How to rationalize the denominator of sqrt(a)/sqrt(a)-2

The quotient of the rational expressions given is obtained by converting the division to multiplication by the reciprocal leading to \((\frac{3x \times (x+5)}{2x \times (2x+5)})\). The final simplification depends on specific values of x.

The question asks to find the quotient of the following rational expressions: \((\frac{3x}{2x+5}) \div (\frac{2x}{x+5})\). To solve this, we first recall that dividing by a fraction is the same as multiplying by its reciprocal. Therefore, the problem becomes \((\frac{3x}{2x+5}) \times (\frac{x+5}{2x})\).

We simplify this further by multiplying the numerators together and the denominators together: \((\frac{3x \times (x+5)}{2x \times (2x+5)})\).

To solve the more difficult problem, as hinted, we might consider factoring or simplifying further. However, the key calculation here reveals that the certain factors in the numerator and denominator might not simplify directly in this expression, leading to a more nuanced understanding of rational expressions. As such, detailed simplification depends on the specific values of x and further factorization may not lead to a simpler form without more context.

The correct answer is: [tex]\text { B. } \frac{3 x+15}{4 x+10}[/tex].

To find the quotient of the given rational expressions, you divide the first rational expression by the second one.

[tex]\frac{\frac{3 x}{2 x+5}}{\frac{2 x}{x+5}}=\frac{3 x}{2 x+5} \cdot \frac{x+5}{2 x}[/tex]

Now, you can simplify this expression:

[tex]=\frac{3 x(x+5)}{2 x(2 x+5)}[/tex]

Cancel out x from both numerator and the denominator.

[tex]=\frac{3 (x+5)}{2 (2 x+5)}[/tex]

[tex]=\frac{3x+15}{4x+10}[/tex]

This value matches with the option B. Thus, B is the correct answer.

Complete Question:

Which of the following is the quotient of the rational expressions shown below?

[tex]\frac{3 x}{2 x+5} \div \frac{2 x}{x+5}[/tex]

[tex]\text { A. } \frac{4 x^2+10 x}{3 x^2+15 x}[/tex]

[tex]\text { B. } \frac{3 x+15}{4 x+10}[/tex]

[tex]\text { C. } \frac{6 x^2}{2 x^2+15 x+25}[/tex]

[tex]\text { D. } \frac{3}{4}[/tex]

Answer:

The triangle is reflected across the x-axis and then translated 1 unite to the right , 1 unit up

Step-by-step explanation:

* Lets revise some transformation

- If point (x , y) reflected across the x-axis

 then the new point = (x , -y)

- If point (x , y) reflected across the y-axis

 then the new point = (-x , y)

- If the point (x , y) translated horizontally to the right by h units

 then the new point = (x + h , y)

- If the point (x , y) translated horizontally to the left by h units

 then the new point = (x - h , y)

- If the point (x , y) translated vertically up by k units

 then the new point = (x , y + k)

- If the point (x , y) translated vertically down by k units

 then the new point = (x , y - k)

* Now lets solve the problem

- A triangle has three vertices

- The vertices are B (-3 , 0) , C(2 , -1) , D (-1 , 2)

- The images of the vertices are B" (-2 , 1) , C" (3 , 2) , D" (0 , -1)

 after two steps of transformations

- After comparing the points with their images we find

# The x-coordinates of the points are added by 1

∴ There is translation to the right

# The y-coordinates of the points not add or subtracted by the same

   number, that means there is a transformation before the translation

   for the y-coordinates

# The sign of y-coordinates of the points are changed , that means

   there is a reflection across the x-axis

∴ B' is (-3 , 0) , C' is (2 , 1) , D' is (-1 , -2)

- After comparing the 1st image with the 2nd images we find

# The x-coordinates of the points are added by 1 and the

   y-coordinates are add by 1

∴ B" is (-2 , 1) , C" is (3 , 2) , D" is (0 , -1)

- From all above

* The triangle is reflected across the x-axis and then translated 1 unite

  to the right , 1 unit up

Answer:

[tex]x=5[/tex] and [tex]y=2[/tex].

Step-by-step explanation:

We have been given a system of equations. We are asked to solve our given system.

[tex]x+y=7...(1)[/tex]

[tex]2x+3y=16...(2)[/tex]

From equation (1), we will get:

[tex]x=7-y[/tex]

Upon substituting this value in equation (2), we will get:

[tex]2(7-y)+3y=16[/tex]

[tex]14-2y+3y=16[/tex]

[tex]14+y=16[/tex]

[tex]14-14+y=16-14[/tex]

[tex]y=2[/tex]

Now, we will substitute [tex]y=2[/tex] in equation (1).

[tex]x+2=7[/tex]

[tex]x+2-2=7-2[/tex]

[tex]x=5[/tex]

Therefore, the point [tex](5,2)[/tex] is solution for our given equation.

Answer:

The answer is (5,2)

Step-by-step explanation:

I took the test and got it correct.

Answer:

38

Step-by-step explanation:

The 38-deg angle and <6 are vertical angles, so they are congruent.

m<6 = 38 deg

Answer:

38 degrees. Congruent inside angles. <3

Step-by-step explanation:

Answer:

(x+5) /-15 = f^-1(x)

Step-by-step explanation:

f(x)= -15x-5

y = -15x-5

Exchange x and y

x = -15y -5

Solve for y

Add 5 to each side

x+5 = -15y -5+5

x+5 = -15y

Divide by -15

(x+5)/-15 = -15y/-15

(x+5) /-15 = y

(x+5) /-15 = f^-1(x)

Answer:

Nearly 23.8%

Step-by-step explanation:

The word ALGEBRA consists of letters A, L, G, E, B, R and A (2 letters A and 5 letters other than A).

The probability that the first letter chosen will be other than A is

[tex]\dfrac{5}{7}[/tex]

Then 2 letters A and 4 letters other than A left (6 letters in total). The probabilty that the second letter chosen is A is

[tex]\dfrac{2}{6}=\dfrac{1}{3}[/tex]

Hence, the probability of choosing a letter other than A and then choosing an A is

[tex]\dfrac{5}{7}\cdot \dfrac{1}{3}=\dfrac{5}{21}\approx 23.8\%[/tex]

The probability of first drawing a letter other than 'A' and then drawing an 'A' from the word ALGEBRA is approximately 23.81%. This is determined by calculating the individual probabilities and then multiplying them together.

The word ALGEBRA has 7 letters. The probability of choosing a letter other than 'A' means we are considering 5 valid letters out of the 7. Therefore, the probability is 5/7. Then, looking for an 'A' within the remaining 6 letters (after one letter is already taken), we see there are 2 'A' letters left, so the probability is 2/6 or simplified as 1/3.

Therefore, the combined probability of both these events happening is the product of their individual probabilities. We multiply the fractions: 5/7 * 1/3 = 5/21 = 0.238095. Translating that fraction into a percentage, we multiply by 100% to get approximately 23.81%.

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Answer:

x = 1

Step-by-step explanation:

Given

5x - (4x - 1) = 2 ← distribute the parenthesis by - 1

5x - 4x + 1 = 2

x + 1 = 2 ( subtract 1 from both sides )

x = 1

Answer:

21 1/3

Step-by-step explanation:

Write this as an expression

3/4 x = 16

Now it is a one step equation

divide 3/4 from its self and 16

3/4 divided by 16 = 21 1/3

x= 21 1/3

Answer:

[tex]21\frac{1}{3}[/tex] or 21.3

Step-by-step explanation:

We are given that the product of [tex] \frac { 3 } { 4 } [/tex] and a number is [tex] 1 6 [/tex].

Assuming that number to be [tex] x [/tex], we can write it as:

[tex] \frac { 3 } { 4 } \times x = 1 6 [/tex]

Taking the denominator to the other side of equation and multiplying it to get:

[tex] 3 x = 1 6 \times 4 [/tex]

[tex] 3 x = 6 4 [/tex]

Isolating the variable to get:

[tex] x = \frac { 6 4 } { 3 } [/tex]

[tex] x = 21\frac{1}{3}[/tex] or [tex] x = 21.3[/tex]

The expression log 5 + log 2 can be combined using the property that the logarithm of a product is equal to the sum of the logarithms, which simplifies to log 10. Since log 10 is equal to 1, the solution to the expression is 1.

To solve the expression log 5 + log 2 using the logarithmic theorems, we can apply one of the fundamental properties of logarithms. Specifically, the logarithm of a product of two numbers is equal to the sum of the logarithms of those two numbers (log xy = log x + log y). Applying this property to our expression, we can combine the two logarithms as follows:

log 5 + log 2 = log (5 x 2)

Now, we can easily calculate the combined term:

log (5 x 2) = log 10

Since the base of the logarithm is not specified, we can assume it is 10 (common logarithm). Therefore, we can simplify further:

log 10 = 1

Thus, the solution to the given expression is 1.

Answer:

B

Step-by-step explanation:

Given

f = c / λ ( multiply both sides by λ )

fλ = c

You simply have to multiply both sides by lambda:

[tex]f = \dfrac{C}{\lambda} \iff f\cdot \lambda = \dfrac{C}{\lambda}\cdot \lambda \iff C = f\lambda[/tex]

Answer:

1/21.

Step-by-step explanation:

To solve the problem 'How long would they take to do the job working together?'  you would work in fractions of the job done / minute:

1/21 + 1/28 = 1/x.

So for person 1 the expression is 1/21.

Answer:

|x-22| < 2

The < has an underline below it

The absolute value inequality that represents the average height range of a golden retriever (20-24 inches) is |h - 22| <= 2, where h represents any height within the range. This means that the difference between any height in the range and the midpoint (22 inches) is at most 2 inches.

To write the average height range of a golden retriever as an absolute value inequality, we need to first find the midpoint of the range and then use that to express the distance from any height in the range to that midpoint. In this case, the average height range for a golden retriever is 20-24 inches. The midpoint of the range is (20 + 24) / 2 = 22 inches. So, any height h in the range satisfies |h - 22| <= 2. This is because the maximum difference between any height in the range and the midpoint is 2 inches (24 - 22 or 22 - 20).

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Answer:

1. y= -1/4

Step-by-step explanation:

y/5 + 3/10 = (y+2)/7

Get a common denominator for the left hand side

2y/10 + 3/10 = (y+2)/7

(2y+3)/10 = (y+2)/7

Using cross products

(2y+3) *7 = 10*(y+2)

Distribute

14y+21 = 10y+20

Subtract 10y from each side

14y-10y +21 = 10y-10y+20

4y+21 =20

Subtract 21 from each side

4y+21-21 = 20-21

4y = -1

Divide by 4

4y/4 = -1/4

y = -1/4

Answer:

24

Step-by-step explanation:

Varies directly means that something constant times l equals V or V=kl where k is that constant.  

Plug in point given 288=k(12) and solve the constant of proportionality by dividing both sides by 12.

k=288/12=24

Answer:

B, D

Step-by-step explanation:

Angles 1 and 2 combined form a right angle, so the sum of their measures is 90°.  Therefore, they are complementary angles.

Since they share a common side and vertex and don't overlap, they are also adjacent angles.

Answer:

the answer is yes because on a graph, even if it has a x intercept, it will eventually intercept the y intercept b/c the line will continue infinitely.

Step-by-step explanation:

Please mark brainliest and have a great day!

no

no becuase not every line acrosses along the whole graph and the y intercept gose up and down

Answer:

○ The graph of y = -2x² + 3 opens downward and is shifted up.

Step-by-step explanation:

According to the Quadratic Equation, y = Ax² + Bx + C, C acts like a y-intercept, and in this case, since both graphs shift up [because both are positive values], we do not pay attention to those. Now, we come over to our A, which makes a BIG difference because both graphs open in opposite directions [one negative, one positive]. With this being stated, we have our answer.

If you are ever in need of assistance, do not hesitate to let me know by subscribing to my You-Tube channel [USERNAME: MATHEMATICS WIZARD], and as always, I am joyous to assist anyone at any time.

** Extended information on Parabolas

Opens down → -A

Opens up → A

Answer:

Step-by-step explanation:

Pay = 10.35 * h   where h is the number of hours that he works. The other 2 questions should have been included with this one. I'll answer the second one here.

Distance = 65 * h

distance = 65 * h miles.

Answer:

5x + 11

Step-by-step explanation:

We are subtracting function g(x) from function f(x).

Write f(x) as is:  f(x) = x + 8.

Then change all of the signs of g(x):  -g(x) = 4x + 3.

Now combine like terms for the sum (f - g)(x) = x + 8 + 4x + 3.  We get

(f - g)(x) = 5x + 11

Answer:

When x <= 8000

The cost remains constant at 0.35 when x increases from 0 to 8000

The slope of cost function over this part is 0

When 8000 < x <= 20000

The cost remains constant at 0.75 when x increases from 8000 to 20000

The slope of cost function over this part is 0

When 20000 < x <= 42000

The cost decreases when x increases from 20000 to 42000

The slope of cost function

[tex]m = \frac{y2 - y1}{x2 - x1} \\ = \frac{(0.83 - \frac{40000}{200000}) - (0.83 - \frac{20000}{200000})}{40000 - 20000} [/tex]

m= -5 × 10^-6

Answer:

Option B

Step-by-step explanation:

. If a function is given in the form of f(x) = ax² + bx + c, then value of x is "input" and the value ... Now from the given table output values of the function are 8, 3, -5, -2 and 12. ... Therefore Option B

Answer:

3a x 9ac

=(3a×9a)(c)

27a2 ×c

7de×3de2

=(7×3)(de ×de2)

= 21de3

Answer:

1/3(n) - 3 < 6; n < 27

Step-by-step explanation:

" 3 subtracted from one third of a number"

= 1/3(n) - 3

given that this is less than 6:

1/3(n) - 3 < 6

1/3(n) < 6 +3

1/3(n) < 9

n < 27

Inequality and solution is  1/3(n) - 3 < 6 and n < 27.

The correct option is (A).

A statement of an order relationship—greater than, greater than or equal to, less than, or less than or equal to—between two numbers or algebraic expressions.

Given statement is : " 3 subtracted from one third of a number less than 6"

or, 1/3(n) - 3 < 6

or, 1/3(n) < 6 +3

or, 1/3(n) < 9

or, n < 27

Hence, Inequality and solution is  1/3(n) - 3 < 6 and n < 27.

Learn more about inequality here:

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Answer:

4 units

Step-by-step explanation:

We have to calculate the distance between two points first to check the correct answer.

so,

The formula for distance is:

[tex]d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}\\Here\\(x_{1},y_{1})=(3,3)\\(x_{2},y_{2})=(7,3)\\Putting\ Values\\d=\sqrt{(7-3)^{2}+(3-3)^{2}}\\d=\sqrt{(4)^{2}+(0)^{2}}\\d=\sqrt{16+0} \\d=\sqrt{16} \\Taking\ Square\ Root\ will\ give\ us:\\d=4 units\\So\ the\ distance\ between\ two\ points\ is\ 4\ units ..[/tex]

Yes, I can solve that inequality.

Any value of ' r ' that's less than 1/3 is a solution.

Answer:

r < 1/3

Step-by-step explanation:

21r<7

Divide each side by 21

21r/21<7/21

r < 1/3

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Have A Great Day!

Answer:

A. 0

Step-by-step explanation:

Nothing can be equal to itself if you subtract 9 from it.

Final answer:

The equation X = X - 9 yields a contradiction upon simplification (0 = -9), indicating that there are no solutions.

Explanation:

To determine the number of solutions for the given equation, we start by inspecting it closely:

X = X - 9

If we attempt to solve for X, we will subtract X from both sides of the equation:

0 = -9

This is a contradiction since 0 does not equal -9. Therefore, the given equation has no solution. Other examples of equations can have one solution, such as X = 1, or have two solutions, such as quadratic equations like x = 4.133 or 9.435. But in this case, the equation does not balance and therefore has no solutions.

The correct answer is:

A. 0

Answer:

x=2

Step-by-step explanation:

Vertical angles are equal

m ZE = m ZF

9x+12 = 3x+24

Subtract 3x from each side

9x -3x +12 = 3x-3x+24

6x+12 = 24

Subtract 12 from each side

6x+12 -12 = 24-12

6x = 12

Divide each side by 6

6x/6 =12/6

x = 2

Answer: C

Step-by-step explanation: The table represents absolute value

For this case we must find a function of the form:

[tex]y = f (x)[/tex]that complies with the relation of the table.

We note that for the first two values of x, the function yields the same value but with a positive sign.

So:

[tex]f (x) = | x |\\y = | -2 | = 2\\y = | -1 | = 1\\y = | 0 | = 0\\y = | 1 | = 1\\y = | 2 | = 2[/tex]

Answer:

Option C

Answer:

[tex]\large\boxed{y=2x^2+4}[/tex]

Step-by-step explanation:

The vertex form of a parabola:

[tex]y=a(x-h)^2+k[/tex]

(h, k) - vertex

We have the vertex at (0, 4) → h = 0 and k = 4.

Substitute:

[tex]y=a(x-0)^2+4=ax^2+4[/tex]

The point (-1, 6) is on the parabola. Put the coordinates of the point to the equation:

[tex]6=a(-1)^2+4[/tex]            subtract 4 from both sides

[tex]2=a\to a=2[/tex]

Finally:

[tex]y=2x^2+4[/tex]