The linear homogeneous constant-coefficient equation is y'' - 4y' + 2y = 0
A linear homogeneous constant-coefficient equation is of the form:
ay'' + by' + cy = 0
where a, b, and c are constants, and y is a function of x. The general solution of this equation can be written as:
[tex]y(x) = c_1e^{(r_1x)} + c_2e^{(r_2x)}[/tex]
where r₁ and r₂ are the roots of the characteristic equation:
ar² + br + c = 0
In our case, the general solution is given as:
y(x) = [tex]Ae^{2x}[/tex] + Bcos(2x) + Csin(2x)
To find the linear homogeneous constant-coefficient equation that has this general solution, we need to first write the general solution in the form:
[tex]y(x) = c_1e^{(r_1x)} + c_2e^{(r_2x)}[/tex]
where r₁ and r₂ are the roots of the characteristic equation. Comparing this form with the given general solution, we see that r₁ = 2, r₂ = 2i, and r3 = -2i.
Therefore, the characteristic equation is:
a(r-2)(r-2i)(r+2i) = 0
Expanding this equation, we get:
a(r³ - 2r²i - 2ri² + 4r² + 8i²r - 8i) = 0
Simplifying and grouping the real and imaginary terms, we get:
a(r³ - 4r) + 2a(r² + 4) = 0
Dividing by a, we get:
r³ - 4r + 2(r² + 4) = y'' - 4y' + 2y = 0
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Complete Question:
Find a linear homogeneous constant-coefficient equation with the given general solution
y(x)=Ae²ˣ+Bcos(2x)+Csin(2x)
The circumference of a circle is five pi centimeters what is the area in square centimeters express your answer in terms of pi
Answer:
Step-by-step explanation:
circumference c=2πr=5π
[tex]r=\frac{5\pi }{2\pi } =\frac{5}{2}~cm \\area=\pi r^{2} =\pi (\frac{5}{2} )^2\\=\frac{25}{4} \pi \\=6.25 ~cm^2[/tex]
What are the coordinates of point G on the coordinate grid below?
A
(-4,3)
(4,-3)
-2
4
2
O
-2
4
B
AY
D
2
(4,3)
(-4,-3)
4
G
XA
Answer:
The answer is G = (4,3)
Step-by-step explanation:
G = (4,3)
18 Look at the two inequalities below.
3x + 2 > 14
10-x≤7
Which value of x makes both inequalities true?
(A) 2
(B) 3
(C) 4
(D) 5
ANSWER :
Answer is D.
Thanks for the question today.
Find the generating functions and the associated sequences of: (x+4) ^ 4
Using binomial theorem, the generating function is G(x) = x^4 + 16x^3 + 96x^2 + 256x + 256 while the associated sequence of (x+4)^4 is {1, 16, 96, 256, 256}.
What is the generating functions and associated sequences of the functionTo find the generating function of (x+4)^4, we expand it using the binomial theorem:
[tex](x+4)^4 = C(4,0)x^4 + C(4,1)x^3(4) + C(4,2)x^2(4^2) + C(4,3)x(4^3) + C(4,4)(4^4)[/tex]
where C(n,k) denotes the binomial coefficient "n choose k".
Simplifying the terms, we get:
[tex](x+4)^4 = x^4 + 16x^3 + 96x^2 + 256x + 256[/tex]
Therefore, the generating function of (x+4)^4 is:
[tex]G(x) = x^4 + 16x^3 + 96x^2 + 256x + 256[/tex]
The associated sequence can be read off by finding the coefficients of each power of x:
The coefficient of x^k is the k-th term of the sequence.In this case, the sequence is given by the coefficients of G(x):a₀ = 256a₁ = 256a₂ = 96a₃ = 16a₄ = 1To find the generating function of (x+4)^4, we expand it using the binomial theorem:
(x+4)^4 = C(4,0)x^4 + C(4,1)x^3(4) + C(4,2)x^2(4^2) + C(4,3)x(4^3) + C(4,4)(4^4)
where C(n,k) denotes the binomial coefficient "n choose k".
Simplifying the terms, we get:
(x+4)^4 = x^4 + 16x^3 + 96x^2 + 256x + 256
Therefore, the generating function of (x+4)^4 is:
G(x) = x^4 + 16x^3 + 96x^2 + 256x + 256
The associated sequence can be read off by finding the coefficients of each power of x:
The coefficient of x^k is the k-th term of the sequence.
In this case, the sequence is given by the coefficients of G(x):
a₀ = 256
a₁ = 256
a₂ = 96
a₃ = 16
a₄ = 1
Therefore, the associated sequence of (x+4)^4 is {1, 16, 96, 256, 256}.
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A box containing 5 balls costs $8.50. If the balls are bought individually, they cost $2.00 each. How much cheaper is it, in percentage terms, to buy the box as opposed to buying 5 individual balls?
Answer: The total cost of buying 5 balls individually is $2.00 x 5 = $10.00.
The box costs $8.50, which means it is $10.00 - $8.50 = $1.50 cheaper to buy the box.
To calculate the percentage difference, we can use the formula:
% difference = (difference ÷ original value) x 100%
In this case, the difference is $1.50, and the original value is $10.00.
% difference = ($1.50 ÷ $10.00) x 100%
% difference = 0.15 x 100%
% difference = 15%
Therefore, it is 15% cheaper to buy the box than to buy 5 individual balls.
Step-by-step explanation:
there are 3/4 as many boys as girls in a class of fith-graders. if there are 35 students in the class how many are girls?
Therefore, there are 20 girls in the class of fifth-graders.
What is fraction?In mathematics, a fraction represents a part of a whole or a division of two numbers. It is written in the form of a numerator and a denominator separated by a horizontal bar (also known as a fraction bar). The numerator represents the number of equal parts being considered, while the denominator represents the total number of equal parts in the whole or the divisor of the division.
Here,
Let the number of girls in the class be represented by 'g'.
According to the problem, the number of boys is 3/4 as many as the number of girls. This means:
Number of boys = (3/4) * number of girls
Number of boys = (3/4) * g
The total number of students in the class is given as 35. So we have:
Number of girls + Number of boys = Total number of students
g + (3/4)g = 35
Simplifying the equation, we get:
(7/4)g = 35
Dividing both sides by 7/4, we get:
g = 20
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certain can of soda contains 60 milligrams of caffeine. The caffeine is
eliminated from the body at a rate of 15% per hour. What is the half-life of the
caffeine? That is, how many hours does it take for half of the caffeine to eliminated
from the body?
Answer:
3.33 hours
Step-by-step explanation:
you have a total of 60mg, it is asked how much hours it takes to remove half the caffeine which is 30mg
1 hour removes 15%, 15% of 60 is 9g
(15/100×60) = 9
now it removes only 9g in an hour, we have to find out how long (hours) it'll take to remove 30g
30÷9= 3.33 hours
In circle � W, � � = 3 WX=3 and m ∠ � � � = 8 0 ∘ ∠XWY=80 ∘ . Find the length of � � ⌢ XY ⌢ . Express your answer as a fraction times � π. X W Y Answer: Length of X Y ⌢ = Length of XY ⌢ =
The Length of arcXY is 4π/3
Define the term Circle identities?The Circle identities refer to a set of fundamental identities in trigonometry that relate the six trigonometric functions of an angle in a right-angled triangle.
Given, (radius) WX =3, it means, r = 3
given, m∠XWY = 80°
So, the length of arc XY;
arcXY = [tex]\frac{80}{360}* 2\pi r[/tex] (by formula: length of arc)
arcXY = [tex]\frac{2}{9}* 2\pi *3[/tex]
arcXY = [tex]\frac{4}{3}\pi[/tex]
Therefore, the Length of arcXY is 4π/3
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Complete question-
What is the degree measure of the smaller angle between the hour and minute hands of the clock at 5:10 PM?
F. 110°
G. 105°
H. 100°
J. 95°
K. 90°
Answer:
At 5:10 PM, the minute hand is at the 2-minute mark, and the hour hand is between the 5 and 6 marks, closer to the 5.
The minute hand is at the 2-minute mark, which is 1/30th of the way around the clock face. So, the minute hand has traveled 1/30th of a full circle, which is 360°, or 12°.
The hour hand is between the 5 and 6 marks, closer to the 5. It has passed the 5 and is 1/6th of the way to the 6. Since the clock face is divided into 12 hours, 1/6th of the way from 5 to 6 is 5 + 1/6 = 5.166... hours.
Each hour mark represents 30 degrees, so the hour hand has traveled 5.166... × 30 = 155 degrees from the 12 o'clock position.
The smaller angle between the hands is the angle between the hour hand and the minute hand that is less than 180 degrees. We can find this angle by subtracting the smaller angle between the hands from 360 degrees.
To find the smaller angle between the hands, we can subtract the angle traveled by the hour hand from the angle traveled by the minute hand:
12° - 155° = -143°
The negative sign indicates that the angle is measured clockwise from the 12 o'clock position. To find the positive angle between the hands, we add 360 degrees:
360° - 143° = 217°
Therefore, the degree measure of the smaller angle between the hour and minute hands of the clock at 5:10 PM is approximately 217 degrees. Since this value is not one of the answer choices, we can round it to the nearest choice, which is F. 110°
How do I solve? I don’t understand
Step-by-step explanation:
Use the 110 to find the 70 degree angle (they form a straight line = 180°)
then 70 + 64 + R angle = 180° ( sum of angles of a triangle)
then : R angle = 46°
then the R angle + 2x-10 = 90° ( because the two lines are perpendicular)
(2x -10)° + 46 ° = 90 °
x = 27
generally, cold fronts move fast er than warm fronts generally, cold fronts have steeper slopes generally, precipitation cover s a much broader area with a cold front especially in winter, cumuliform clouds are more often associated with cold fronts
Cold fronts generally move faster than warm fronts because cold air is denser and thus, moves more quickly. Precipitation with a cold front typically covers a broader area, especially during the winter.
On the other hand, warm fronts move more slowly as they are characterized by the gradual lifting of warm air over colder air. Cold fronts also typically have steeper slopes than warm fronts. This is because the leading edge of a cold front is more abrupt.
With a steep rise in the cold air mass. In contrast, the leading edge of a warm front has a gentler slope as the warm air gradually rises over the colder air.
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PLEASE HELP MEEE
whoever answers right gets brainliest!!!
Answer:
[tex]30\leq x[/tex] AND [tex]x \leq 106[/tex]
Notice the valid answer is the one with the AND since need to be both at the same time.
Step-by-step explanation:
Is the one is market, you add 6 to each side and you obtain that answer
[tex]24 \leq x-6\leq 100[/tex]
[tex]24 +6\leq x\leq 100+6[/tex]
[tex]30\leq x\leq 106[/tex]
in a class if 108 students, 60 like football, 53 like Tennis and 10 like neither. calculate the number of students who like football but not tennis
Answer:
60 - 10 = 50
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can someone help me please
△CDE∼△PQR. CD=9 m, EC=15 m, PQ=15 m. What is the length of RP?
Answer:
RP = 25
Step-by-step explanation:
since the triangles are similar then the ratios of corresponding sides are in proportion, that is
[tex]\frac{RP}{EC}[/tex] = [tex]\frac{PQ}{CD}[/tex] ( substitute values )
[tex]\frac{RP}{15}[/tex] = [tex]\frac{15}{9}[/tex] ( cross- multiply )
9 RP = 15 × 15 = 225 ( divide both sides by 9 )
RP = 25
What is the gravitational force acting on a 70.0 kg object standing on the Earth’s surface?
Answer:
What is the gravitational force acting on a 70.0 kg object standing on the Earth’s surface?
Step-by-step explanation:
The gravitational force acting on a 70.0 kg object standing on the Earth’s surface can be calculated using the formula:
F = G * (m1 * m2) / r^2
Where:
F = gravitational force
G = gravitational constant (6.6743 x 10^-11 N*m^2/kg^2)
m1 = mass of the first object (in this case, the mass of the Earth)
m2 = mass of the second object (in this case, the mass of the object, 70.0 kg)
r = distance between the centers of the two objects (in this case, the radius of the Earth, which is approximately 6,371 km)
Using the values above, we can calculate the gravitational force as follows:
F = (6.6743 x 10^-11 N*m^2/kg^2) * (5.97 x 10^24 kg * 70.0 kg) / (6,371,000 m)^2
F = 686.6 N
Therefore, the gravitational force acting on a 70.0 kg object standing on the Earth’s surface is approximately 686.6 N.
A shop is having a 40% sale. A jumper originally costs £50. How much will it be in sale?
Answer:
30 pounds
Step-by-step explanation:
Since the jumper costs 50 pounds, the decimal for the number will be:
50.
To find ten percent of the number, you move the decimal one time to the LEFT.
Therefore, 10% of 50 pounds will be 5 pounds. To find 40%, we simply will multiply the 10% amount by 4.
40% of 50 will be:
20
Therefore, the jumper will have 20 pounds off.
Since the original cost is 50 pounds, we simply will just subtract 50 - 20:
50 - 20 = 30
The jumper will cost 30 pounds with the sale deal.
I need help with this question.. :')
The equation of the line passing through A and B is y = (4/5)x - (2/5).
What is the line example's equation?A straight line's general equation is y = mx + c, where m is the gradient and y = c is the value at which the line intersects the y-axis. The y-axis intercept is denoted by the number c. A straight line with gradient m and intercept c on the y-axis has the equation y = mx + c.
The point-slope form of a linear equation can be used to find the equation of the line passing through points A and B:
y - y1 = m(x - x1) (x - x1)
where m denotes the slope of the line, (x1, y1) denotes the coordinates of point A or B, and (x, y) denotes the coordinates of any other point on the line.
To calculate the slope, we can use points A (3, 2) and B (8, 6).
m = (y2 - y1) / (x2 - x1) = (6 - 2) / (8 - 3)\s= 4 / 5
So the equation for the line connecting A and B is:
y - 2 = (4/5)(x - 3) (x - 3)
This equation can be simplified by multiplying both sides by 5:
5y - 10 = 4x - 12
Then we can rearrange it to form the slope-intercept equation, y = mx + b:
5y = 4x - 2
y = (4/5)x - (2/5)
As a result, the equation for the line connecting A and B is y = (4/5)x - (2/5).
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Amelia knits 7 sweaters and 9 scarves to give to her friends. She uses 9 balls of yarn to make each sweater and 4 balls of yarn to make each scarf. How many balls of yarn does Amelia use in all?
Amelia uses 99 balls of yarn in all to make the sweaters and scarves.
What is addition ?
Addition is a basic mathematical operation that combines two or more quantities to find a total or a sum. It is denoted by the symbol "+" and is usually taught in elementary school. In addition, the order of the numbers does not matter, meaning that you can add them in any order and still get the same result. For example, 3 + 5 is the same as 5 + 3, and the answer is 8 in both cases.
According to the question:
To find the total number of balls of yarn that Amelia uses, we can start by finding the total number of balls of yarn she uses to make the sweaters and scarves separately, and then add those amounts together.
To make 7 sweaters, Amelia uses 7 x 9 = 63 balls of yarn.
To make 9 scarves, Amelia uses 9 x 4 = 36 balls of yarn.
Adding these together, we get:
63 + 36 = 99
So, Amelia uses 99 balls of yarn in all to make the sweaters and scarves.
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Ursula is looking at data about educators’ salaries in different cities in the country (in thousands of dollars). For which data set would the mean be the best measure of center? 48, 44, 41, 50, 38, 42, 92, 39 48, 14, 42, 53, 49, 51, 55, 43 55, 52, 60, 65, 56, 63, 68, 58 72, 78, 67, 81, 80, 76, 29, 70
There are no very large or small numbers inside the range of 38 to 92. The first data set [tex]48, 44, 41, 50, 38, 42, 92, 39.[/tex] is the one for which the mean would serve as the most accurate indicator of the center.
What are mean, median, and average?By adding every value in a collection and dividing by the total number of values, you may get the mean. The median is the value that falls in the center when a group of values are arranged from smallest to greatest. The value that appears the most frequently in the set is the mode.
How does the mean become used?The mean may be used as a benchmark for all observations since it can be used to reflect the typical value.
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Milan bought packs of pencils. Each pack has pencils. Write an equation to represent the total number of pencils that Milan bought.
Answer:
total number of pencils = p × n
total number of pencils = 3 × 12 = 36
Step-by-step explanation:
Let's break down the problem into its components. Milan bought packs of pencils, so we can represent the number of packs he bought with the variable "p". We also know that each pack has "n" number of pencils, so the total number of pencils Milan bought can be found by multiplying the number of packs with the number of pencils in each pack.
In mathematical notation, this can be written as:
total number of pencils = number of packs × number of pencils in each pack
or, using the variables we defined:
total number of pencils = p × n
So, if Milan bought 3 packs of pencils and each pack had 12 pencils, the total number of pencils he bought would be:
total number of pencils = 3 × 12 = 36
STUDY SKILLS 7. State ONE way in which each of the examination writing skills below could effectively assist you when writing your examinations. 7.1 Read the question 7.2 Plan the response 7.3 Answer the questions (3 x 1) (3)
Examination writing skills refer to a set of abilities that are essential for effective and successful writing in an exam context.
These skills include reading the question carefully, planning your response, writing a clear and concise answer, using appropriate terminology, providing evidence to support your argument, and presenting your answer in a logical and organized manner.
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Complete question:
State one way in which each of the examination writing skills below could effectively assist you when writing your examinations:
Read the questionPlan the responseAnswer the questionstrue/false. all of the following are things you can do when faced with any ethical dilemma as a managerial accountant except: always contact the board of directors first.
The statement " all of the following are things you can do when faced with any ethical dilemma as a managerial accountant except: always contact the board of directors first" is false because contacting the board of directors is not always necessary or appropriate when faced with an ethical dilemma as a managerial accountant
Contacting the board of directors is not always necessary or appropriate when faced with an ethical dilemma as a managerial accountant. While the board of directors may need to be informed in some situations, there are other steps that can be taken depending on the specific circumstances. For example, seeking guidance from a supervisor, consulting with a professional organization, or seeking legal advice may be more appropriate in some cases. The appropriate course of action will depend on the specific situation and ethical principles involved.
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Use the substitution method to solve the following system of equations: 3x + 5y = 3
x + 2y = 0
(1, 1)
(6, −3)
(6, 1)
(1, −3).
One side of the triangle is 4 cm, and the sum of the other two sides is equal to a whole number of cm. What is the smallest possible perimeter of the triangle?
F. 9 cm
G. 10 cm
H. 11 cm
J. 15 cm
K. 17 cm
Answer:
9 cm
Step-by-step explanation:
By the Triangle Inequality, any two sides of a triangle must be greater than the remaining side.
In order to minimize the perimeter, we will assume that 4 cm is the longest side.
Thus, the two remaining sides must be greater than 4.
Since we are given that the sum of the two remaining sides is a whole number, the smallest whole number value greater than 4 is 5.
Hence, the smallest perimeter possible 9 cm.
There is a total of 360 students and teachers at a school. A trip is organized and 65% of the students and teachers bought tickets to go on this trip.
(a) Work out how many of the students and teachers bought tickets to go on the trip.
The number of teachers,the number of male students and the number of female students who bought tickets to go on the trip are in the ratios 1:3:5 .
(b) Calculate the number of female students who bought tickets to go on the trip.
All the male students who bought tickets went on the trip but 4 of the female students who bought tickets did not go on the trip.
(c) Find the ratio of the number of male students who went on the trip to the number of female students who went on the trip.
Answer: Below :)
Step-by-step explanation:
(a) The percentage of students and teachers who bought tickets to go on the trip is 65%. So the number of students and teachers who bought tickets can be found by multiplying the total number of students and teachers by 65%:
65/100 x 360 = 234
Therefore, 234 students and teachers bought tickets to go on the trip.
(b) Let the number of teachers who bought tickets be x. Then the number of male students who bought tickets is 3x, and the number of female students who bought tickets is 5x.
The total number of students and teachers who bought tickets is:
x + 3x + 5x = 9x
We know that 234 students and teachers bought tickets, so we can set up the following equation:
9x = 234
Solving for x, we get:
x = 26
So the number of teachers who bought tickets is 26, the number of male students who bought tickets is 78 (3x), and the number of female students who bought tickets is 130 (5x).
However, 4 of the female students who bought tickets did not go on the trip, so the actual number of female students who went on the trip is:
130 - 4 = 126
(c) The ratio of the number of male students who went on the trip to the number of female students who went on the trip is:
78/126
Simplifying this ratio by dividing both numerator and denominator by 6, we get:
13/21
Therefore, the ratio of the number of male students who went on the trip to the number of female students who went on the trip is 13:21.
Milly took a loan of N$900 with simple interest for as many years as the rate of interest. If she paid N$324 as
interest at the end of the loan period, what was the rate of interest?
Answer:
Let's assume that the rate of interest is r (in decimals), and the time period is also r years. Then we can use the simple interest formula:
I = P * r * t
where I is the interest paid, P is the principal amount (the loan amount in this case), r is the rate of interest per year, and t is the time period in years.
Substituting the given values, we get:
324 = 900 * r * r
Simplifying, we get:
r² = 324/900
r² = 0.36
Taking the square root of both sides, we get:
r = ±0.6
Since the rate of interest cannot be negative, we can take r = 0.6. Therefore, the rate of interest is 0.6 or 60% per year.
Which statement explains how you could use coordinate geometry to prove that
quadrilateral ABCD is a rectangle?
You could use coordinate geometry to prove that ABCD is a rectangle by checking the side lengths, slope, and diagonal lengths using the distance and slope formulas.
What is a rectangle?
A rectangle is a quadrilateral with four right angles, meaning each of its angles measures 90 degrees. It is a type of parallelogram where opposite sides are parallel and equal in length. The opposite sides of a rectangle also have equal length, and its diagonals bisect each other. In addition to its geometrical properties, rectangles are commonly used in various fields for their symmetry, regularity, and practicality, such as in architecture, engineering, and graphic design.
To prove that quadrilateral ABCD is a rectangle using coordinate geometry, one possible method is to use the properties of a rectangle. Specifically, a rectangle is a quadrilateral with all four angles equal to 90 degrees and opposite sides equal in length.
So, to use coordinate geometry to prove that ABCD is a rectangle, you could:
Assign coordinates to the four vertices of ABCD, say A(x1,y1), B(x2,y2), C(x3,y3), and D(x4,y4).
Use the distance formula to calculate the lengths of each side of the quadrilateral, i.e., AB, BC, CD, and DA.
Use the slope formula to find the slopes of the four sides. Since opposite sides of a rectangle are parallel, their slopes will be equal.
Use the distance formula again to calculate the diagonal lengths, AC and BD.
Check that the opposite sides are equal in length and parallel, and that the diagonals are equal in length. If all these conditions are satisfied, then ABCD is a rectangle.
Therefore, you could use coordinate geometry to prove that ABCD is a rectangle by checking the side lengths, slope, and diagonal lengths using the distance and slope formulas.
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find the lenght of each side of a rohmbus with permeter of 40 meters
Answer:
10 metres
Step-by-step explanation:
All sides are equal in Rhombus
So, the length of one side= 40m÷4
=10 metres
write each expression without the absolute value symbol
The piecewise function forms of the absolute values are, respectively:
Case 1 (|(x - 19) + 11|)
x - 8 for x > 8
0 for x = 8
8 - x for x < 8
Case 2 (|14 - (x - 6)|)
- x + 22 for x < 22
0 for x = 22
x - 22 for x > 22
Case 3 (|14 - (6 - x)|)
x + 8 for x > - 8
0 for x = - 8
- x - 8 for x < - 8
How to rewrite absolute value functionsIn this problem we find three cases of absolute values represented in a single expression. Herein we need to determine an alternative form without "bar" operators.
This form is represented by a piecewise function derived from definition of absolute value:
|x - a| = x - a for x > a|x - a| = a - x for x < a|x - a| = 0 for x = a.Then, we find the following definitions for each of the three cases:
Case 1
|(x - 19) + 11| = |x - 8|
|x - 8| = x - 8 for x > 8
|x - 8| = 0 for x = 8
|x - 8| = 8 - x for x < 8
Case 2
|14 - (x - 6)| = |- x + 22|
|- x + 22| = - x + 22 for x < 22
|- x + 22| = 0 for x = 22
|- x + 22| = x - 22 for x > 22
Case 3
|14 - (6 - x)| = |x + 8|
|x + 8| = x + 8 for x > - 8
|x + 8| = 0 for x = - 8
|x + 8| = - x - 8 for x < - 8
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