problem solving involving rational equation
1. The reciprocal of 5 plus the reciprocal of 7 is the reciprocal of what number?

2. The reciprocal of the product of two consecutive intergers is 1/72

The new equation will be:

[tex]g(x) =\frac{1}{x+4}+4[/tex]

OR

[tex]g(x) = \frac{x+5}{x+4}[/tex]

Step-by-step explanation:

The function transformations are defined as:

Shifting function upwards b units

[tex]f(x) => g(x) = f(x)+b[/tex]

Shifting functions to the left b units

[tex]f(x) => g(x) = f(x+b)[/tex]

Given function is:

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

Shifting the function 4 units to the left will give us:

[tex]g(x) => g(x+b)\\= \frac{1}{x+4}[/tex]

Now shifting the function 4 units upwards

[tex]g(x) = \frac{1}{x+4} + 4[/tex]

Simplifying

[tex]g(x) = \frac{1+x+4}{x+4}\\= \frac{x+5}{x+4}[/tex]

The new equation will be:

[tex]g(x) =\frac{1}{x+4}+4[/tex]

OR

[tex]g(x) = \frac{x+5}{x+4}[/tex]

Keywords: function, transformations

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Answer:

[tex]\frac{2}{15}[/tex]

Step-by-step explanation:

Number of students participating in the competition = 13.

Total number of students in the school = 15.

Thus ,

the number of students who did not make to this competition =

Total number of students - number of students participated

= 15-13 = 2.

Now ,

probability of the student who do not makes it to the competition

= fraction of students who did not make to this competition from the total number of students .

=[tex]\frac{2}{15}[/tex].

Answer:

The slope of f(x) is 1.5

Step-by-step explanation:

step 1

Find the slope of the linear function that has an x-intercept of 2/3 and a y-intercept of -1

so

we have the points

(2/3,0) and (0,-1)

The formula to calculate the slope between two points is equal to

[tex]m=\frac{y2-y1}{x2-x1}[/tex]

substitute the values

[tex]m=\frac{-1-0}{0-2/3}[/tex]

[tex]m=\frac{-1}{-2/3}[/tex]

[tex]m=\frac{3}{2}=1.5[/tex]

step 2

Find the slope of the function graphed

take the points

(0,-1) and (3,6)  approximately

substitute in the formula

[tex]m=\frac{6+1}{3-0}[/tex]

[tex]m=\frac{7}{3}=2.33[/tex]

step 3

we know that

f(x) represents the function with the smaller slope

The smaller slope is 1.5

therefore

The slope of f(x) is 1.5

Answer:

15i+15

Step-by-step explanation:

(6+3i)(3+i)

18+9i+6i+3i^2

18+15i+3(-1)

18+15i-3

15i+15

After multiplyinh complex numbers (6+3i)(3+i), the final answer will be 15+15i.

To multiply complex numbers, we can use the distributive property.

First, distribute 6 to both terms of (3+i), giving us 18+6i.

Then, distribute 3i to both terms of (3+i), resulting in 9i+3i².

We can simplify 3i² by using the fact that i²=-1, so 3i²=-3.

Putting it all together, we have (6+3i)(3+i)

= 18+6i+9i-3

= 15+15i.

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Answer:

1][tex]measure\ angle\ DCA=measure\ angle\ CAB=18\ degree[/tex]

2][tex]measure\ angle\ A+measure\ angle\ B+measure\ angle\ C=180\ degree[/tex]

3][tex]18\ degree+3x+3x=180\ degree[/tex]

4][tex]x=27[/tex]

Step-by-step explanation:

Given:-measure angle DCA = 18 degree

line AB II line DC

Solution:-

1] measure angle DCA and measure angle CAB are interior  alternate angles and interior alternate angles are equal.

[tex]measure\ angle\ DCA=measure\ angle\ CAB=18\ degree[/tex]------------------ (Equation 1)

Therefore the equation for statement number 1 is [tex]measure\ angle\ DCA=measure\ angle\ CAB=18\ degree[/tex]

2]Sum of interior angles of triangle is equal to 180 degree.

In Triangle ABC,

measure angle A+measure angle B+measure angle C=180 degree

Therefore the equation for statement number 2 is measure angle A+measure angle B+measure angle C=180 degree

3] In Triangle ABC,

[tex]measure\ angle\ B=3x---------------(Given)[/tex]

[tex]measure\ angle\ C=3x---------------(Given)[/tex]

[tex]measure\ angle\ A=18\ degree---------------(from\ equation\ [tex]measure\ angle\ A+measure\ angle\ B+measure\ angle\ C=180\ degree[/tex]

Now substituting the value of angles we get,

[tex]18\ degree+3x+3x=180\ degree-------------(Equation\ 2)[/tex]

Therefore the equation for statement number [tex]3\ is\ 18\ degree+3x+3x=180\ degree[/tex]

4] After solving the equation 2 we get,

[tex]18+3x+3x=180[/tex]

[tex]6x=180-18[/tex]

[tex]6x=162[/tex]

[tex]x=27\ degree[/tex]

Therefore value of [tex]x=27\ degree[/tex]

Answer:

The two integers are -47 and -48.

Step-by-step explanation:

Let the two consecutive integers be [tex]$ (n - 1) $[/tex] and [tex]$ n $[/tex].

Given that their sum is -95.

⇒ (n - 1) + n = - 95

⇒ 2n - 1 = - 95

⇒ 2n = - 94

Dividing by 2 throughout, we get:

n = - 47

⇒ n - 1 = - 47 - 1 = - 48

Therefore, the two integers should be - 47 and - 48.

Let X = the number of hours the server works.

The equation for the amount Nick spends is 15x + 47

The next job pays 14x + 54

Set the equations to equal and solve for x:

15x + 47 = 14x +54

Subtract 14x from both sides:

X + 47 = 54

Subtract 47 from both sides:

X = 7

Now replace x with 7 in the second equation:

14x + 54 = 14(7) + 54 = 98 +54 = 152

The job would pay $152 and be 7 hours long.

Answer:

It's your friend at school. 8360 is your answer.

Step-by-step explanation:

88 + 88

= 176 + 88

= 264 + 88

= 352 + 88

= 440 + 88

= 528 + 88

= 616 + 88 =

704 x 88

= 8360

Answer:

8,360

Step-by-step explanation:

88 + 88 + 88 + 88 + 88 + 88 + 88 + 88 × 88 is our problem.

Instead of adding all of these 88's together, we will just multiply 88 by 8 because there are 8 of these 88's.

That will leave us with: 704

But that is not our answer.

We still have one more step, and it is to multiply 704 by 8.

After we do that we will have: 8,360 as our answer.

I hope this helps! :>

Answer:

410 / 23 = 17.8260869565

Lydia would need about 18 rolls of wallpaper to paper the walls of her kitchen.

Step-by-step explanation:

The answer will be 380 inches as of what I have done

Answer:

BC=16.992

Step-by-step explanation:

Answer:

16.992

Step-by-step explanation:

Answer:

-85

Step-by-step explanation:

Answer:

-85

Step-by-step explanation:

Answer:

Account C will give George at least a 4% annual yield.

Step-by-step explanation:

This problem can be solved by using simple maths formula given below

Year rate = (1+i)^n-1

where i = compounding rate and n = interval period

Now checking annual rate for each option using above formula

Account A)

Yr = 3.8% ( n= 12 and i= 3.75/12)

Account B)

Yr = 3.9 % ( n= 4 and i= 3.85/4)

Account C)

Yr = 4 % ( n= 365 and i= 3.95/365)

Answer:

Step-by-step explanation:

buying the windows and having them installed costs $ 7699.25

installation alone costs 3325

therefore, the cost of the windows is : 7699.25 - 3325 = 4374.25

and if they are paying 230.25 per window, then there is

4374.25 / 230.25 = 18.997 windows

let me check something...

18 * 230.25 = 4144.50

19 * 230.25 = 4374.75

if ur numbers are written correctly, they can only buy 18 windows.....they would be short 50 cents from buying 19 windows

To determine the number of windows the Rayhills purchased, the total cost minus the installation cost was divided by the cost per window, leading to an estimate of approximately 19 windows.

To find out how many windows Rayhills bought, we need to subtract the installation cost from the total cost and then divide the result by the cost per window. The total cost of buying the windows and having them installed is $7,699.25, and the company charges $3,325 for the installation. If each window costs $230.25, we can calculate the number of windows as follows:

Now let's perform the division:

$4,374.25 / $230.25 ≈ 19 windows (rounded to the nearest whole number since you can't buy a fraction of a window).

Therefore, the Rayhills bought approximately 19 windows.

Answer:

y-4=-2(x+4)

Step-by-step explanation:

Perpendicular means negative reciprocal of the slope.

y-y1=m(x-x1)

y-4=-2(x-(-4))

y-4=-2(x+4)

The equation of the line that passes through the point (-4, 4) and is perpendicular to the line [tex]y = \frac{1}{2}x + 1[/tex] is [tex]y = -2x - 4[/tex].

To find the equation of the line that passes through the point (-4, 4) and is perpendicular to the line [tex]y = \frac{1}{2}x + 1[/tex], we need to do the following steps:

Determine the slope of the given line:
The slope of the given line [tex]y = \frac{1}{2}x + 1[/tex] is [tex]\frac{1}{2}[/tex].

Find the slope of the perpendicular line:
The slopes of two perpendicular lines are negative reciprocals of each other. Therefore, if the slope of the given line is [tex]\frac{1}{2}[/tex], the slope of the perpendicular line will be [tex]-2[/tex] (since [tex]-1 \div \frac{1}{2} = -2[/tex]).

Use the point-slope form to find the equation:
The point-slope form of a line is [tex]y - y_1 = m(x - x_1)[/tex], where [tex](x_1, y_1)[/tex] is a point on the line, and [tex]m[/tex] is the slope.
Here, the point is [tex](-4, 4)[/tex] and the slope [tex]m = -2[/tex].

Substituting these values into the point-slope form, we get:
[tex]y - 4 = -2(x + 4)[/tex]

Simplify the equation:
Distribute the slope on the right side:
[tex]y - 4 = -2x - 8[/tex]

Add 4 to both sides to solve for [tex]y[/tex]:
[tex]y = -2x - 4[/tex]

Answer:

Percent change is −31.19%.

Step-by-step explanation:

It is given that "a price of a dirt bike dropped from $872 to $600".

Initial price = $872

New price = $600

We need to find the Percent change.

[tex]\text{Percent change}=\dfrac{\text{New price - Initial price}}{\text{Initial price}}\times 100[/tex]

[tex]\text{Percent change}=\dfrac{600-872}{872}\times 100[/tex]

[tex]\text{Percent change}=\dfrac{-272}{872}\times 100[/tex]

[tex]\text{Percent change}=-31.19266[/tex]

[tex]\text{Percent change}\approx -31.19[/tex]

Therefore, the percent change is −31.19%.

Final answer:

The chance change in the price of the dirt bike is-31.2, indicating a  drop in price from$ 872 to$ 600.

Explanation:

To calculate the chance change in the price of a dirt bike that dropped from$ 872 to$ 600, we follow this formula given in the reference  Chance change = (( change in value)/( original value)) × 100  

First, we find the change in value, which is the new price abated from the original price $ 600-$ 872 = -$ 272  also, we divide this change by the original price  -$ 272/$ 872 ≈-0.312  

Eventually, to find the chance change, we multiply this bit by 100  -0.312 × 100 = -31.2  This means the price of the dirt bike  dropped by31.2.

You know that m<TSU=m<PSR because they make a vertical angle pair.

So...m<PSR=4x.

If, m<QSR=3x then m<PSQ=4x-3x=x because they are adjacent.

And, m<PQS=180-68=112 degrees (linear pair).

Now we know all the interior angles in triangle PQS so we can say that...

x+x+112=180 (sum of interior angles in triangle)

2x=68

x=34 degrees

The requried value of x is 34 in each case, as of the given conditions

The triangle is a geometric shape that includes 3 sides and the sum of the interior angle should not be greater than 180°

Here,
As given in the question,

PU ∩ RT = S, Q ∈ PR

Since the sum of the remote interior angles is equal to the sum of the remote exterior angles.
∠SPQ + ∠PSQ = ∠SQR
∠PSQ = 68 - x

Now, When two lines intersect each other, their alternate angles are equal,
∠TSU = ∠PSR
4x = ∠PSQ + ∠QSR
4x = 68 - x + 3x
2x = 68
x = 34

Thus, the requried value of x is 34.

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Answer:

4/3

Step-by-step explanation:

m=(y2-y1)/(x2-x1)

m=(-20-(-8))/(-12-(-3))

m=(-20+8)/(-12+3)

m=-12/-9

m=12/9

simplify

m=4/3

Answer:

Tax on her normal wage is $ 6755

She can work 176 hours in a year and still remain in the same tax category

Step-by-step explanation:

Becky earns $633.81 for a 40 hour week.

Her normal monthly  wage is $ 633.81 *4 = $ 2535.24

Her annual income is $ 2535.2 *12=  $ 30,422.88

Tax on her annual income is  $30,422.88- $ 20,700= $9722.88

Tax on $ 20,700 is $ 3060

Tax on remaining $ 9722.88 is 0.38*9722.88=  $36,95

Total tax is $ 3060 + $ 3695= $ 6755

She can earn up to ( $ 36,000- $ 30, 423=) $ 5577 to remain in the same tax bracket.

She earns  $633.81/40= $ 15.84 in one hour

She gets twice the wage for the overtime  $ 31. 69

No of hours she can work = $5577/ $ 31.69 = 175.98 almost 176 hours

The equivalent expressions to [tex]5^2/5^8[/tex] are [tex]5^(^-^6^)[/tex] and [tex]\frac{1}{5^6}[/tex]

The expression[tex]5^2/5^8[/tex]can be simplified using the properties of exponents. When you have the same base raised to different exponents and they are being divided, you can subtract the exponents.

So, [tex]5^2/5^8 = 5^(^2^-^8^)[/tex] = 5^(-6).

Therefore, the equivalent expressions are [tex]5^(^-^6^)[/tex] and [tex]\frac{1}{5^6}[/tex]

In mathematics, equivalent expressions are algebraic expressions that have the same value for all possible values of the variables involved

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The expression 5^2/5^8 simplifies to 5^-6 due to the laws of exponents that allow subtracting of exponents when dividing like bases.

The expression 5^2/5^8 can be simplified by understanding the laws of exponents. When dividing like bases, you subtract the exponent in the denominator from the exponent in the numerator. By applying this law, the expression 5^2/5^8 simplifies to 5^(2-8) which equals 5^-6.

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Answer:

[tex]\text{Area of quadrilateral QRST}=216[/tex]

Step-by-step explanation:

Please find the attached image for the quadrilateral QRST.

We have been given that [tex]QS=18 \text{ and } RT=24[/tex]. We are asked to find the area of quadrilateral QRST.

Upon looking at the image of quadrilateral QRST, we can see that quadrilateral QRST is a kite as it has one pair of diagonally opposite equal angles (98 degrees).

We know that area of kite is equal to half the product of its diagonal.

We can see from our image that QS and RT are diagonal of kite, so the area of quadrilateral QRST would be:

[tex]\text{Area of quadrilateral QRST}=\frac{QS\cdot RT}{2}[/tex]

[tex]\text{Area of quadrilateral QRST}=\frac{18\cdot 24}{2}[/tex]

[tex]\text{Area of quadrilateral QRST}=\frac{432}{2}[/tex]

[tex]\text{Area of quadrilateral QRST}=216[/tex]

Therefore, the area of quadrilateral QRST is 216 square units.

Answer:

Step-by-step explanation:

[tex]1+3+9+27+81\\\\a_1=1,\ a_2=3,\ a_3=9,\ a_4=27,\ a_5=81\\\\r=\dfrac{a_{n+1}}{a_n}\Rightarrow r=\dfrac{a_2}{a_1}=\dfrac{a_3}{a_2}=\dfrac{a_4}{a_3}=\dfrac{a_5}{a_4}\\\\r=\dfrac{3}{1}=\dfrac{9}{3}=\dfrac{27}{9}=\dfrac{81}{27}=3[/tex]

Good evening ,

Answer:

Step-by-step explanation:

Since it’s a geometric series then

a₁ = 1 ( because 1 is the first term of the series)

3/1 = 9/3 = 27/9 = 81/27 = 3 then r=3.

:)

Answer:

x = -8

Step-by-step explanation:

-2x-16+0 = 0

Combine like terms.

-2x-16 = 0

Add 16 to both sides (to isolate the -2x from -16)

-2x = 16

Divide both sides by -2 to isolate the x from -2

x = -8

well 4 is basically 1/5 of 20, so that would be about 20%

and since 4 x 4 is 16, my guess is that 16 is 80% of 20

Answer: 80%

Step-by-step explanation: To answer the question what percent of 20 is 16, we translate the question into an equation.

So we have "what percent" that's x, "of 20" means times 20, "is 16" = 16. So now we have the equation (x)(20) = 16.

Now to solve for x, since x is being multiplied by 20, we need to divide by 20 on both sides of the equation. On the left side of the equation the 20's cancel and we have x and on the right side of the equation we have 16 divided by 20 which is 0.8.

Now, remember that we want to write our answer as a percent so we need to convert 0.8 into a percent.

We can convert a decimal into a percent by moving the decimal point two places to the right so we can change 0.8 to a percent by simply moving the decimal point two places to the right and we have 80%.

This means that 80% of 20 is 16.

Answer:

No, it's not a counterexample, because 48 is a multiple of 8

Step-by-step explanation:

I think like 4 x 3 + 8 x 2 = 28 is a counterexample, because 28 is not a multiple of 8.

Perl's conjecture that the sum of a multiple of 4 and a multiple of 8 must be a multiple of 8 is incorrect. In fact, the sum would definitely be a multiple of 4 but not necessarily a multiple of 8, since the sum of integers multiplied by 4 does not guarantee that it's a multiple of 8.

The subject of this question is Mathematics, specifically dealing with number theory and the properties of multiples and factorization. Perl's conjecture is that the sum of a multiple of 4 and a multiple of 8 must be a multiple of 8. We can examine this by understanding how multiples work.

A multiple of 4 can also be considered as 4n, where n is an integer. Similarly, a multiple of 8 is 8m, where m is integer. If we add both, we get 4n+8m. Now, as per distributive property of multiplication over addition, the expression becomes 4n+4(2m), which can be further simplified as 4(n+2m).

We can now see that the sum is actually 4 times the sum of some integer, so it must indeed be a multiple of 4. However, since it's not necessarily the case that 'n+2m' itself is a multiple of 2, we cannot say that the overall sum is guaranteed to be multiple of 8.

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Answer:

Step-by-step explanation:

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To calculate the probability of a student scoring 75% or above, count how many scores are 75 or higher, and then divide by the total number of scores. In this case, the probability is 6 out of 15, or 0.4.

The question asks about the probability that a student picked at random scored 75% or above based on a list of exam scores.

To find this probability, one must count the number of scores that are 75% or higher and divide by the total number of scores.

Out of the provided scores (61, 74, 60, 70, 83, 90, 90, 80, 100, 102, 28, 75, 98, 21, 49), we see that there are 6 scores that are 75 or above (83, 90, 90, 80, 100, 102, 98, 75).

Since there are 15 scores in total, the probability is calculated as -

= 6/15

= 0.4.

Answer:

y=-2/3x+20/3

Step-by-step explanation:

m=(y2-y1)/(x2-x1)

m=(4-6)/(4-1)

m=-2/3

y-y1=m(x-x1)

y-6=-2/3(x-1)

y=-2/3x+2/3+6

y=-2/3x+2/3+18/3

y=-2/3x+20/3