Solve the inequality and express your answer in interval notation. x^2-12x+3<0

Answer:

[tex](f+g)(x)=8x^{2} +4x-4[/tex]

Step-by-step explanation:

[tex](f+g)(x)=3x^{2} +5x^{2} -2x+6x+4-8=8x^{2} +4x-4[/tex]

Hope this helps

The polynomial functions for which (x+3) is a factor

A polynomial of degree n is a function of the form

f(x) = anx

n + an−1x

n−1 + . . . + a2x

2 + a1x + a0

where the a is a real number (sometimes called the coefficients of the polynomial). Although

this general formula might look quite complicated, particular examples are much simpler. For

example,

f(x) = 4x

3 − 3x

2 + 2

is a polynomial of degree 3, as 3 is the highest power of x in the formula. This is called a cubic

polynomial, or just cubic.

Theorem Polynomial Division: Suppose d(x) and p(x) are nonzero polynomials were

the degree of p is greater than or equal to the degree of d. There exist two unique polynomials,

q(x) and r(x), such that p(x) = d(x) q(x) + r(x), where either r(x) = 0 or the degree of r is

strictly less than the degree of d.

so, if we put x=-3 then f(x) must be equal to zero

here no function satisfies of none of the option is correct for [x+3] factor

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(x +3) is a factor of the function f(x)= x³ +5x²-3x³-10x²+2x-3.

Here, we are given 4 functions. If (x+3) is a factor of any of these equations, then x = -3 and y = 0 must be a solution. Let us check each function one by one-

1. f(x) = x²-12x³ +54x² - 108x+81

putting x = -3, we get-

f(-3) = -3²-12(-3)³ +54(-3)² - 108(-3)+81

= 9+324+486+324+81

= 1,224

Thus, (x +3) is not a factor.

2. f(x)=x²-3x³-x+3

putting x = -3, we get-

f(-3) = -3²-3(-3)³-(-3)+3

= 9+81+9+3

= 102

Thus, (x +3) is not a factor.

3. f(x)=x+2x¹-23x³-60x²

putting x = -3, we get-

f(-3) = -3+2(-3)¹-23(-3)³-60(-3)²

= -3-6+621+540

= 1,152

Thus, (x +3) is not a factor.

4. f(x)= x³ +5x²-3x³-10x²+2x-3

putting x = -3, we get-

f(-3) = -3³ +5(-3)²-3(-3)³-10(-3)²+2(-3)-3

= -27+45+81-90-6-3

= 0

Thus, (x +3) is a factor of the function f(x)= x³ +5x²-3x³-10x²+2x-3.

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Answer:

Nate can make 60 packages.

Step-by-step explanation:

7 1/2 is equivalent to 7 4/8, or 60/8

Divide 60/8 by 1/8 and you get 60.

Answer:

160 mm.

Step-by-step explanation:

the average pen is 149mm. the average pen is 14.9cm. 160mm is closest.

It would have to be the 160mm.

This is because store-bought pens average 130 and 140mm/ 13 – 14 cm. This is closely followed by pens in the 140 – 150mm range.

Hope this helps!

Have a great day and God bless! :)

The table shows a proportional relationship between x and y.

In this question, we have been given a table which contains values of x and y.

We need to decide whether the table shows a proportional relationship between x and y.

x             y

17         2 5/6

5/6       5/36

6            1/6

19          3 1/6

First we convert the improper fraction in the y-column to proper fraction.

2 5/6 = 17/6

and 3 1/6 = 19/6

so, given table would be,

x             y

17          17/6

5/6       5/36

6            1/6

19          19/6

Consider the ratio x/y of each row:

x             y                x/y

17          17/6            1/6

5/6       5/36            1/6

6            1/6              1/6

19          19/6             1/6

As the ratio x/y is constant, x and y are directly proportional.

Therefore, the table shows a proportional relationship between x and y.

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The expression f(x) = 3 · x² - 2 · x + 4 evaluated at x = a + h is equal to the expression f(x) = 3 · a² + 6 · a · h + 3 · h² - 2 · a - 2 · h + 4.

In this question we find a quadratic equation, which has to be evaluated at x = a + h and simplified afterwards. Then, the complete procedure is shown below:

f(x) = 3 · (a + h)² - 2 · (a + h) + 4

f(x) = 3 · (a² + 2 · a · h + h²) - 2 · a - 2 · h + 4

f(x) = 3 · a² + 6 · a · h + 3 · h² - 2 · a - 2 · h + 4

The expression f(x) = 3 · x² - 2 · x + 4 evaluated at x = a + h is equal to the expression f(x) = 3 · a² + 6 · a · h + 3 · h² - 2 · a - 2 · h + 4.

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Answer:

(0, 21)

Step-by-step explanation:

Stationary points occur when the gradient of a graph is zero.

Therefore, to find the x-coordinate(s) of the stationary points of a function, differentiate the function, set it to zero and solve for x.

Differentiate the given function:

[tex]\begin{aligned}y & = x^3+ax^2+b\\\implies \dfrac{\text{d}y}{\text{d}x}&=3 \cdot x^{3-1}+2 \cdot ax^{2-1}+0\\\implies \dfrac{\text{d}y}{\text{d}x}& = 3x^2+2ax\end{aligned}[/tex]

Set the differentiated function to zero:

[tex]\begin{aligned} \dfrac{\text{d}y}{\text{d}x}& =0\\ \implies 3x^2+2ax & = 0 \\ \implies x(3x+2a)&=0\end{aligned}[/tex]

Therefore the stationary points occur when:

[tex]\implies x=0[/tex]

[tex]\implies 3x+2a=0 \implies x=-\dfrac{2}{3}a[/tex]

There is a stationary point at (4, -11), therefore substitute x = 4 into the expression for x and solve for a:

[tex]\begin{aligned}x & = 4\\\implies -\dfrac{2}{3}a & = 4\\a & = -\dfrac{3}{2}(4)\\a & = -6\end{aligned}[/tex]

Substitute the found value of a and the point (4, -11) into the function and solve for b:

[tex]\begin{aligned}y & = x^3 + ax^2 + b\\\implies -11 & = (4)^3+(-6)(4)^2+b\\ -11 & = 64-96+b\\b & = -11-64+96\\ b & = 21\end{aligned}[/tex]

Therefore, the function is:

[tex]\boxed{y=x^3-6x^2+21}[/tex]

The other stationary point is when x = 0.  Therefore, to find the coordinates of this point, substitute x = 0 into the function:

[tex]\begin{aligned}y & =x^3-6x^2+21\\x=0\implies y & =(0)^3-6(0)^2+21\\ y & =21 \end{aligned}[/tex]

Therefore, the coordinates of the other stationary point are (0, 21).

To determine if a stationary point is minimum or maximum, differentiate the function again:

[tex]\begin{aligned}y & =x^3-6x^2+21\\\implies \dfrac{\text{d}y}{\text{d}x} & = 3x^2-12x\\\implies \dfrac{\text{d}^2y}{\text{d}x^2} & = 6x-12\\\end{aligned}[/tex]

Substitute the x-coordinate of the stationary point into the second derivative:

[tex]\begin{aligned} \dfrac{\text{d}^2y}{\text{d}x^2} & = 6x-12\\x=0 \implies \dfrac{\text{d}^2y}{\text{d}x^2} & = 6(0)-12= -12 \\\end{aligned}[/tex]

[tex]\textsf{As $\dfrac{\text{d}^2y}{\text{d}x^2} < 0$, stationary point $(0,21)$ is a maximum.}[/tex]

Differentiation Rules

[tex]\boxed{\begin{minipage}{4.8 cm}\underline{Differentiating $ax^n$}\\\\If $y=ax^n$, then $\dfrac{\text{d}y}{\text{d}x}=nax^{n-1}$\\\end{minipage}}[/tex]

[tex]\boxed{\begin{minipage}{4cm}\underline{Differentiating a constant}\\\\If $y=a$, then $\dfrac{\text{d}y}{\text{d}x}=0$\\\end{minipage}}[/tex]

According to the solving the total and the difference of the numbers:

total = 136 + 64

difference  = 64

Mathematical difference is the outcome of one of the key operations, which is the subtraction of two numbers. It reveals the margin of difference between two numbers.

Subtract the largest number from the lowest number to determine the difference between two integers. The difference between these two numbers is the product of this sum. As a result, there are 55 between the numbers 45 and 100.

Let the numbers be x and y

Given,

x+y = 200_____(1)

x-y = 72_____(2)

Add above 2 eq.

2x= 272

x= 136

Substitute the value of x in eq 1,we get

136+y = 200

y = 64

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Answer:

the answer is x=3

Step-by-step explanation:

Answer:

Please mark my answer as the brainliest, it would mean a lot to me

x = 5

Step-by-step explanation:

4x + 3 = 18 - x

3x + 3 = 18

3x = 15

x = 5

The measurements are precise. Therefore, the correct option is D.

It should be noted that precision simply means how close a test is when it's repeated.

On the other hand, it should be noted that accuracy simply means how close the results are to the standard value.

Therefore, when the student does an experiment in a lab on the boiling point of water. In the 3 trials, the student gets the temperatures of 99.2, 99.5, 100.2 degrees, they're precise.

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Answer:

1672.50 = 37.75x + 1068.50

x = 16

There were 16 players brought to the tournament.

Step-by-step explanation:

y = mx + b

y= the total cost

m = cost of each meal

x = the number of players

b = cost of the bus

Plug in what you know and solve for x

1672.50 =  37.75x + 1068.50  Subtract 1068.50 from both sides.

604 = 37.75x  Divide both sides by 37.75

16 = x

Equation: 1068.50 + 37.75x =1,672.50

Answer: X= 16

If 1,672.50 is our total it will go behind the equal sign. 37.75x and 1068.50 must be added together to help us receive our total.

Our written equation should look like this

1068.50 + 37.75x =1,672.50

X is the number of players on the team.To find out we must first subtract the amount of money spent on the bus, from the total amount spent.

1,672.50-1068.50 is 604. This means our equation now looks like this

37.75x=604. The next and final step is to divide 37.75 into 604.

604/ 37.75 is 16. Which means that there are 16 players on the team.

I hope this helps & Good luck.

The independent variable in these equations based on how they are written is x, since y is a function of x here so y depends on the value of x, while x can be changed freely.

So, for each of these questions, plug in the given value of the independent variable for x.

Independent variable equal to 10: 0.1*10 + 5 = 6

Independent variable equal to 50: 0.1*50 + 5 = 10

Independent variable equal to 120: 0.1*120 + 5 = 17

Answer: -30 degrees fahrenheit

Step-by-step explanation: 18 + (-12) = -30

2(14m-8)=180

24m-16=180

24m=196

m= 8 and 1/6

Answer:

[tex]\textsf{a)} \quad f(t) = 4+0.2t \:\: \textsf{ and }\:\:A(t)=2(1.03)^t[/tex]

b)  77 years

c)  86 years

d)  110 years

Step-by-step explanation:

Given:

As the food supply grows at a constant rate of being adequate for an additional 0.2 million people per year, it can be expressed as a linear function:

[tex]f(t) = 4+0.2t[/tex]

where:

Given:

As the annual attendance increases by 3% per year, it can be expressed as an exponential function:

[tex]A(t)=2(1.03)^t[/tex]

where:

Graph the two functions (see attached) and locate the value of t for which A(t) > f(t) for the first time.

From inspection of the graph, the two functions intersect at t ≈ 77.  So after approximately 77 years the food supply will not be enough for the number of people attending the amusement park.  Therefore, after approximately 77 years, the park will first experience shortages of food.

If the park doubles its initial food supply and maintains the rate of increase of 0.2 million people per year, the new equation would be:

[tex]f(t) = 8+0.2t[/tex]

Again, graph the new function against A(t) and find the point where the two functions intersect.  A(t) = f(t) at approximately t = 86 so the park will first experience food shortages after 86 years.  So doubling the initial food supply delays the eventual food shortage by only c. 9 years.

If the park doubled the rate at which its food supply increases in addition to doubling its initial food supply, the new equation would be:

[tex]f(t) = 8+0.4t[/tex]

Again, graph the new function against A(t) and find the point where the two functions intersect.  A(t) = f(t) at approximately t = 110 so the park will first experience food shortages after 110 years.  So doubling the initial food supply and doubling the rate delays the eventual food shortage by only c. 33 years compared to the initial parameters.  Shortages would still occur, but it would be later in approximately 110 years' time.

120 + 4w is the expression to show the total number of songs on Iris's playlist after w weeks.

She total ahs 120 songs already

She downloads or adds 4 songs every week

An expression to show the total number of songs on Iris's playlist after w weeks is:

120 + 4w

An expression or mathematical expression is a limited combination of symbols that is well-formed according to context-dependent norms. To help identify the sequence of operations and other features of logical syntax, mathematical symbols can denote numbers (constants), variables, operations, functions, brackets, punctuation, and grouping.

Many authors differentiate between an expression and a formula, with the former referring to a mathematical item and the latter referring to a statement about mathematical objects.

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Reason:

7 units to the left means x becomes x-7

2 units up means y becomes y+2

Therefore, the old preimage (x,y) translates or shifts to the new image (x-7, y+2)

Example:

[tex](\text{x},\text{y}) = (3,5)\\\\(\text{x},\text{y})\to(\text{x}-7,\text{y}+2)\\\\(3,5)\to(3-7,5+2)\\\\(3,5)\to(-4,7)\\\\[/tex]

The preimage (3,5) moves to (-4,7) after shifting 7 units left and 2 units up.

W is the point which is non coplanar to plane k .

A group of points that are in the same plane are said to be coplanar. A coplanar line connects any two or three points. It's possible that four or more points are coplanar.

The coplanar points A, B, C, and D are displayed on the left side of the above figure. Additionally, there are other sets of coplanar points in the box to the right. For instance, the left side of the box's left side is the plane that contains the coplanar points P, Q, X, and W. There are four coplanar points on each of the box's six faces, however they are not the only coplanar point groups. Points Q, X, S, and Z, for instance, are coplanar despite the fact that the plane containing them isn't depicted.

Non-coplanar points are a collection of points that are not all located on the same plane.

Points P, Q, X, and Y in the previous figure are not coplanar. Q, X, and Y are located on the box's top, and P, Q, and X are located on the box's left side, however no flat surface has all four points.

since from the figure we can find out that except two points W , Y all the points are on the plane K .

so according to question and given figure we conclude that one of the point which in non coplanar to K is W.

points highlighted in the figure is non coplanar to point k .

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Answer:

Hello,

Step-by-step explanation:

Only one solution: x=-2, y=3,z=-5

Answer:

consistent, independent; one

Step-by-step explanation:

The solution is (−2, 3, −5). Because the system has one solution, it is consistent and independent.

Answer:

0.36

Step-by-step explanation:

12) Z ∪ N = {Set of Integers}

13)  H ∪ Q = R which is a set of real numbers

We are told that;

R = U represents the set of real numbers

N is the set of natural numbers

W is the set of whole numbers

Z is the set of integers

Q is the set of rational numbers

H is the set of irrational numbers

12) We want to find Z ∪ N. This is a set union of the set of integers and the set of natural numbers.

A rational number is a number that is expressed as the ratio of two integers, where the denominator should not be equal to zero. Thus, an integer is also a rational number and all natural numbers are positive integers and as such;

Z ∪ N = {Set of Integers}

13) We want to find H ∪ Q. This is a set union of the irrational numbers and the set of rational numbers. Now, we know that all real numbers are classified as either rational or irrational numbers

Thus; H ∪ Q = R which is a set of real numbers

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Answer:

Step-by-step explanation:

Given an assistant bricklayer laying 50 bricks an hour has worked 2 hours longer than a master bricklayer laying 80 bricks an hour, you want an equation stating they have laid the same number of bricks. You also want to know how long each has worked.

The output of each bricklayer is the product of the rate at which they lay bricks and the time for which they do it. If t is the time spent by the master bricklayer, their output will be 80t. The time spent by the assistant will be (t+2) hours, and their output will be 50(t+2).

Here is the equation that says their output is the same:

  80t = 50(t+2)

Solving the equation for t, we have ...

  80t = 50t +100

  30t = 100 . . . . . . . . . subtract 50t

  t = 3 1/3 . . . . . . . divide by 30

The master bricklayer has worked 3 1/3 hours when they have laid the same number of bricks.

The assistant bricklayer has worked 5 1/3 hours at that time.

__

Additional comment

Each will have laid (80)(3 1/3) = 266 2/3 bricks.

Answer:

Reflect over the x axis and than turn 90 degrees clockwise about the origin.

Answer:

the number is -15

Step-by-step explanation:

if you turn the word problem into an equation it would be

4x+19=-86-3x

with x being the unknown number

the next step is to move the -3x to the other side by adding it to the 4x

your equation should now be 7x+19=-86

next you should move the 19 to the other side by subtracting it from the -86

this leaves you with the equation

7x= -105

lastly you must divide the -105 by seven which leaves you with the answer x= -15

Answer: an=3n-1    a₅=14   a₁₀₀=299

Step-by-step explanation:

a=2     d=3

[tex]\boxed {a_n=a_1+d(n-1)}\\\\a_n=2+3(n-1)\\\\a_n=2+3n-3\\\\a_n=3n-1\\\\a_{100}=3(100)-1\\\\a_{100}=300-1\\\\a_{100}=299\\\\a_5=3(5)-1\\\\a_5=15-1\\\\a_5=14[/tex]

Answer:

Jojo is on a low carb diet

Answer:

-8

Step-by-step explanation:

2x/7 =

2(12)/-3 =

24/-3 =

-8

Answer:

-8

Explanation:

Given expression:

[tex]\rightarrow \sf \dfrac{2x}{y}[/tex]

when  x = 12, y = -3

Substitute values inside expression:

[tex]\rightarrow \sf \dfrac{2(12)}{-3}[/tex]

multiply

[tex]\rightarrow \sf \dfrac{24}{-3}[/tex]

simplify

[tex]\rightarrow \sf -8[/tex]

The solution is (x, y, z) = (-1, 0, -2)

Given equations are:

2x−y+2z=−6 ---------------(1)

−3y+z=−2 ------------------(2)

2x−3z=4 -------------------(3)

From (2), z = -2+3y

Substitute this into (1) and (3)

(1)⇒ 2x-y+2(-2+3y) = -6

⇒ 2x+5y-4=-6

⇒2x+5y = -2

and

(3) ⇒ 2x-3(-2+3y)=4

⇒ 2x-9y+6 = 4

⇒ 2x-9y=-2

Now solve 2x+5y = -2 and 2x-9y=-2

Subtracting the equations above,

14y = 0

⇒y=0

Then from 2x-9y=-2, x = -1

From (2),z = -2+3y

⇒ z = -2

So the solution is (x, y, z) = (-1, 0, -2)

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The average rate of change of the given function is 12.

The average rate of change of function f on interval [a,b] is:

f(b) -f(a)/b-a

we have interval [x₁,x₂]

[x₁,x₂] = [3,5]

f(3) = x²₊4x

f(3) = 3²+4(3)

f(3) = 21

f(5) = x²₊4x

f(5) = 5²+4(5)

f(5) = 45

f(x₂) -f(x₁)/x₂-x₁

⇒ 45-21/5-3

= 24/2

= 12

The average rate of change of function f(x) = x²+4x on interval [3,5]

is 12.

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