6TH GRADE MATH FIND THE SLOPE OF THE TABLE, TYSM

Suppose you look by eye at a star near the edge of a dusty interstellar cloud. The star will look dimmer than it would if it were outside the cloud.

Interstellar clouds are vast, dense regions of dust and gas that exist between stars. These clouds contain tiny solid particles of dust, which scatter and absorb light as it passes through them.

When we observe a star that is located near the edge of an interstellar cloud, the light from that star has to pass through a greater amount of dust before it reaches our eyes or telescopes. The dust scatters and absorbs some of the light, causing the star to appear dimmer than it would if it were outside the cloud.

This effect is similar to how the sun appears dimmer when it is viewed through a thick layer of clouds or fog. In both cases, the amount of light that reaches us is reduced due to the presence of a barrier that scatters and absorbs some of the light.

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The expression's fully factored form is:[tex]72x^{3} + 72x^{2} + 18x = 18x(4x^{2} + 1)(x + 1)[/tex]

Factored Value, also known as "trended value," is the base annual value plus a yearly inflation factor based on a variation in the cost if live that is not to exceed 2% and is set by the State Agency of Equalization.

Rewriting an expression as the sum of factors is referred to as factor expressions or factoring. For instance, 3x + 12y may be expressed as 3 (x + 4y), which is a straightforward equation. The computations get simpler in this method. Three or (x + 4y) were examples of factors.

We can factor out [tex]18x[/tex] from each term to simplify the expression:

[tex]72x^{3} + 72x^{2} + 18x = 18x(4x^{3} + 4x^{2} + 1)[/tex]

An expression enclosed in parentheses can now be calculated by grouping or factoring.

[tex]4x^{3} + 4x^{2} + 1 = (4x^{2} + 1)(x + 1)[/tex]

The expression's properly factored version has the following result,

[tex]72x^{3} + 72x^{2} + 18x = 18x(4x^{2} + 1)(x + 1)[/tex]

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When 3 < x < 5, y can be expressed as y = 3x - 6 without absolute value notation.

What is absolute value notation ?

Absolute value notation is a mathematical notation used to represent the magnitude or distance of a real number from zero. It is denoted by vertical bars or pipes around the number. For example, the absolute value of x is written as |x|.

When 3 < x < 5, the expression |x-3| evaluates to x-3, the expression |x+2| evaluates to x+2, and the expression |x-5| evaluates to 5-x. Therefore, we can rewrite the expression y = |x-3| + |x+2| - |x-5| as:

y = (x-3) + (x+2) - (5-x)

Simplifying this expression, we get:

y = 3x - 6

Therefore, when 3 < x < 5, y can be expressed as y = 3x - 6 without absolute value notation.

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Answer: first 4 are false last one is true

Step-by-step explanation:

1. Time taken to fill one community pool is 2hours 24 minutes.

2. Time taken to audit 30 files is 4 hours 41 minutes.

3. The time taken to erect is 54/7 hours or 7 hours 43 minutes

Time and work is a branch of mathematics that deals with the calculation of the amount of time required to complete a job or task by a worker or a group of workers working at a certain rate.

1. Job: Fill one community pool       Rate        Time         Work Completed

  Reserve water tower                     1/6            x                  x/6

  City water pipes                             1/4            x                  x/4

Solution: Fill one community pool;   [tex]\frac{x}{6} + \frac{x}{4} = 1[/tex]

Simplify, x = 12/5 hours or 2hours 24 minutes

Time taken to fill one community pool is 2hours 24 minutes.

2. Job: Work together       Rate        Time         Work Completed

  Mr. Dupree                       12/5           x                  12x/5

  Ms. Carmichael                16/4           x                  16x/4

Solution: If they work together, time taken to audit one file is

[tex]\frac{12x}{5} + \frac{16x}{4} = 1[/tex]

Simplify, x = 5/32 hours

So, time taken to audit 30 file, (5/32)×30 = 75/16 hours or 4 hours 41 minutes

Time taken to audit 30 files is 4 hours 41 minutes.

3. Job: Work together            Rate         Time         Work Completed

  Macon Construction           100/8           x                100x/8

  Thomson Masonry              200/12        x                 200x/12

Solution: If they work together, time taken to complete a work

[tex]\frac{100x}{8} + \frac{200x}{12} = 1[/tex]

Simplify, x = 6/175

Macon Construction 2 hours before so,

remaining work = 250 - (100×2hour/8) = 225 feet

then, the time taken to erect 225 feet rock facing by together is

6/175 × 225 = 54/7 hours or 7 hours 43 minutes

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1. Time taken to fill one community pool is 2hours 24 minutes. 2. Time taken to audit 30 files is 4 hours 41 minutes. and 3. The time taken to erect is 54/7 hours or 7 hours 43 minutes.

An audit is an independent review of financial statements and related documents in order to confirm their accuracy and compliance with relevant regulations.

1. Job: Fill one community pool       Rate        Time         Work Completed
 Reserve water tower                     1/6            x                  x/6
 City water pipes                             1/4            x                  x/4
Solution: Fill one community pool;  
Simplify, x = 12/5 hours or 2hours 24 minutes
Time taken to fill one community pool is 2hours 24 minutes.

2. Job: Work together       Rate        Time         Work Completed
 Mr. Dupree                       12/5           x                  12x/5
 Ms. Carmichael                16/4           x                  16x/4
Solution: If they work together, time taken to audit one file is
Simplify, x = 5/32 hours
So, time taken to audit 30 file, (5/32)×30 = 75/16 hours or 4 hours 41 minutes
Time taken to audit 30 files is 4 hours 41 minutes.

3. Job: Work together            Rate         Time         Work Completed
 Macon Construction           100/8           x                100x/8
 Thomson Masonry              200/12        x                 200x/12
Solution: If they work together, time taken to complete a work
Simplify, x = 6/175
Macon Construction 2 hours before so,
remaining work = 250 - (100×2hour/8) = 225 feet
then, the time taken to erect 225 feet rock facing by together is
6/175 × 225 = 54/7 hours or 7 hours 43 minutes

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Answer: f(-1/2) = 16/3

Step-by-step explanation:

Substituting -1/2 for x in the given function:

f(-1/2) = (-2/3)(-1/2) + 5

f(-1/2) = 1/3 + 5

f(-1/2) = 16/3

Therefore, f(-1/2) = 16/3.

Answer:

The simplest possible equation for the line on the graph would be x = - 2

Answer:

Step-by-step explanation:

Let x be the number of points scored by Team B.

Then, Team A scored twice as many points, or 2x.

The total number of points scored by both teams is 12, so we can set up the equation:

x + 2x = 12

Combining like terms, we get:

3x = 12

Dividing both sides by 3, we get:

x = 4

So Team B scored 4 points, and Team A scored twice as many, or 8 points.

(A) This means that the  systolic blood pressure 100 is less than average blood pressure and 150is higher than the average blood pressure.

(B)  The average size of such errors is about 125 mmHg, and 95% of the predictions we make using regression will be correct to within about 90 mmHg.

(C) He is 60 years old, which means that he is 1.7857 SD(s) above the average age of men in the practice.

(D) The percentile corresponding to -0.6 as27.43. this means that 27.43% of people with blood pressure reading above 125.

(A) We have to find the z statistics of systolic blood pressure 100 and 150.

We have:

z₁₀₀ = 100 -125/14

      = -1.7857

And, Z₁₅₀ = 150-125/14

               = 1.7857

So, the  systolic blood pressure 100 is -1.7857 standard deviation to the left of the mean 125.

And, the  systolic blood pressure 150 is 1.7857 standard deviation to the left of the mean 125.

This means that the  systolic blood pressure 100 is less than average blood pressure and 150is higher than the average blood pressure.

(B) The above predictions all are subject to error. The average size of such errors is about 125 mmHg, and 95% of the predictions we make using regression will be correct to within about 90 mmHg.

             -2.5 = x -125/14

           ⇒ x = -2.5 × 14 +125

           ⇒ x = 90.

(C)  A man is selected at random from the practice. He is 60 years old, which means that he is 1.7857 SD(s) above the average age of men in the practice. Another way of expressing his relative age is that he is at the 96% percentile of age among all men in the practice.

(D) The z score is: z = 0.6

The percentile corresponding to -0.6 as27.43. this means that 27.43% of people with blood pressure reading above 125.

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Answer:

Step-by-step explanation:

To calculate the interest earned by Linda for the first year, we can use the formula:

A = P(1 + r/n)^(nt)

Where A is the amount after t years, P is the principal amount, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the time in years.

For the first year, we have:

A = $50,000(1 + 0.06/1)^(1*1) = $53,000

So, the interest earned by Linda for the first year is:

Interest = $53,000 - $50,000 = $3,000

For the second year, we can use the same formula with t = 2:

A = $50,000(1 + 0.06/1)^(1*2) = $56,180

Interest = $56,180 - $53,000 = $3,180

For the third year, we can use the same formula with t = 3:

A = $50,000(1 + 0.06/1)^(1*3) = $59,468.80

Interest = $59,468.80 - $56,180 = $3,288.80

Now, to calculate the interest earned by Bob for each of the first three years, we can use the formula:

Interest = Prt

Where P is the principal amount, r is the annual interest rate, and t is the time in years.

For the first year, we have:

Interest = $50,0000.061 = $3,000

For the second year, we have:

Interest = $50,0000.061 = $3,000

For the third year, we have:

Interest = $50,0000.061 = $3,000

As we can see, Linda earns more interest than Bob for each year, as her interest is compounded annually, while Bob's interest is simple interest. Therefore, the answer is:

Linda earns more.

Answer:

Linda earns $9550.8 interest and bob earns $9000 interest

Step-by-step explanation:

Linda takes compound interest: C.I. = Principal (1 + Rate)Time − Principal

interest= 50,000(1+6/100)³

=59550.8 - 50000

Linda earns $9550.8 interest in 3 years.

bob takes simple interest: S.I = prt/100

interest = 50,000*6*3/100

Bob earns $9000 in 3 years.

thus, Linda earns more interest than bob.

Step-by-step explanation:

Speed =  distance / time    

  you are given distance = 23 miles  and time = .5 hr

      distance / time =  23 miles / .5 hr =   46 mph

Answer: 9x - 3x* - 3y + y* square meters

Step-by-step explanation:

(a) The expanded form of (a + b) 2 is:

(a + b) 2 = a2 + 2ab + b2

(b) The area of the rectangular mat is:

Area = Length × Breadth

Given that the length is 3x - y meters and the breadth is 3 - * meters.

So, the area of the rectangular mat can be calculated as:

Area = (3x - y) × (3 - *)

= 9x - 3x* - 3y + y*

Therefore, the area of the rectangular mat is 9x - 3x* - 3y + y* square meters.

Answer:

Step-by-step explanation:

In response to the stated question, we may state that As a result, the probability of correctly answering none of the 20 questions is roughly 0.0000262, or 0.00262%.

Probabilistic theory is a branch of mathematics that calculates the likelihood of an event or proposition occurring or being true. A risk is a number between 0 and 1, with 1 indicating certainty and a probability of around 0 indicating how probable an event appears to be to occur. Probability is a mathematical term for the likelihood or likelihood that a certain event will occur. Probabilities can also be expressed as numbers ranging from 0 to 1 or as percentages ranging from 0% to 100%. In relation to all other outcomes, the ratio of occurrences among equally likely alternatives that result in a certain event.

The likelihood of getting any one question accurate is 1/4 if you merely guess on each question. The binomial distribution formula may be used to calculate the chance of correctly answering none of the 20 questions:

P(X=0) = (n choose X) * pX * (1-p) (n-X)

[tex]If n=20, X=0, and p= 1/4\\P(X=0) = (20 pick 0) (20 choose 0) * (1/4)^0 * (3/4)^20\\P(X=0) = 1 * 1 * 0.0000262\\P(X=0) = 0.0000262[/tex]

As a result, the probability of correctly answering none of the 20 questions is roughly 0.0000262, or 0.00262%.

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Answer:

$31122.00

Step-by-step explanation:

We know

She works 35 hours a week, earning $17.10 an hour.

17.10 x 35 = $598.50 a week

How much does she earn in one year?

We Take

598.50 x 52 = $31122.00

So, she earns $31122.00 one year.

The Function of maximum or minimum for t is infinity.

While the principal subordinate can let us know if the capability is expanding or diminishing, the subsequent subsidiary. tells us in the event that the primary subsidiary is expanding or diminishing. On the off chance that the subsequent subsidiary is positive, the first.

To analyze the function R(t) = 2501² / e(3t), we can take the first and second derivatives:

R'(t) = -7503 * 2501² / e(3t)

R''(t) = 22509 * 2501² / e(3t)

To find the inflection point, we can set R''(t) = 0 and solve for t:

22509 * 2501² / e(3t) = 0

t = ln(0) / -3 = undefined

Since there is no real solution to this equation, there is no inflection point for this function.

To find the maximum or minimum, we can set R'(t) = 0 and solve for t:

-7503 * 2501² / e(3t) = 0

t = infinity

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The equation is saying that -12 is equal to -3 multiplied by y. To solve for y, divide both sides by -3. This would give an answer of 4.

An equation is a mathematical statement that expresses the equality or inequality of two values or expressions. It consists of two expressions connected by an equals sign, inequality sign or other relational operator. Equations can involve numbers, variables, and operations such as addition, subtraction, multiplication, division and exponentiation. An equation can be used to solve problems related to mathematics, science, engineering, finance, and many other disciplines. Equations can also be used to model and describe real-world phenomena.

t = 30

a = 12.5

p = -5.5

x = 2/3

y = 4.

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a. t = 30/2; To solve this equation, divide both sides by 2/5. The resulting equation is t = 30/2.

An equation is a mathematical statement that expresses the equality of two expressions by using an equals sign (=). It states that the two expressions on either side of the equals sign are equal in value. An equation is an example of a mathematical problem, which can be used to solve real-world problems.

b. a = 4.5; To solve this equation, add 8 to both sides. The resulting equation is a = 4.5.
c. p = -7/2; To solve this equation, add 3 to both sides. The resulting equation is p = -7/2.
d. x = 2; To solve this equation, divide both sides by 3. The resulting equation is x = 2.
e. y = 4; To solve this equation, divide both sides by -3. The resulting equation is y = 4.

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Answer:

340 seats in Theatre 2

Step-by-step explanation:

let n be the number of seats in Theatre 3 then the number of seats in theatre 2 is n + 150

summing and equating gives

27 0 + n + 150 + n = 800

420 + 2n = 800 ( subtract 420 from both sides )

2n = 380 ( divide both sides by 2 )

n = 190

then

number of seats in Theatre 2 = n + 150 = 190 + 150 = 340

Answer:

Theatre 3 has 190 seats, and Theatre 2 has 190 + 150 = 340 seats.

Step-by-step explanation:

Let x be the number of seats in Theatre 3.

Then the number of seats in Theatre 2 is x + 150.

And the total number of seats in the multiplex is 270 + x + (x + 150) = 800.

Simplifying the equation, we get

2x + 420 = 800

2x = 380

x = 190

Therefore, Theatre 3 has 190 seats, and Theatre 2 has 190 + 150 = 340 seats.

Answer:

The given function is an exponential growth function, not an exponential decay function because as the exponent x increases, the value of y also increases instead of decreasing.

To describe its end behavior using limits, we need to find the limit of the function as x approaches infinity and as x approaches negative infinity.

As x approaches infinity, the exponent -x approaches negative infinity, and the base (1/6) is raised to increasingly larger negative powers, causing the function to approach zero. So, the limit as x approaches infinity is 0.

As x approaches negative infinity, the exponent -x approaches infinity, and the base (1/6) is raised to increasingly larger positive powers, causing the function to approach infinity. So, the limit as x approaches negative infinity is infinity.

Therefore, the end behavior of the function is that it approaches zero as x approaches infinity and approaches infinity as x approaches negative infinity.

Answer: Supergirl = 19 and Wonder Woman = 8

Step-by-step explanation:

Let g represent the quantity of Supergirl keychains and w represent the quantity of Wonder Woman keychains.

                            Qty       Cost

Supergirl                 g         $3g

Wonder Woman     w       $2w

Total                       27       $73

Qty:     g +  w  = 27  →   -2(g +  w  = 27)    →    -2g - 2w = -54

Cost: 3g + 2w = 73  →     1(3g + 2w = 73)  →     3g + 2w =  73

                                                                           g          =  19

Input g = 19 into one of the original equations to solve for w:

g  + w = 27

(19) + w = 27

        w = 8

a. The value of P(2) is 0.022

b. The value of P(2 or fewer) is 0.027

From the question; n = 15, p = 0.4

a. We have to determine P(2).

P(X = x) = ⁿCₓ·Pˣ·(1 - P)ⁿ⁻ˣ

P(X = 2) = ¹⁵C₂·(0.4)²·(1 - 0.4)¹⁵⁻²

We can write ⁿCₓ = [tex]\frac{n!}{x!(n - x)!}[/tex]

P(X = 2) = [tex]\frac{15!}{2!(15 - 2)!}[/tex] · (0.16) · (0.6)¹³

P(X = 2) = [tex]\frac{15\times14\times13!}{2\times1\times13!}[/tex] · (0.16) · (0.6)¹³

P(X = 2) = (15 × 7) · (0.16) · (0.6)¹³

After simplification

P(X = 2) = 0.022(approx)

b. We have to determine P(2 or fewer).

P(x ≤ 2) = P(x = 0) + P(x = 1) + P(x = 2)

P(x ≤ 2) = ¹⁵C₀·(0.4)⁰·(1 - 0.4)¹⁵⁻⁰ + ¹⁵C₁·(0.4)¹·(1 - 0.4)¹⁵⁻¹ + ¹⁵C₂·(0.4)²·(1 - 0.4)¹⁵⁻²

After simplification like above

P(x ≤ 2) = 0 + 0.005 + 0.022

P(x ≤ 2) = 0.027

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The function f(x,y) has only minimum value at (1,0) is -1 and maximum value does not exist.

The given function is f(x,y)=x²+y²-2x

First find the partial derivative with respect to x and y

f'(x)=2x-2

f'(y)=2y

f'(x)=0=f'(y)

2x-2=0

x=1

and y=0

Now we will cheak maxima and minima at (1,0)

f''(x,y)=2 and f"(x,y)=2 and f"(x,y)=0( derivative of first order of x with respect to y)

We know that

rt-s²≥0 and r positive then f is minimum and  r negative maximum

r=2 , t=2 and s=0

rt-s²≥0 and r is positive so f(x,y) is minimum at (1,0)

f(1,0)=1-2=-1

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Answer:

Step-by-step explanation:

Since we know that x + 1 is a factor of f(x), we can use synthetic division to find the other factor and then solve for the remaining zeros.

We set up synthetic division as follows:

-1 | 1 -3 16 20

| -1 4 -20

|_____________

1 -4 20 0

The last row of the synthetic division gives us the coefficients of the quadratic factor, which is x^2 - 4x + 20. We can use the quadratic formula to find its roots:

x = (-(-4) ± sqrt((-4)^2 - 4(1)(20))) / (2(1))

= (4 ± sqrt(-64)) / 2

= 2 ± 2i√2

Therefore, the three zeros of f(x) are -1, 2 + 2i√2, and 2 - 2i√2.

Answer:

See step by step.

Step-by-step explanation:

lets define the events:
A: cuban festival        C: tropical Garden
B: street art show      D: african festival

a) theoretically the probability is

[tex]P(A)=P(B)=P(C)=P(D)= \frac{1}{4} = 0.25 \\[/tex]
This is 25% (for each one, equally)

b) The experimental probability is given by:

[tex]P(A)= \frac{32}{150} =0.2133[/tex]

[tex]P(B)= \frac{38}{150} =0.2533[/tex]

[tex]P(C)= \frac{35}{150} =0.2333[/tex]
[tex]P(D)= \frac{45}{150} =0.3000[/tex]

c) The theoretically probabilities are all equally, the experimental probabilities are close to 25% each one, but differ lightly each one, since is an experiment and the result is random.  

The standard deviation for the given mean length x is found to be: 2.138.

The term "standard deviation" (or "") refers to a measurement of the data's dispersion from the mean. A low standard deviation implies that the data are grouped around the mean, whereas a large standard deviation shows that the data are more dispersed.

Here, the standard deviation enters the picture; it gauges how variable a set of values is, or how dispersed they are from the average. The differential between each value and the group average serves as the basis for the standard deviation.

Given data:

mean μ = 86

standard deviation σ = 8

number of sample n = 14

average body length  x = 91.1

The population standard deviation is divided by that of the square root of both the sample size to calculate the standard deviation of the sampled distribution of the sample mean.

So,

σₓ = σ / √n

σₓ = 8 / √14

σₓ = 2.138

Thus, the standard deviation for the given mean length x is found to be: 2.138.

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The complete question is-

Deer mice (Peromyscus manicul.atus) are small rodents native of North America. Their adult body lengths (excluding tail) are known to vary approximately Normally, with mean 86 mm and standard deviation 8 mm. Deer mice are found in diverse habitats and exhibit different adaptations to their environment. A random sample of 14 deer mice in a rich forest habitat gives an average body length of x = 91.1 mm. Assume that the standard deviation σ of all deer mice in this area is also 8 mm.

What is the standard deviation of the mean length x?

Rumiya's total earnings can be represented by the inequality: [tex]85000 + 0.1x > 100000[/tex] and she would need to make sales of at least $150,000 to earn over $100,000 for the year.

What do you mean by commission and inequality ?

A commission is a percentage of sales that a salesperson earns on top of their base salary. In this case, Rumiya earns a 10% commission on sales she makes for the year. An inequality is a statement that compares two values, indicating whether one is greater than, less than, or equal to the other. It is used to represent that Rumiya needs to make sales that exceed a certain amount in order to earn a desired amount.

Finding the minimum amount of sales :

Rumiya's total earnings for the year will be the sum of her base salary and commission on sales. We can represent this as an inequality:

[tex]85000 + 0.1x > 100000[/tex]

To solve for [tex]x[/tex], we first need to isolate the variable on one side of the inequality. We can do this by subtracting 85000 from both sides:

[tex]0.1x > 15000[/tex]

Next, we can solve for [tex]x[/tex] by dividing both sides by 0.1:

[tex]x > 150000[/tex]

Therefore, Rumiya would need to make sales of at least $150,000 to earn over $100,000 for the year. This means that her commission on these sales would be $15,000 (10% of $150,000).

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Answer:

Step-by-step explanation:

Answer:

To convert the equation y - 7 = 3(x - 4) to standard form Ax + By = C, we need to rearrange it so that it has the form Ax + By = C, where A, B, and C are constants.

y - 7 = 3(x - 4)

y - 7 = 3x - 12 (distribute the 3)

3x - y = 7 - 12 (move y to the left-hand side)

3x - y = -5

Now we have the equation in standard form, where:

A = 3

B = -1

C = -5

Therefore, the values of A, B, and C are 3, -1, and -5, respectively.

This is the absolute value function shifted to the left 2 units:

| x+2 | = {

x+2 if x >= -2,

-(x+2) if x < -2

}

In mathematics, a function is a relationship between a set of inputs and a set of possible outputs with the property that each input is related to exactly one output. It is a rule or a set of rules that assigns each input value exactly one output value. Functions can be represented using equations, graphs, or tables. They are used to model real-world phenomena and solve problems in various fields such as science, engineering, economics, and finance.

Here,

The absolute value function is defined as:

| x | = {

x if x >= 0,

-x if x < 0

}

To shift the function to the left 2 units, we can replace x with (x+2):

| x+2 | = {

x+2 if x+2 >= 0,

-(x+2) if x+2 < 0

}

Simplifying further, we get:

| x+2 | = {

x+2 if x >= -2,

-(x+2) if x < -2

}

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Complete question:

The absolute value function is defined as:

| x | = {

x if x >= 0,

-x if x < 0

}

Find the function when shifted to the left 2 units.