Use the equation below to answer these questions.
[tex]x+2=\sqrt{3x+10}[/tex]

1.) What is the solution to the equation?

2.) What is the extraneous solution? Why?

3.) In general, what is an extraneous solution?

Answer:

C.) 1/169

Step-by-step explanation:

There are 4 jacks in a deck of 52 cards

The probability the first card is a jack

4/52 = 1/ 13

Since we replace the card

The probability the second card is a jack

4/52 = 1/ 13

To get the probability that we get a jack then a jack, we multiply the probabilities

1/13 * 1/13 = 1/169

Answer:

9.6667

Step-by-step explanation:

3t=29

Divide each side by 3

3t/3 = 29/3

t = 29/3

Three goes into 29  nine times  (3*9 = 27 ) with 2 left over

t =  9 2/3

We know 1/3 = .3 repeating

so 2/3  = .6 repeating

t = 9 .6 repeating

Answer: 3.0650

Step-by-step explanation:

K-12

Answer:

10.3 units

Step-by-step explanation:

From the given graph, we can see the coordinates of the points B (-2, 4) and D (3, -5).

To use the Pythagorean Theorem, we must have a right angled triangle. So we can select a point X (3, 4) to calculate the distance between B and D using the Pythagoras Theorem.

BX = 5, XD = 9

BD = [tex]\sqrt{BX^2+XD^2}[/tex]

BD = [tex]\sqrt{(5)^2+(9)^2} =\sqrt{106} =10.29[/tex]

Therefore, the points B and D are 10.3 units apart.

Answer:

∠DBC = 40°

Step-by-step explanation:

We are given a figure where we know that the angle m∠1 = 65°, m∠2 = 60° and m∠6 = 85°. With the help of these given measures of the angles, we are to find the measure of the angle m∠DBC.

Since the sum of angles in a triangle is equal to 180 degrees, so:

∠1 + ∠2 + ∠3 = 180

∠3 = 180 - (65 + 60)

∠3 = 55°

Also ∠6 and ∠B are alternate interior angles so if ∠6 = 85° then ∠B is also = 85°.

Now that we know ∠3 and ∠B, we can find ∠DBC:

∠DBC = 180 - (85 + 55)

∠DBC = 40°

Answer:40

Step-by-step explanation:

See angle 1 = 65°

Angle 2 = 60°

We know in a triangle all angle count 180°

So angle 3 = 55°

Now in a straight line all angle count 180°

So angle DBC + angle 3 + remaining angle along the line m will count 180°

Now angle 6 and remaining angle along the line m will be equal as 'm' ?and 'b' are parallel lines and t is intersecting them so it subtend equal angles.

Angle 6 = 85° so remaining angle along the line m is also 85°

We know angle DBC + angle 3 + rem. angle = 180°

Or 55° + 85° + angle DBC = 180°

Therefore, angle DBC = 40°

Hope it helps!!!

Answer: 691

Step-by-step explanation:

There are 3 different ways to find the remainder.  I am not sure which method you are supposed to use, so I will solve using all 3 methods.

Long Division:

         10x³ + 24x² + 77x + 230

x - 3 ) 10x⁴ -  6x³  + 5x²   - x  +  1

      - (10x⁴ - 30x³)   ↓       ↓      ↓

                   24x³ + 5x²     ↓      ↓

               - (24x³ - 72x²)   ↓      ↓

                              77x²  - x       ↓

                         -  (77x² - 231x)  ↓

                                        230x + 1

                                     - (230x - 690)

                                                    691 ← remainder

Synthetic Division:

x - 3 = 0   ⇒  x = 3

3  |  10     -6        5       -1        1

   |   ↓      30     72    231    690

      10      24     77    230   691 ← remainder

Remainder Theorem:

f(x) = 10x⁴ - 6x³ + 5x²  - x  +  1

f(3) = 10(3)⁴ - 6(3)³ + 5(3)² - (3)  +  1

     =  810   - 162    + 45    -  3  +  1

     = 691

Answer: Fertilizer A = 20, Fertilizer B = 56

Step-by-step explanation:

Step 1: Set up the equations

[tex]\begin{array}{c|c|c|c}& FertilizerA&FertilizerB&Quantity Required\\Nitrogen&8&5&440\\Phosphorous&2&5&260\\Potassium&4&5&360\\\end{array}[/tex]

Nitrogen: 8x + 5y ≥ 440

Phosphorous: 2x + 5y ≥ 260

Potassium: 4x + 5y ≥ 360

Step 2: Find the vertices

It is easiest to graph the equations to find the vertices. (see attachment).  You can also solve each system of equations to find the intersected points.

The following satisfy the "greater than or equal to" requirement:

Step 3: Use vertices in cost function C(x) to find the minimum

C(x) = $30x + $20y

(0, 88): $30(0) + $20(88) = $1760

(20, 56): $30(20) + $20(56) = $1720    ← This is the minimum!

(50, 32): $30(50) + $20(32) = $2140

(130, 0): $30(130) + $20(0) = $3900

The minimum cost occurs when 20 bags of Fertilizer A and 56 bags of Fertilizer B are purchased.  

This is a linear programming problem. The farmer needs to solve the system of inequalities that define the minimum nutrient requirements for the field and the cost function of the fertilizers to determine the least expensive way to meet the nutrient requirements.

The subject of this problem lies in the realm of linear programming—a mathematical method for determining a way to achieve the best outcome in a given mathematical model.

To solve this problem, we need to find the quantities of Fertilizer A and Fertilizer B that meets the minimum requirements for the field at minimum cost. Let x be the quantity of Fertilizer A and y be the quantity of Fertilizer B. Then, based on the information provided in the problem, we can set up the following inequalities:

Given that a 50-lb bag of Fertilizer A costs $30 and a 50-lb bag of Fertilizer B costs $20, the total cost of the fertilizers can be represented by the equation C = 30x + 20y, where C is the total cost.

The goal is to minimize the cost C = 30x + 20y, subject to the constraints above.
Using graphing or mathematical software, we can construct a graph of these inequalities and find the solution that lies on the feasible region where the cost is minimized.

The farmer needs to solve this system to determine the least amount of each fertilizer to use while still meeting the minimum soil nutrient requirements.

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Answer:

3 Dollars

Step-by-step explanation:

$4+.50cents=4.50

6 quarters is $1.50

4.50-1.50=3 dollars

So, I got this question for my Imagine Math, and here is what I got;

4+0.50−1.50.

Why; I got this because 4+0.50 is 4.50 and then 4.50-1.50 is 3.

Hopes this helps.

Answer:

(1, 3)

(-1, 3)

Step-by-step explanation:

h(x) = 3x^2

Let x =1

h(1) = 3 * (1)^2

    = 3(1)

    = 3

So if the input is 1 the output is 3

h(-1) = 3 * (-1)^2

     = 3 *1

If the input is -1 the output is 3

Answer:(1,3) and (-1,3)

Step-by-step explanation:

Answer: Second, third, fifth and sixth options are correct.

Step-by-step explanation:

Since we have given that

[tex]\frac{3}{4}+m=\frac{-7}{4}[/tex]

Now, we will solve it for the value of m :

[tex]\frac{3}{4}+m=\frac{-7}{4}\\\\m=\frac{-7}{4}-\frac{3}{4}\\\\m=\frac{-7-3}{4}\\\\m=\frac{-10}{4}[/tex]

Hence, the value of m is

[tex]\frac{-10}{4}[/tex]

and we can also apply m=[tex]\frac{-5}{2}[/tex]

if [tex]\frac{-5}{4}+m=\frac{-15}{4}\\\\m=\frac{-15}{4}+\frac{5}{4}\\\\m=\frac{-10}{4}\\\\m=\frac{-5}{2}[/tex]

And

[tex]m+2=-0.5\\\\m=-0.5-2\\\\m=-2.5\\\\m=\frac{-5}{2}[/tex]

Therefore, Second, third, fifth and sixth options are correct.

Answer:

Standard form: [tex]\left(x-\dfrac{9}{2}\right)^2+(y+5)^2=\dfrac{121}{4}.[/tex]

This equation represents the circle with the center at the point [tex]\left(\dfrac{9}{2},-5\right)[/tex] and the radius [tex]r=\dfrac{11}{2}.[/tex]

Step-by-step explanation:

Consider expression [tex]x^2+y^2-9x+10y+15=0.[/tex]

First, form perfect squares:

[tex](x^2-9x)+(y^2+10y)+15=0,\\ \\\left(x^2-9x+\dfrac{81}{4}\right)-\dfrac{81}{4}+(y^2+10y+25)-25+15=0,\\ \\\left(x-\dfrac{9}{2}\right)^2+(y+5)^2=10+\dfrac{81}{4},\\ \\\left(x-\dfrac{9}{2}\right)^2+(y+5)^2=\dfrac{121}{4}.[/tex]

This equation represents the circle with the center at the point [tex]\left(\dfrac{9}{2},-5\right)[/tex] and the radius [tex]r=\dfrac{11}{2}.[/tex]

B) 87 feet

The mnemonic SOH CAH TOA reminds you the relationship between angle, adjacent, and opposite sides is ...

... Tan = Opposite/Adjacent

In this geometry, the side adjacent to the angle is marked 150 ft, and the side opposite the angle is the height we want to find. This means ...

... tan(30°) = height/(150 ft)

Multiplying by 150 ft, we get ...

... height = (150 ft)·tan(30°) ≈ 87 ft

8.

   Statement                                           Reason

1. A'F is perpendicular bisector of JK    1. Given

2. JA ≅ KA                                               2. Dfinition of perp bisector

3. ∠JAF and ∠KAF are right angles      3. Dfinition of perp bisector

4. A'F ≅ A'F                                              4. Reflexive Property

5. ΔJFA ≅ ΔKFA                                      5. SAS Theorem

6.

(ANGLE): It is given that ∠AMD ≅ ∠EDM

(SIDE): MD ≅ MD by the Reflexive Property

Which angle would satisfy AAS?   ∠MAD ≅ ∠ DEM

NOTE: If you chose the other angle, it would satisfy ASA

Answer:

[tex]\frac{40}{19}\text{ or }2\frac{2}{19}[/tex] cases per hour.

Step-by-step explanation:

We are told that Jerry hears 5 cases every [tex]2\frac{3}{8}[/tex] hours.

To find the number of cases that Jerry hears per hour let us divide 5 by [tex]2\frac{3}{8}[/tex].

[tex]\text{Jerry hears cases per hour}=5\div 2\frac{3}{8}[/tex]

Let us convert our mixed fraction into improper fraction.

[tex]\text{Jerry hears cases per hour}=5\div \frac{19}{8}[/tex]

Since dividing a number by a fraction is same as multiplying the number by the reciprocal of the fraction.

[tex]\text{Jerry hears cases per hour}=5\times \frac{8}{19}[/tex]

[tex]\text{Jerry hears cases per hour}=\frac{40}{19}[/tex]

[tex]\text{Jerry hears cases per hour}=2\frac{2}{19}[/tex]

Therefore, Jerry hears [tex]\frac{40}{19}\text{ or }2\frac{2}{19}[/tex] cases per hour.

Answer:

4000

Step-by-step explanation:

Davila can job 400 feet in 8 min

I'm assuming you meant to say "graph" instead of "table".

The function rule is y = x+2 because the y intercept is 2, where the graph crosses the y axis. The slope is 1 meaning we move 1 unit up and 1 unit to the right each time. You can use the slope formula to determine the slope, or simply make this observation of rise vs run.

-------------------------------

Because this line doesn't go through the origin, and because it's not in the form y = k*x, this means we do not have a direct variation equation.

We do not have an inverse variation equation either because it is not in the form y = k/x or x*y = k. A visual indication of this is that the graph isn't a curved hyperbola.

Therefore, this function is neither direct variation nor inverse variation

-------------------------------

Plug in y = 10 and solve for x

y = x+2

x+2 = y

x+2 = 10

x+2-2 = 10-2

x = 8

The value of x is 8

So if x = 8, then y = 10 meaning that (x,y) = (8,10) is on this blue diagonal line.

Answer:

Correct choice is B

Step-by-step explanation:

All three functions are increasing when x is increasing. You can find values of each function when x is "enough large". For example, at x=100,

1. linear function: [tex]y=10\cdot 100=1,000.[/tex]

2. exponential function: [tex]y=5^{100}\approx 0.8\cdot 10^{70}.[/tex]

3. quadratic function: [tex]y=4\cdot 100^2+5\cdot 100=40,500.[/tex]

As you can see,  when x approaches positive infinity, the exponential function will exceed both the linear and the quadratic functions.

Also you can use graphical method to get from the attached diagrams result. When x approaches positive infinity, the exponential function will exceed both the linear and the quadratic functions.

Correct choice is B.

Answer: D. (6, 9)

Step-by-step explanation:

Midpoint is the "average" of the x's and y's:

Given: (5, 13) and (7, 5)

Midpoint: [tex](\dfrac{5+7}{2},\dfrac{13+5}{2})[/tex]

             = [tex](\dfrac{12}{2},\dfrac{18}{2})[/tex]

              = (6, 9)

Answer: (6, 9)

Step-by-step explanation:

9514 1404 393

Answer:

  (b)  f(x) = 2^x -3

Step-by-step explanation:

The horizontal asymptote is y=-3, eliminating the first and last choices.

The curvature is upward, eliminating the third choice.

The graph is a representation of ...

  f(x) = 2^x -3

Answer:

The simplest form that represents the difference is ( 2w-3).

Step-by-step explanation:

In this sum after w weeks subscription sold by me = 3w+4

and the subscription sold by my friend =5w+1

so the difference between the numbers of my friend and me

⇒(5w+1)-(3w+4)

⇒5w+1-3w-4

⇒2w-3

So the simplest form is (2w-3).

Final answer:

The difference in the number of subscriptions sold by the friend and the student after w weeks is represented by the simplified expression 2w - 3.

Explanation:

To find the difference between the number of subscriptions your friend has sold and the number you have sold, you simply subtract your sales from your friend's sales. The expression for your sales is (3w + 4) subscriptions and for your friend's sales is (5w + 1) subscriptions. So, the expression representing the difference will be:

(5w + 1) - (3w + 4)

Distributing the negative sign across the terms in the second set of parentheses gives us 5w + 1 - 3w - 4. Combining like terms, which are the 'w' terms and the constant terms separately, we get:

(5w - 3w) + (1 - 4) which simplifies to 2w - 3.

This expression 2w - 3 is the simplified form representing the difference between the number of subscriptions sold by your friend and you after w weeks.

Final answer:

To set up a profitable computer repair business, assuming $50 for parts per repair and a service fee of $150, the weekly expenses equation is y = 50x + 500, and income equation is y = 150x, where x is the number of repairs.

Explanation:

When starting a computer repair business with estimated expenses of $500 per week, it's crucial to set average costs for replacement parts and fees for services to ensure profitability. Let's assume an average cost of $50 for parts per computer repaired and a service fee of $150 per repair. If x represents the number of computers you repair each week, your total weekly expenses for parts would be 50x plus the fixed expenses of $500, resulting in a total expense equation of y = 50x + 500. Your total weekly income can be modeled by the equation y = 150x, with each repair contributing $150 to the income.

To break even or make a profit, the total income must exceed total expenses, leading to an analysis based on the number of repairs done weekly. This practical application of linear equations in a business setting helps in budgeting, pricing strategies, and understanding economic principles behind running a service-based business. Monitoring these variables is key to achieving sustainability.

Answer:

The even functions are options 2, 3, and 5

Step-by-step explanation:

Please, see the attached file.

Thanks.

Answer:

Options B, C and E are even functions.

Step-by-step explanation:

If f(x) = f(-x) then function is called to be even.

A). f(x) = ∛8x

f(-x) = ∛8(-x) = (∛8)(∛(-x) = 2∛(-x)

Therefore f(x) ≠ f(-x)

So function is not an even function.

B). [tex]f(x)=log_{9}x^{6}[/tex]

[tex]f(-x)=log_{9}(-x)^{6}[/tex]

[tex]=log_{9}(x)^{6}[/tex]

f(x) = f(-x)

So this function is even.

C). [tex]f(x)=\frac{1}{x^{8}+7x^{7}}[/tex]

[tex]f(-x)=\frac{1}{(-x)^{8}+7(-x)^{6}}[/tex]

            = [tex]\frac{1}{x^{8}+7x^{6}}[/tex]

f(x) = f(-x)

Therefore given function is even.

D). f(x) = [tex]e^{x^{8}-x }[/tex]

[tex]f(-x)=e^{(-x)^{8}-(-x)}=e^{x^{8}+x}[/tex]

Therefore f(x) ≠ f(-x)

So the given function is not even.

E). f(x) = |8x| - 3

f(-x) = |8(-x)| - 3

      = |8x| - 3

f(x) = f(-x)

Therefore, function is even.

F). [tex]f(-x)= -9(-x)^{10}+5(-x)^{4}-12(-x)[/tex]

[tex]f(-x)= -9(x)^{10}+5(x)^{4}+12(x)[/tex]

f(x) ≠ f(-x)

Therefore the given function is not an even function.

Options B, C and E are even functions.

Answer:

The last graph

Step-by-step explanation: On edge.

Answer:

D or the last graph

Step-by-step explanation:

Edge 2022

Answer:

Center (1,3) and radius 6

Step-by-step explanation:

We must complete the square to find the center and radius of the circle.

First make sure the x and y squared terms have 1 as their coefficients. We also make sure x and y terms together.

[tex]x^2-2x+y^2-6y=26[/tex]

We now create space between the x and y terms with parenthesis.

[tex](x^2-2x)+(y^2-6y)=26[/tex]

We complete the square by taking the middle terms -2x and the -6y - divide each and square them.

[tex]\frac{-2}{2} =(-1)^{2} =1[/tex]

[tex]\frac{-6}{2} =(-3)^{2} =9[/tex]

We add the squares to both sides.

[tex](x^2-2x+1)+(y^2-6y+9)=26+1+9[/tex]

Simplify.

[tex](x^2-2x+1)+(y^2-6y+9)=36[/tex]

And write the quadratics in factored form.

[tex](x-1)^{2} +(y-3)^{2} =36[/tex]

The center is (h,k) or (1,3). The radius is the square root of 36 which is 6.

Answer:

C.p(v)=5000/v

Step-by-step explanation:

Got it right on the test.

The equation that can be used to find the pressure of the gas when the volume is changed is P(v) = 500/v

Given:

p(v) = pressure of a gas

v = volume of the gas

P(v) varies inversely with v

let

k = constant of proportionality

The equation:

P(v) = k/v

If P(v) = 25 kg/cm² and v = 200cm²

Therefore,

P(v) = k/v

25 = k / 200

25 × 200 = k

k = 5,000

substitute the value of k into the equation

So,

P(v) = 500/v

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When a 20-foot length of wire is cut into 6 pieces of equal length, each piece would be approximately 3.33 feet long. Rounding to the nearest whole numbers, this means the length of each piece will fall between 3 feet and 4 feet.

To find the length of each piece of wire when a 20-foot length of wire is cut into 6 pieces of equal length, you divide the total length by the number of pieces. So, you calculate:

Since we're looking for whole-number lengths that the length of each piece falls between, we can round 3.33 feet down to the nearest whole number (3 feet) and up to the next whole number (4 feet). Thus, the length of each piece will fall between 3 feet and 4 feet.

Answer:

The function is:

21000 * (1 - [tex]\frac{16}{100}[/tex])[tex]^{n}[/tex]

The price after 3 years would be D) $12,447

Step-by-step explanation:

Because the value is depreciating, the price will decrease.

The function formula is:

Amount x (1 ± [tex]\frac{percentage}{100}[/tex])[tex]^{n}[/tex]

Where n is the amount of years and the ± is a + if the value is increasing, and a - if the value is depreciating.

So plug the values in, with a minus:

21000 * (1 - [tex]\frac{16}{100}[/tex])[tex]^{n}[/tex]

The price after 3 years would simply be the following equation:

21000 * (1 - [tex]\frac{16}{100}[/tex])[tex]^{3}[/tex]

Which gives the result 12446.7 or 12447 to 1.d.p

This means that the answer is D) $12,447!

Answer:

The required domain is [tex]x\ne -1[/tex]

Step-by-step explanation:

The given function is

[tex]f(x)=\frac{1-x}{x}[/tex]

We need the inverse of this function.

We first of all have to let [tex]y=f(x)[/tex].

This implies that,

[tex]y=\frac{1-x}{x}[/tex]

Next, we interchange [tex]x[/tex] and [tex]y[/tex] to obtain,

[tex]x=\frac{1-y}{y}[/tex]

We make y the subject to get,

[tex]xy=1-y[/tex]

[tex]xy+y=1[/tex]

[tex](x+1)y=1[/tex]

[tex]y=\frac{1}{1+x}[/tex]

The inverse function is

[tex]f^{-1}(x)=\frac{1}{1+x}[/tex]

The domain of this function is

[tex]x+1\ne 0[/tex]

[tex]\Rightarrow x\ne -1[/tex]

The correct answer is D

Answer:

see explanation

Step-by-step explanation:

For words per minute divide 228 by 4

words per minute = [tex]\frac{228}{4}[/tex] ( = 57 )

Final answer:

To find the number of words Carlos types per minute, set up an equation and solve for 'x'.

Explanation:

To find the number of words Carlos types per minute, we can set up an equation using the given information. Let's assume that the number of words Carlos types in 1 minute is 'x.' We know that Carlos can type 228 words in 4 minutes. So, the equation would be:

228 words in 4 minutes = x words in 1 minute

To solve this equation, we can cross-multiply and divide:

(228 words) x (1 minute) = (4 minutes) x (x words)

Simplifying further:

x = (228 words) / (4 minutes)

Therefore, the equation that represents the number of words Carlos types per minute is:

x = 228 / 4