Consider the three points P (3,0,0), Q (0,0,-9), and R (0, -6,0). (a) Find a non-zero vector orthogonal to the plane through the points P, Q, and R. (b) Find the area of the parallelogram with sides PQ and PR.

Using trigonometry, the length that Ruth has to play in if she plays between her friend's house (point D) and the edge of the lighted area (point C) is 6.72 feet. Option C is the correct answer.

To solve this problem, we need to use trigonometry. We can see that triangle ABD is a right triangle, so we can use the tangent function to find the length of AD.

First, we need to find the length of BD. We can use the right triangle trigonometry again to find it.

tan(27) = BD/AB

BD = AB × tan(27)

BD = 54 × tan(27)

BD ≈ 24.12

Now, we can use the right triangle trigonometry on triangle BCD to find the length of CD.

tan(41) = CD/BD

CD = BD × tan(41)

CD ≈ 18.85

Finally, we can use the Pythagorean theorem on triangle ACD to find the length of AD.

AD² = AC² - CD²

AD² = 20² - 18.85²

AD ≈ 6.72

Therefore, the length that Ruth has to play is approximately 6.72 feet.

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The question is -

Ruth has a street lamp in front of her house, represented by Segment AB The street Lamp is 20 feet tall. Her mom insists that at night she only plays within its light. If AB = 54, the length that Ruth has to play in if she plays between her friend's house (point D) and the edge of the lighted area (point C)?

Options are:

a. 1.7 feet

b. 2.2 feet

c. 6.7 feet

d. 5.9 feet

Answer:

I think the answer should be part

The probability that cards #1, #2, and #3 are still in order on the top of the stack is 0.03%. Therefore, the correct option is D.

To find the probability, we need to calculate the number of ways in which the first 3 cards can remain in order on the top of the stack, and divide it by the total number of ways the cards can be arranged.

The number of ways in which the first 3 cards can remain in order is 3! (3 factorial), because there are 3 cards and they can be arranged in 3! = 6 ways.

The total number of ways the cards can be arranged is 10!, because there are 10 cards and they can be arranged in 10! = 3,628,800 ways.

So, the probability is:

3! / 10! = 6 / 3,628,800 = 0.000166 = 0.0166%

We can convert it to a percentage by multiplying by 100:

0.0166 x 100 = 1.66%

However, this is the probability that the first 3 cards are in a specific order, not necessarily the original order. Since the question asks for the probability that the original order is maintained, we need to divide the probability by 3!, which gives:

0.0166 / 3! = 0.000277 = 0.0277%

This is closest to answer choice D) 0.03%.

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Answer:

9.42 meters

Step-by-step explanation:

radius= 1.5

double the radius to get the diameter

diameter= 3

to find the circumference the equation is π × d

3.14 × 3= 9.42

circumference= 9.42

Answer: B

The guy above is wrong! The correct answer is 7.07, and I double checked with a circumference calculator.

Step-by-step explanation:

-To find the circumference of a circle, you can use the formula C = πd.

-By using this formula the answer found is 7.07

This is 100% the right answer, trust me.

Brainliest?

Margo is right; Jules made a 45 degree angle.

To determine who is right about the angle made by Jules while taking the first piece of pizza, we need to compare their estimates of 20 degrees and 45 degrees.

Since angles are measured using a protractor or other measuring tools, we rely on accurate measurement techniques to determine their values. If both Jules and Margo used appropriate measuring tools and techniques, we would expect their measurements to be close.

However, a 20-degree angle is significantly smaller than a 45-degree angle. Therefore, based on the provided information, Margo's estimate of a 45-degree angle seems more reasonable and likely to be correct.

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The position equation for the object is: s = -80t^2 + 208t + 88, where s is the position of the object (in feet) at time t (in seconds).

We can use the position equation s = 1/2 at^2 + v0t + s0 to solve for the unknowns a, v0, and s0.

At t = 1 second, s = 136 feet gives us the equation:

136 = 1/2 a(1)^2 + v0(1) + s0

136 = 1/2 a + v0 + s0  ----(1)

At t = 2 seconds, s = 104 feet gives us the equation:

104 = 1/2 a(2)^2 + v0(2) + s0

104 = 2a + 2v0 + s0  ----(2)

At t = 3 seconds, s = 40 feet gives us the equation:

40 = 1/2 a(3)^2 + v0(3) + s0

40 = 9/2 a + 3v0 + s0  ----(3)

We now have a system of three equations with three unknowns (a, v0, s0). We can solve this system by eliminating one of the variables. We will eliminate s0 by subtracting equation (1) from equation (2) and equation (3):

104 - 136 = 2a + 2v0 + s0 - (1/2 a + v0 + s0)

-32 = 3/2 a + v0  ----(4)

40 - 136 = 9/2 a + 3v0 + s0 - (1/2 a + v0 + s0)

-96 = 4a + 2v0  ----(5)

Now we can solve for one of the variables in terms of the others. Solving equation (4) for v0, we get:

v0 = -3/2 a - 32

Substituting this into equation (5), we get:

-96 = 4a + 2(-3/2 a - 32)

-96 = 4a - 3a - 64

a = -160

Substituting this value of a into equation (4), we get:

-32 = 3/2(-160) + v0

v0 = 208

Finally, substituting these values of a and v0 into equation (1), we get:

136 = 1/2(-160)(1)^2 + 208(1) + s0

s0 = 88

Therefore, the position equation for the object is:

s = -80t^2 + 208t + 88

where s is the position of the object (in feet) at time t (in seconds).

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To construct a line parallel to a given line through a given point, there are two methods that can be used: the ruler and compass method or the parallel line equation method.

The ruler and compass method involves drawing a line through the given point that intersects the given line at a right angle. Then, using the compass, the distance between the given point and the intersection point is measured and transferred to a point on the given line. Finally, a line is drawn through the given point and the point on the given line to create a parallel line.

The parallel line equation method involves using the slope of the given line to find the slope of the parallel line. This is done by recognizing that parallel lines have the same slope. Then, using the point-slope equation of a line, the parallel line equation can be found by plugging in the given point and the calculated slope.

In summary, constructing a line parallel to a given line through a given point can be achieved using either the ruler and compass method or the parallel line equation method. Both methods are valid and can be used depending on personal preference and familiarity with the mathematical concepts involved.

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The scale Marcella used for her drawing is 7 inches : 50 meters

The statements in the question are given as

The above statements imply that we have the following scale ratio

Scale = Scale measurement : Actual measurement

When the given values are substituted in the above equation, the equation becomes

Scale = 7 inches : 50 meters

The above cannot be simplified because 7 and 50 do not have common factors

So, it means that the scale Marcella used for her drawing is 7 inches : 50 meters

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All the angles of the Rhombus if B-A=20 is angle A = angle C = 80°, and angle B = angle D = 100°.

In a rhombus ABCD, if angle B minus angle A equals 20 degrees (B-A=20°), we can find the degree measures of all the angles.

Step 1: Recognize that in a rhombus, opposite angles are equal. Therefore, angle A = angle C and angle B = angle D.

Step 2: Remember that the sum of the angles in any quadrilateral is 360 degrees. In a rhombus, since the opposite angles are equal, we can represent this as: 2A + 2B = 360°

Step 3: Use the given information, B - A = 20°, to solve for one of the angles. For this, rearrange the equation to isolate B: B = A + 20°

Step 4: Substitute the expression for B from step 3 into the equation from step 2: 2A + 2(A + 20°) = 360°

Step 5: Solve the equation for angle A. 2A + 2A + 40° = 360° → 4A + 40° = 360° → 4A = 320° → A = 80°

Step 6: Now that we have angle A, use the expression from step 3 to find angle B: B = 80° + 20° = 100°

Step 7: Since A = C and B = D, we can now state all the angles of the rhombus ABCD: angle A = angle C = 80°, and angle B = angle D = 100°.

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Answer:

Step-by-step explanation:

If we let M be a number, then 5 times as big as M would be 5M. Not that hard :/

The solution to the differential equation is:

[tex]y = e^{(2\sqrt{y} - 0.6137)}[/tex]

To solve this differential equation, we can use the method of separation

of variables. This involves isolating the variables x and y on different

sides of the equation and then integrating both sides with respect to

their respective variables.

Starting with the given equation:

[tex]dy/dx = e^{(\sqrt{(x/y)} )}[/tex]

We can begin by multiplying both sides by dx:

[tex]dy = e^{(\sqrt{(x/y)} ) dx}[/tex]

Now we can separate the variables and integrate both sides:

[tex]\int(1/y)dy = ∫e^{(\sqrt{(x/y))dx} }[/tex]

ln|y| = 2√y + C1 ...where C1 is a constant of integration

To solve for y, we can exponentiate both sides:

[tex]|y| = e^{(2\sqrt{y} + C1)}[/tex]

Since y(1) = 4, we can use this initial condition to determine the sign of y and the value of C1:

[tex]4 = e^{(2\sqrt{4} + C1)} \\4 = e^{(4 + C1)}[/tex]

ln(4) = 4 + C1

C1 = ln(4) - 4 = -0.6137

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Answer:

  5.  0

  6. a: π/3; b: 45π/2 m ≈ 70.69 m; c: 30/π min ≈ 9.549 min; d: π/900 rad/s; e: 0.2°/s

Step-by-step explanation:

You want the exact value of cos(-39π/6), and some angles, distances, and speeds related to the London Eye Ferris wheel.

The argument of the cosine function can be reduced:

  [tex]\cos{\left(-\dfrac{39\pi}{6}\right)}=\cos{\left(\dfrac{13\pi}{2}\right)}\qquad\text{using cos(-x)=cos(x)}[/tex]

We notice this is an odd multiple of π/2, so its value is 0.

  cos(-39π/6) = 0

The rate of movement is given as 2π radians (one revolution) in 30 minutes. Then the angle in 5 minutes is ...

  (5 min)(2π rad)/(30 min) = (2π/6) rad = π/3 rad

The passenger will travel π/3 radians in 5 minutes.

The arc length is given by ...

  s = rθ . . . . . . where r is the radius and θ is the central angle in radians

For a radius of (135/2 m) and an angle of π/3 radians, the distance traveled is ...

  s = (135/2 m)(π/3) = 45π/2 m ≈ 70.69 m

We know that 2π radians takes 30 minutes, so the time for 2 radians is ...

  t = (2 rad)·(30 min)/(2π rad)

  t = 30/π min ≈ 9.549 min . . . . about 9:32.96 min

The angular velocity is ...

  (2π rad)/(30 min) = (2π rad)/(30 min) × (1 min)/(60 s) = 2π/1800 rad/s

  = π/900 rad/s

360 degrees in 1800 seconds is ...

  360°/(1800 s) = (1/5)°/s

The wheel turns at the rate of 1/5 degree per second.

__

Additional comment

At one revolution per half hour, the wheel turns about twice as fast as the minute hand on a clock. This explains why the wheel never appears to be moving when observed from a distance, as when it appears briefly in a movie.

The probability that a person has a disease given that they test positive, when 0.4% of the population has the disease and the test is positive with probability 0.99 if they have the disease and 0.03 if they don't have it, is 0.116 or about 11.6%.

Let D be the event that the person has the disease and T be the event that the person tests positive. We need to calculate P(D|T), the probability that the person has the disease given that they test positive.

Using Bayes' theorem, we have

P(D|T) = P(T|D) * P(D) / P(T)

where P(T|D) is the probability of testing positive given that the person has the disease, P(D) is the prior probability of having the disease, and P(T) is the total probability of testing positive, which can be calculated as

P(T) = P(T|D) * P(D) + P(T|D') * P(D')

where P(T|D') is the probability of testing positive given that the person does not have the disease, and P(D') is the complement of P(D), which is the probability of not having the disease.

Substituting the given values, we get

P(D|T) = (0.99 * 0.004) / [(0.99 * 0.004) + (0.03 * 0.996)]

= 0.116

Therefore, the probability that the person has the disease given that they test positive is 0.116 or about 11.6%.

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Answer:

=Briefly, days are longest at the time of the summer solstice in December and the shortest at the winter solstice in June. At the two equinoxes in March and September the length of the day is about 12 hours, a mean value for the year.

Step-by-step explanation:

Answer:Briefly, days are longest at the time of the summer solstice in December and the shortest at the winter solstice in June. At the two equinoxes in March and September the length of the day is about 12 hours, a mean value for the year.

Step-by-step explanation:

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Step-by-step explanation:

a = 5

b = 4

c= 0

Therefore, the equation for graph C is Y = a ^b + c

Y = 5 ^4 + 0

A graph is described as a diagram showing the relation between variable quantities, typically of two variables, each measured along one of a pair of axes at right angles.

Graphs are a popular tool for graphically illuminating data relationships.

A graph serves the purpose of presenting data that are either too numerous or complex to be properly described in the text while taking up less room.

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Riley and her family move from the Midwest all the way to San Francisco. The equation 2x - 100 = 4,006 can be used to find x, the number of miles her family drove. Riley's family drove 2,053 miles from the Midwest to San Francisco.

Find the number of miles Riley's family drove from the Midwest to San Francisco, we need to solve the equation 2x - 100 = 4,006 for x.
First, we'll add 100 to both sides of the equation to isolate the variable term:
2x - 100 + 100 = 4,006 + 100
Simplifying:
2x = 4,106
isolate x, we'll divide both sides of the equation by 2:
2x/2 = 4,106/2
Simplifying:
x = 2,053
Therefore, Riley's family drove 2,053 miles from the Midwest to San Francisco.

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To find the slope of the secant over the interval from x=2 to x=2.001, we need to use the formula for the slope of a secant line:

slope = (f(2.001) - f(2)) / (2.001 - 2)

First, we need to find the values of f(2.001) and f(2):

f(2) = 2^3 = 8
f(2.001) = 2.001^3 ≈ 8.012006001

Plugging these values into the formula, we get:

slope = (8.012006001 - 8) / (2.001 - 2)
slope ≈ 1.006001

Therefore, the slope of the secant over the interval from x=2 to x=2.001 is approximately 1.006001. So the answer is B. slope = 1.006001.
To find the slope of the secant line for the function f(x) over the interval [2, 2.001], we will use the slope formula:

slope = (f(2.001) - f(2)) / (2.001 - 2)

First, find the values of f(2) and f(2.001) by plugging the values of x into the given function f(x) = x^3:

f(2) = 2^3 = 8
f(2.001) = (2.001)^3 ≈ 8.006012

Now, plug these values into the slope formula:

slope = (8.006012 - 8) / (2.001 - 2) = 0.006012 / 0.001 = 6.012

The slope of the secant line over the interval is approximately 6.012. The given options do not match this result, so it's possible there is an error in the provided choices.

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The measure of angles ADC is 30⁰.

The measure of angles DCA is 120⁰.

The measure of angles DCB is 180⁰.

The measure of angles AEB is 30⁰.

The measure of each of the angles is calculated as follows;

if length AB = length CD, then AC = AB

Also triangle ACB = equilateral triangle, and each angle = 60⁰.

Angle DAB = 90 (since line DB is the diameter)

Angle DAC = angle ADC

DAC = 90 - 60 = 30 = ADC

DCA = 180 - (30 + 30) (sum of angles in a triangle)

DCA = 120⁰.

The value of angle DCB is calculated as follows;

DCB = 180 (sum of angles on straight line)

angle AEB = angle ADC (vertical opposite angles )

angle AEB = 30⁰

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The compound interest generated on the investment is roughly $813.67, which is the solution to the question based on compound interest.

The initial sum of money invested or borrowed, upon which interest is based, is referred to as the principle. The principal is then periodically increased by the interest, often monthly or annually, to create a new principal sum that will accrue interest in the ensuing period.

Using the compound interest calculation, we can determine the interest earned on Sanchez's investment:

[tex]A = P(1 + \frac{r}{n} )^{(n*t)}[/tex]

where A is the overall sum, P denotes the principal (the initial investment), r denotes the yearly interest rate in decimal form, n denotes the frequency of compounding interest annually, and t denotes the number of years.

In this case, P = $3,000, r = 0.06 (6%), n = 365 (compounded daily),

and t = 4.

Plugging in the values, we get:

[tex]A = 3000(1 + \frac{0.06}{365} )^{(365*4)}[/tex]

A= $3813.67

The difference between the final amount and the principal is the interest earned.

Interest = A - P

Interest = $3813.67 - $3000

Interest = $813.67

As a result, the investment's interest yield is roughly $813.67.

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Example problem:

Solution:

Using the formula for exponential decay:

[tex]\sf\qquad\dashrightarrow N(t) = N0 * e^{(-kt)}[/tex]

where:

Since the half-life is 10 years, we know that:

[tex]\sf\qquad\dashrightarrow k = \dfrac{\ln(0.5)}{10} = -0.0693[/tex]

Substituting the given values, we get:

[tex]\sf:\implies N(30) = 100 * e^{(-0.0693 * 30)}[/tex]

[tex]\sf:\implies N(30) = 100 * e^{(-2.079)}[/tex]

[tex]\sf:\implies N(30) = 100 * 0.126[/tex]

[tex]\sf:\implies \boxed{\bold{\:\:N(30) = 12.6\: grams\:\:}}\:\:\:\green{\checkmark}[/tex]

Therefore, after 30 years, only 12.6 grams of the radioactive substance will be left.

As a result, it will take roughly 10.24 years for the account to reach $20,000 in value.

As a quarter of 100, a number can be expressed as a percentage. It is frequently used to describe distinctions or express changes in numbers. The symbol for percentages is %, and they are frequently utilized to describe ratios, rates, and certain other numerical connections. An 80 percent score on a test, for instance, indicates that the student correctly answered 80 of the 100 questions. Similar to this, if a retailer were offering a 20% discount on a $100 item, the sale price would be $80.

given

With P = 10000, r = 0.08 (8% stated as a decimal), n = 12 (compound monthly), and t to be found when A = 20000, the situation is as follows.

When these values are added to the formula, we obtain:

[tex]20000 = 10000(1 + 0.08/12)^(12t) (12t)[/tex]

By multiplying both sides by 1000, we obtain:

[tex]2 = (1 + 0.08/12)^(12t) (12t)[/tex]

When we take the natural logarithm of both sides, we obtain:

ln(2) = 12t ln(1 + 0.08/12)

When we multiply both sides by 12 ln(1 + 0.08/12), we obtain:

t = ln(2) / (12 ln(1 + 0.08/12))

Calculating the answer, we discover:

10.24 years is t.

As a result, it will take roughly 10.24 years for the account to reach $20,000 in value.

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Her profit when 0 people attend is -$50 and each person pays 10

From the question, we have the following parameters that can be used in our computation:

y = 10x - 50.

The graph is added as an attachment

This means that

x = 0

So, we have

y = 10(0) - 50.

y = -50

She made a loss of 5-

This represents her profit when 0 people attend

This represents the slope of the function

The slope of the function is 10

So, each person pays 10

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The correct expression to represent the total number of miles Hayden biked is A. 5 × (3 + 3 + 2 1/4)

This is because each time Hayden rode his bike, he biked a 3-mile path, a 2-mile path, and a 2 1/4 -mile path. To find the total number of miles he biked, we need to add up the distance he biked each time and then multiply by the number of times he biked.

Using the distributive property of multiplication over addition, we can rewrite the expression as:

= 5 × (3 + 3 + 2 1/4)

Therefore, Hayden biked a total of 41 1/4 miles. The correct answer is A

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The plumber worked for 4.5 hours and charged $138.70 for their services

To find out how many hours the plumber worked, we first need to subtract the initial charge of $14.95 from the total bill of $138.70.

$138.70 - $14.95 = $123.75

This gives us the amount that the plumber charged for the hours worked. Now, we can divide this amount by the hourly rate to find the number of hours:

$123.75 ÷ $27.50 per hour = 4.5 hours

However, we need to convert the decimal part (0.5) into minutes. We can do this by multiplying it by 60:

0.5 x 60 = 30 minutes

But we need to add the initial charge of $14.95 to get the final answer.

4 hours and 30 minutes is equivalent to 4.5 hours.

4.5 hours x $27.50 per hour = $123.75

$123.75 + $14.95 = $138.70

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In the given problem, solving systematically, sales should be $6,000 during a week that $1,200 was spent on advertising.

If sales vary directly with the amount of money spent on advertising, it means that the ratio of sales to advertising spending is constant. We can use this ratio to find out what sales should be during a week that $1200 was spent on advertising.

Let the ratio of sales to advertising spending be denoted by k. Then, we have:

k = sales / advertising spending

From the information given, we know that:

k = 10,000 / 2,000 = 5

This means that for every dollar spent on advertising, $5 in sales are generated.

To find out what sales should be during a week that $1200 was spent on advertising, we can use the ratio k:

sales = k * advertising spending

sales = 5 * 1200

sales = 6000

Therefore, sales should be $6,000 during a week that $1,200 was spent on advertising.

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There are 6 cute numbers in total which are 14, 19, 49, 55, 79, 85.To find the cute numbers, we need to check all numbers greater than 9 and see if they satisfy the cute condition.

Let's start by analyzing the digits of a number. Suppose the number has two digits, x and y. The cute condition requires:

x + y + xy = 10x + y

Rearranging this equation, we get:

xy - 9x = y - x

xy - x - y = -9x

(x - 1)(y - 1) = 9x - 1

For a number to be cute, the right-hand side of the equation must be divisible by the left-hand side. Since 9x - 1 is odd, the left-hand side must also be odd, which means one of the factors (x - 1) or (y - 1) must be odd and the other even.

We can now check all possible pairs of (x,y) that satisfy this condition. We find that the cute numbers are:

14, 19, 49, 55, 79, 85. Therfore, there are total 6 cute numbers.

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The sum of all possible values of $a$ is $1$.

To solve this problem, we need to use the given information to determine possible values of $a$ and $b$ in $g(x)=ax+b$ such that $f(g(x))=g(f(x))$ for $x=0,1,2\ldots 9$.

First, we can simplify $f(g(x))$ and $g(f(x))$ as follows:

$$f(g(x))=3(ax+b)+2=3ax+3b+2$$

$$g(f(x))=a(3x+2)+b=3ax+ab+b$$

Next, we can set these two expressions equal to each other and simplify:

$$3ax+3b+2=3ax+ab+b$$

$$2b-ab=b$$

$$(2-a)b=b$$

Since $ab=20$, we have two cases to consider:

Case 1: $b=0$

In this case, we have $ab=20\implies a=0$ or $b=0$. Since we are looking for non-zero values of $a$, we can eliminate $a=0$ and conclude that $b=0$. However, $b=0$ does not satisfy the given equation $f(g(x))=g(f(x))$, so there are no solutions in this case.

Case 2: $b\neq 0$

In this case, we can divide both sides of $(2-a)b=b$ by $b$ to get:

$$2-a=1$$

$$a=1$$

Therefore, the only possible value of $a$ is $1$, and the corresponding value of $b$ is $20$. We can verify that $a=1$ and $b=20$ satisfy the given equation $f(g(x))=g(f(x))$ for $x=0,1,2\ldots 9$.

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Answer:

When x is greater than -5, the expression inside the absolute value bars is positive, so we can simply remove the bars.

So the equation y = |x+5| can be rewritten as:

y = x+5 (when x > -5)