how I got 1 1/2 cups of flour.
(Sorry if wrong!)
Answer:
1 1/2
Step-by-step explanation:
Which of the following numbers are between 2/5 and 7/8
15/16
3/4
3/8
1/3
What is your FAVORITE color? The best answer earns Brainliest. ANSWER IN COMMENTS, then if it is good, answer for real.
my favorite color is grey with green. the colors look immaculate together, brown is also one of my favorites. if im being honest, i think all colors are beautiful, but grey,green,and brown are some of my favorites. thank you for the points luv, god bless you hun.
I need some help. Could someone explain it to me?
A relation between two variables a and b is a rule that takes some values a from a domain set and returns some value b in the range set.
The table says that, if you give 0, the relation returns 2. If you give 1, you get 6. If you give 4, you get -9. If you give 8, you get 8 back.
The domain is the set of values the relation can take, so here it is the set {0, 1, 4, 8}.
From the given choices, only 0 belongs to the domain.
f (x) = 4x + 7. Find the inverse of f(x).
O A. f-1 (x) = -74
OB. f-'(x) = 7 - 4x
O c. f-1(x) = 11
O D. f ? (x) = 4x – 7
Anybody please ? I suck at math !
Answer:
C. f(x) = x - 7 all over 4
Step-by-step explanation:
NB: Let f(x) = y
Exchange X and Y
Make y the subject
f(x) = 4x + 7
y = 4x + 7
x = 4y + 7
x - 7 = 4y
x - 7 all over 4 = 4 ÷ 4
y = x - 7 all over 4
Bonnie is baking cookies and making icing. She notices that 2 ounces of food coloring creates 9 cups of red icing. If 14 cups of red icing are needed to finish decorating cookies, how many ounces of food coloring are needed? Round to the nearest tenth of an ounce.
Answer:
3.1
Step-by-step explanation:
Write a proportion.
2 / 9 = x / 14
Cross multiply.
9x = 28
Divide.
x ≈ 3.1
5. A man purchased a sewing machine for ₹1250 and spent ₹250 on its transportation. He sold it for ₹1600. Find his profit percent.
Answer:
COST PRICE=1250+250=1500
SELLING PRICE=1600
GAIN=1600-1500=100
GAIN PERCENT=>
Step-by-step explanation:
I HOPE ITS HELP YOU
14. Mr. Feo is paid biweekly. His annual salary is $74,153. What is his biweekly salary, rounded to the
I
nearest cent?
Answer:
1426.00
Step-by-step explanation:
3/4 a - q = k, solve for a
Answer: a = 4k + 4q ÷ 3
Step-by-step explanation:
What multiplication expression is equivalent to 2/3÷1/2
PLEASE HURRY!!! WILL GIVR BRAINLIEST
Answer:
The answer is 39
Step-by-step explanation:
The scale on a map is 2 cm = 15 miles. Dave measures the distance to the next town at 7 cm. how may miles away is the next town?
Answer:
15x3.5 = 52.5
Step-by-step explanation:
Answer:
52.5
Step-by-step explanation:
Multiplication method
What is 5x5 to 4x2.....
Answer:
25, and 8
Step-by-step explanation:
5 x 5 = 25
4 x 2 = 8
Answer:
Step-by-step explanation:
5x5 = 10
4x2 = 8
Four friends made 55 cupcakes and shared them equally.
How many cupcakes did each friend get?
Out of the 55 cupcakes shared equally among four friends each friend will have 13 cupcakes.
What is the unitary method?A unitary method is a mathematical technique for finding the value of a single unit and then deriving the required units by multiplying with the single unit.
To find how many cupcakes each friend gets we can use the unitary method.
Given four friends made 55 cupcakes, therefore each friend can have (55/4) = 13.75 cupcakes, But the no. of cupcakes should be integers.
∴ Each friend can have 13 cupcakes.
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add the following conceptually and using the standard algorithm 79-26
Answer: 53
Step-by-step explanation:
79
-26
53
Of the following statements, which one or ones describe actions helpful to your credit score?
1. Quickly paying off debts
II. Having recent debts
III. Having long-standing lines of credit
a. I only
b. II and III
c. I and HII
d. III only
Answer:
a is the answer
Step-by-step explanation:
Write the equation of a line that passes through two points (3, 4) and (4, -2).
Answer:
y=-6x+22
Step-by-step explanation:
Equation is declining so its a negative
Mid point on x axis between the two is 22
The function m = 30 - 3r represents the amount m (in dollars) of money you have after renting r video games. What is the range of the function?
14 is the GCF of a number M and 210. What are possible values of M?
Answer:
Hello,
we know that to find the GCF we descompose the numbers and then we multiply the common terms with the lowest exponent. So we first descompose 210:
210 | 2
105 | 3
35 | 5
7 | 7
1
210= 2x3x5x7
Now one possible solution is 14 because:
14 | 2
7 | 7
1
As you can see 2 and 7 are common terms, so:
GCF= 2x7
GCF=14 --> It's correct
Then, if we analyze it, we'll realize that we have an infinitive number of solutions. The only requirement is that "M" must have 2 and 7 when we descompose it.
For example:
M= 2²x7x3x11
M=924
924 is another possible solution, it doesn't matter if 2 or 7 repeat, since I said the GCF is the multiplication of the common terms with the LOWEST exponent. That's all.
Step-by-step explanation:
Which degree of rotational symmetry does this figure have
Answers:
A: 120
B:90
C:180
D:270
Answer:
A:120
Step-by-step explanation:
Well it's not B or C or D so the logical conclusion is that it must be A.
Find the value of x.
its 38 (sorry if i get u wrong)
Sales tax is calculated as a percentage of the sales price. If sales tax is 6%, what is the sales tax on clothing that costs $180? (3 points)
Answer:
10.80
Step-by-step explanation:
Sales price x sales tax rate = sales tax
180 x .06 (6%) = 10.80
The amount of sales tax for the clothing that costs $180 is $10.8.
Given that,
Cost of a clothing = $180
We have to find the sales tax amount.
Sales tax is calculated as a percentage of the sales price.
Here,
Sales tax percentage = 6%
In order to find the actual sales tax amount, we have to multiply the cost with the percentage of sales tax.
Sales tax amount = 6% of 180
= (6/100) 180
= 0.6 × 18
= 10.8
Hence the sales tax amount is $10.8.
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what is 78+34 is the same as 80+
Answer:
112 its the same as 80 + 32
The expression is 78+34=80+32
What is equivalence?Equivalent equations are algebraic equations that have identical solutions or roots. Adding or subtracting the same number or expression to both sides of an equation produces an equivalent equation.
Given here, 78+34=112 and 80+32 =112 thus, 78+34=80+32 The equivalence is preserved.
Hence, The expression is 78+34=80+32
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Simplify: b3·b−5·b11
A) b9
B) b15
C) b19
D) b-165
In math what number acts like a switch
