The coordinates of the vertices of parallelogram ABCD are A(-3,2) , B (-2,-1) , C ( 4,1) and D ( 3,4). The slopes of which line segments could be calculated to show that ABCD is a rectangle?
1) AB and DC
2) Ab and BC
3) AD and BC
4) AC and BD
When you say what answer you chose can you explain why that’s the answer please in depth?
Answers
I cannot draw the rectangle here. When you draw the line of x and y axis, plot the numbers, when you connect the dot, you will see the rectangle.I don't know how to draw it here, but that is the answer.
To show that a parallelogram is a rectangle, we need to show that ONE of the interior angles is a right angle.
Therefore we need to find the angle between two adjacent sides.
To find two adjacent sides in a parallelogram with vertices in counter-clockwise, we need two sides with a common vertex, for example,
AB+BC, or BC+CD, or CD+DA, or DA+AB.
Any two sides without a common vertex are opposite sides.
Related Questions
What are the values of a and b?
Answers
Answer:
[tex]A= \frac{400}{21} B=\frac{580}{21}[/tex]
Step-by-step explanation:
You just have to remember the rules of the triangles in order to be able to solve this:
The rule of trigonometry that you will have to use is a is similie to 20, B is simile to 29 and 20 is 20.
So you would have to put them like this:
[tex]\frac{21}{20}= \frac{20}{a}= \frac{29}{b}[/tex]
You just have to clear the equations for A and B:
[tex]a=\frac{20*20}{21}=\frac{400}{21}\\ b=\frac{29*20}{21} =\frac{580}{21}[/tex]
So that would be your answer.
The value of 'a' is 400/21 and the value of 'b' is 580/21 and this can be determined by using the properties of trigonometry.
According to the properties of trigonometry:
[tex]\dfrac{21}{20}=\dfrac{20}{a}=\dfrac{29}{b}[/tex] --- (1)
Now, in order to determine the value of 'a' use the above equation:
[tex]\dfrac{21}{20}=\dfrac{20}{a}[/tex]
Cross multiply in the above equation.
[tex]21a = 20\times 20[/tex]
21a = 400
Divide 400 by 21 in order to get the value of a.
[tex]a = \dfrac{400}{21}[/tex]
Now, to determine the value of 'b' again use the equation (1).
[tex]\dfrac{20}{\dfrac{400}{21}}=\dfrac{29}{b}[/tex]
Cross multiply in the above expression.
[tex]20b = 29\times \dfrac{400}{21}[/tex]
[tex]20b = \dfrac{11600}{21}[/tex]
Now, divide on both sides by 20 in the above expression.
[tex]\rm b = \dfrac{11600}{21\times 20}[/tex]
[tex]\rm b = \dfrac{580}{21}[/tex]
For more information, refer to the link given below:
https://brainly.com/question/19237987
donna received a $70 gift card for a coffee store. she used it in buying some coffee that cost $7.84 per pound. after buying the coffee, she had $30.80 left on her card. how many pounds of coffee did she buy?
Answers
$70 minus the price per pound times the number of pounds equals the amount left on the card ($30.80).
$70 - $7.84p= $30.80
subtract 70 from both sides
-7.84p= -39.20
divide both sides by -7.84
p= 5 pounds of coffee
Hope this helps! :)
Let's solve this step by step!
So, we know that after Donna will have $30.80 remaining after buying a certain number of pounds, and we also know that each pound of coffee costs $7.84.
Subtract the amount that will be left from the gift card.
70 - 30.80 = 39.2
Now, divide this by the cost per pound.
39.2 ÷ 7.84 = 5
A N S W E R:
Donna bought 5 pounds of coffee.
Good day!
NEED HELP ASAP!!!!!!!!
Answers
C = 2 x 3 x 2
C = 12 units
answer
A. 12 units
C= 2(3.14)(2)
C= 12.56
ANSWER: the best estimate is A) 12
Hope this helps! :)
Which of the following is an odd function?
g(x) = x2
g(x) = 5x – 1
g(x) = 3
g(x) = 4x
Answers
The 4th selection is appropriate.
Answer:
g(x) = 4x is an odd function.
Step-by-step explanation:
A function g(x) is odd if it satisfies that, for all x we have g(-x) = -g(x). Then,
[tex]g(x) =x^2[/tex] is not an odd function beacuse only give positive values, then for example if x= 2
[tex]g(-2) =(-2)^2 = 4 \neq -4 = -g(2)[/tex].
g(x) = 5x-1 is not an odd function. I'll also give you a counterexample: for x=1 we have
g(-1) = 5(-1)-1 = -6 ≠ -4 = -(5-1) = -g(1).
g(x) = 3 is not an odd function. I'll also give you a counterexample: for x=1 we have
g(-1) = 3 ≠ -3 = -g(1).
Finally, g(x) = 4x is an odd function because for all x we have g(-x) = -g(x):
g(-x) = 4(-x) = -4x = -g(x).
Evaluate 2LW + 2HL + 2HW for L = 3,W = 2, and H = 4.
Answers
2(3)(2) + 2(4)(3) + 2(4)(2)
12 + 24 + 16 = 52
=2(3)(2) + 2(4)(3) + 2(4)(2)
=12 + 24 + 16
= 52
complete the pattern and find the rule 87,91,95,99,___,___, ___
Answers
how did you find it you can 87-91=4
so that is your rule +4
Answer:
aₙ = a₁ + (n-1).d
103
107
Step-by-step explanation:
This is an arithmetic sequence because the difference (d) between 2 successive numbers is constant.
d = 91 - 87 = 95 - 91 = 99 - 95 = 4
If the first term is a₁, we can find the n term using the following expression.
aₙ = a₁ + (n-1).d
Here, we want to find the fifth and sixth elements.
a₅ = 87 + (5-1).4 = 103
a₆ = 87 + (6-1).4 = 107
y=0.75(1.05)t Determine whether the function represents a exponential growth or decay.Identify the percent rate of change
Answers
The function ff(x) = 0.75(1.05)ˣ is an example of an exponential growth function.
The percent rate of change is 5%
What type of function is the equation
From the question, we have the following parameters that can be used in our computation:
f(x) = 0.75(1.05)ˣ
The function ff(x) = 0.75(1.05)ˣ is an example of an exponential growth function.
In exponential growth functions, the base (in this case, 1.05) is greater than 1, and as x increases, the function increases
Also, we have the percent rate of change to be
Rate = 1.05 - 1
Rate = 0.05
So, we have
Rate = 5%
Hence, the percent rate of change is 5%
Suppose that Bangladesh, India, Pakistan, and Nepal are economically interdependent. The following table shows the exports and imports of each country, with all monetary values given in millions of equivalent US dollars. Country of Origin Exporting to... Amount ($ 2007-13-04-00-00_files/i0260000.jpg 1,000,000) Bangladesh India 2,817 Bangladesh Pakistan 2,369 Bangladesh Nepal 3,039 India Bangladesh 2,023 India Pakistan 3,462 India Nepal 2,184 Pakistan Bangladesh 3,201 Pakistan India 2,336 Pakistan Nepal 2,338 Nepal Bangladesh 2,707 Nepal India 3,363 Nepal Pakistan 2,332 Based on the information in the table, assuming that these four countries trade strictly within the group, what is the value of the greatest trade balance in this group? a. $841 million b. $847 million c. $8,402 million d. $8,516 million
Answers
1. Total trade between Bangladesh and other three nations which are India , Pakistan, Nepal= 2,817 + 2,369 + 3,039=$ 8,225
2. Total trade between India and other three nations which are Bangladesh , Pakistan, Nepal= 2,023 +3,462 + 2,184=$7669
3. Total trade between Pakistan and other three nations which are Bangladesh,India, Nepal=3,201 + 2,336 +2,338= $7875
4. Total trade between Nepal and other three nations which are Bangladesh,India, Pakistan=2,707 +3,363+2,332 = $ 8402
The greatest trade is between Nepal and other three nations which are Bangladesh,India, Pakistan=2,707 +3,363+2,332 = $ 8402→→Option (C)
What are two pair fractions with common denominators for 3/5 and 3/4
Answers
[tex]\bf \cfrac{3}{5}\cdot \cfrac{4}{4}\implies \cfrac{12}{20}\qquad \qquad \qquad \qquad \qquad \cfrac{3}{4}\cdot \cfrac{5}{5}\implies \cfrac{15}{20}[/tex]
Brigitte fostered 14 dogs this year which is 5 less than last year how many dogs did brigitte foster last year
Answers
We then have to make the addition:
14 + 5 = 19
19 dogs.
Answer:
brigitte fostered 19 dogs. last year
Answer:
9 DOGS
Step-by-step explanation:
5 LESS= 14-5
=9
help please!! just solve for y
Answers
Using this information, we can make and solve an equation:
8y - 4 = 60° (we know that it is equal to 60°, since it is an equilateral triangle)
8y = 64° (we added both sides by 4 to get the y's alone on one side)
y = 8° (we divide both sides by 8 to find out what just y is)
-------------------------------------------------------------------------------------------------------------
Answer:
So y is equal to 8°
Jennifer bought 28 tickets to the museum for students and adult chaperones.She paid a total of $350.80, which included an $8.00 donation to the museum.Write a system of equations that can be used to determine the number of student tickets, s, and the number of adult tickets, a, Jennifer purchased. Algebraically solve the system of equations to determine the number of student tickets Jennifer purchased. Show or explain your work.
The price for students is 14.50 and the adults are 19.50
Answers
s+a=28
Total price paid is:
total_student_price + total_adult_price + $8 = $350.80
We know:
student_price = $14.50
adult_price = $19.50
Also:
total_student_price = $14.50*s
total_adult_price = $19.50*a
Now we have:
$14.50*s + $19.50*a + $8 = $350.80
System of equations that we have is:
s+a=28
$14.50*s + $19.50*a + $8 = $350.80
Now we solve first equation and insert it into second one:
s=28-a
14.50*(28-a) + 19.50a + 8 = 350.80
406 - 14.50a + 19.50a + 8 = 350.80
414 + 5a = 350.80
5a = 350.80 - 414
5a = -63.2
a= -12.64
This problem does not have solution for three reasons:
1) we got negative number of persons which is impossible
2) we got decimal number of persons which is impossible
3) total price paid ends in .80 while price per ticket for both types of tickets ends in .50. There is no way to get that final price using whole number of tickets
What equation results from completing the square and then factoring? x^2 + 24x = 33
Answers
Answer:
[tex](x+12)^2=177[/tex]
Step-by-step explanation:
We have been given an equation: [tex]x^2+24x=33[/tex].
To complete the square we change the left hand side of the equation to a perfect square trinomial. For this we add [tex](\frac{b}{2})^2[/tex] to both sides of equation, where b is the coefficient of x.
We can see that coefficient of x for our given equation is 24. So we will add [tex](\frac{24}{2})^2=12^2=144[/tex] to both sides of our equation.
[tex]x^2+24x+144=33+144[/tex]
[tex]x^2+24x+144=177[/tex]
Let us factor left side of our equation as the square of binomial.
[tex](x+12)^2=177[/tex]
Therefore, our resulting equation will be [tex](x+12)^2=177[/tex].
Answer: (x+12)^2=177
Step-by-step explanation:
Find the area of a circle circumscribed about an equilateral triangle whose side is 18 inches long.
Answers
the Area of a circle=pi*r²
the length of a radius of circle circumscribed about an equilateral triangle is
r=b*(√3)/3
where
b---------> length of a side equilateral triangle-------> 18 in
then
r=18*(√3)/3=10.39 in
A=pi*(10.39)²=108*pi=339.29 in²
the answer is 339.29 in²
The points A(12,7), B(12,−3), C(−2,−3), and D(−2,7) form rectangle ABCD. Which point is halfway between A and B? CLEAR CHECK (12,0) (12,5) (12,2) (12,−2)
Answers
coordinates are
A(12,7), B(12,−3), C(−2,−3), and D(−2,7)
i use the formula of midpoint between A and B
Xm=(x1+x2)/2 Ym=(y1+y2)/2
A(12,7) B(12,−3)
Xm=(12+12)/2=12
Ym=(7+(-3))/2=2
the point (12,2) is halfway between A and B
Hilda has $210 worth of $10 and $12 stock shares. The number of $10 shares is five more than twice the number of $12 shares. How many of each type of share does she have?
Answers
y: number of shares of $ 10
x: number of shares of $ 12
We write the system of equations:
10x + 12y = 210
y = 2x + 5
Solving the system:
x = 75/17
y = 235/17
Answer:
she has:
75/17 shares of $ 12
235/17 shares of $ 10
Answer:
15 | 10-dollar shares, 5 | 12- dollar shares
Step-by-step explanation:
Write the equation by adding the total values of each type of share.
10(2t+5)+12t=210
We can solve this equation for t to find the number of $12 shares.
10(2t+5)+12t=210
20t+50+12t=210
32t=160
t=5
So, the number of $12 shares is 5. Since the number of $10 shares is 5 more than twice the number of $12 shares, the number of $10 shares is 2(5)+5=15.
Hilda bought 5 $12 shares and 15 $10 shares.
a bag contains 1p 2p and 5p coins .3/8 of the bag are 1p coins.there are as many 5p coins than 1p coins in the bag . there are 640 cois in total. work out the number of 2p coins in the bag
Answers
The number of 2p coins in the bag is 160
How to get the number of 2p coins
Assuming:
the number of 1p coins is xthe number of 2p coins is y and the number of 5p coins is z.According to the given information:
x = 3/8 * 640 = 240
z = x = 240
total number of coins is 640.
x + y + z = 640
plugging in the values
240 + y + 240 = 640
y + 480 = 640
y = 160
Kareem lives 4/10 of mile from the mall.Write two equivalent fraction that show what fraction of mile Kareem lives from the mall
Answers
1) We can simplify 4/10 by dividing both #s by 2.
4/10 ÷ 2/2= 2/5
2) We can increase both #s by multiplying each by both by the same number. I choose the number 2 (bit this works with any number as long as you increase both numerator and denominator by the same amount).
=4/10 * 2/2
=(4*2)/(10*2)
=8/20
Two equivalent fractions are 2/5 and 8/20.
Hope this helps! :)
A couple has 12 children, what is the probability that child number 11 is female?
Answers
The probability is 1/12.
What is probability?Probability is a measure of the likelihood of an event to occur. Many events cannot be predicted with total certainty. We can predict only the chance of an event to occur i.e. how likely they are to happen, using it.
Given:
Total children = 12
There is possibility of girl number 11.
So, the probability that child number 11 is female
=1/12.
Hence, the probability is 1/12.
Learn more about probability here:
https://brainly.com/question/12629667
#SPJ2
Final answer:
The probability that the 11th child is female is 50%, assuming an equal chance for the birth of either sex and that the gender of the previous children does not affect this outcome.
Explanation:
The question at hand is a basic probability question in the realm of mathematics. It attempts to find the likelihood of an event that is purely random and independent of previous outcomes. The event in question is the sex of a newborn child, which can either be male or female, generally with equal likelihood.
To determine the probability that child number 11 is female, we assume a 50% chance for either sex, since the likelihood of having a boy or girl is typically equal. The sex of the previous children has no influence on the gender of the next child. Therefore, the probability of child number 11 being female is simply 1/2 or 50%.
It is important to understand that each child's sex is determined independently of their siblings. This is akin to flipping a fair coin where each flip is independent of the previous flips.
Consider the function y = 9 - x2, where x ≥ 3. What is the inverse of the function? What is the domain of the inverse? Show all of your work for full credit.
Answers
Please see the attached file.
Thanks.
Answer:
The inverse of the function is [tex]y^{-1}=\sqrt{9-x}[/tex].
The domain of the inverse function is [tex]D:(-\infty,0],\{x|x\in \mathbb{R}\}[/tex]
Step-by-step explanation:
Given : Function [tex]y=9-x^2[/tex] where, [tex]x\geq 3[/tex]
To find : What is the inverse of the function? What is the domain of the inverse?
Solution :
Function [tex]y=9-x^2[/tex]
To find the inverse we interchange the value of x and y,
[tex]x=9-y^2[/tex]
Now, we get the value of y
[tex]y^2=9-x[/tex]
[tex]y=\pm\sqrt{9-x}[/tex]
As [tex]x\geq 3[/tex] so x>0
[tex]y=\sqrt{9-x}[/tex]
The inverse of the function is [tex]y^{-1}=\sqrt{9-x}[/tex].
The domain of the inverse is the range of the original function.
The range is defined as the set of all possible value of y.
As [tex]x\geq 3[/tex]
Squaring both side,
[tex]x^2\geq 9[/tex]
Subtract [tex]x^2[/tex] both side,
[tex]9-x^2\leq 0[/tex]
[tex]y\leq 0[/tex]
The range of the function is [tex]R:(-\infty,0],\{y|y\in \mathbb{R}\}[/tex]
The domain of the inverse function is [tex]D:(-\infty,0],\{x|x\in \mathbb{R}\}[/tex]
Shelly got 60 questions correct on her math test.If she got 75% how many questions were on the test?
Answers
An isosceles triangle has area of 125 ft. If the base is 16 ft, what is the length of each leg? Round your answer to the nearest tenth
Answers
Final answer:
To find the length of each leg of an isosceles triangle with an area of 125 ft² and a base of 16 ft, we first calculate the height and then apply the Pythagorean Theorem. The length of each leg is approximately 17.6 ft.
Explanation:
The question asks about finding the length of each leg of an isosceles triangle given the area and the base. To solve this, we use the formula for the area of a triangle, A = 1/2 × base × height, and apply the Pythagorean Theorem since the height creates a right triangle with the leg of the isosceles triangle and half of the base.
First, we find the height using the given area, A = 125 ft² and base, b = 16 ft. The formula rearranges to height = 2A / base, giving us height = 2(125) / 16 = 15.625 ft.
Next, using the Pythagorean Theorem, a² + b² = c², we let a represent half of the base (8 ft), b be the height (15.625 ft), and c be the leg of the triangle. Solving for c, we find c ≈ 17.6 ft, rounded to the nearest tenth.
Please answer this question and give give a detailed answer. Brainliest to who ever answers first and gets it correct
Answers
note, that ∠GFB=∠CBH=117° (AD||EG!) and ∠x=∠CFG.
For ∠CFG: ∠CFG=117-∠CFB=117-42=75°
Answer: 75°
What’s the answer to the question?
Answers
answer: A. (6;2;0)
Details are in the attachment.
susan has a circular koi pond in a square patio with sides measuring 30 feet each. Find the area of the shaded region to the nearest hundredth.use 3.14 for pi
Answers
d=30 .
Half of the diameter is the radus
30/2 = 15 . r=15
Area of a circle is A=[tex] \pi r^{2} [/tex] .
Plug in the values A = (3.14)(15)^2 = 706.50 ft squared
Give a polynomial-time algorithm that takes a sequence of supply values s1, s2, . . . , sn and returns a schedule of minimum cost. for example, suppose r = 1, c = 10, and the sequence of values is
Answers
Try a rudimentary proposition, do a bubble sort, that gives O(n^2) for worst and average case. It is a polynomial algorithm.
We can also do a quick sort, with worst case O(n^2) and average case O(nlogn), which is already better.
Do we need to sort everything? Not really.
What about a single pass, and store the minimum found, exchange as required, such as:
small=A(0)
for i:1, n {
if A(i)<small small: A(i)
}
return small
This is a linear algorithm, best case n, worst case 2n so O(n)
What can be determined about this data set before finding the range or the interquartile range? 19, 25, 35, 38, 41, 49, 50, 52, 99
Answers
19, 25, 35, 38, 41, 49, 50, 52, 99
Just upon looking at the data set, I can determine that there is an OUTLIER. An outlier is a point that is distant from other points.
The outlier in this data set is number 99.
These data set has the following five number summary:
1) minimum number - 19
2) 1st quartile - 30
3) 2nd quartile or median - 41
4) 3rd quartile - 51
5) maximum number - 99
Interquartile range = q3 - q1 → 51 - 30 = 21
To determine if an observation point is an outlier, we need to determine the lower fence and the upper fence.
lower fence = q1 - 1.5(iqr)
lower fence = 30 - 1.5(21) → 30 - 31.5 = -1.5
upper fence = q3 + 1.5(iqr)
upper fence = 51 + 1.5(21) → 51 + 31.5 = 82.50
Any number outside the lower fence, -1.5, and upper fence, 82.50, is an OUTLIER.
99 is beyond the upper fence. Thus, it is an outlier.
The data has an outlier, but the interquartile range will not change. The answer is c
The following stem-and-leaf plot represents the scores earned by Mr. Roberts's class on their most recent science test.
What is the median of the scores?
78
80
81
86
Answers
[tex] \frac{19+1}{2}=10th [/tex]
Thus the 10th score is the median score for the class of Mr. Robert. The 10th score from the stem and leaf plot is 81.
Thus 81 is the median score of Mr. Robert's Class.
Answer:
im pretty sure its 81
Step-by-step explanation:
In triangle XYZ,XY=15,YZ=21, and XZ=27. What is the measure of angle Z to the nearest degree?
Answers
Final answer:
Using the Law of Cosines, we find that the cosine of angle Z is approximately 0.6333, and therefore, angle Z in triangle XYZ is approximately 51° when rounded to the nearest degree.
Explanation:
To find the measure of angle Z in triangle XYZ, with sides XY = 15, YZ = 21, and XZ = 27, we can use the Law of Cosines. The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. In our case, we can find the cosine of angle Z by the equation:
cos(Z) = (XY² + XZ² - YZ²) / (2 · XY · XZ)
Plugging in the values, we get:
cos(Z) = (15² + 27² - 21²) / (2 · 15 · 27)
Calculating further:
cos(Z) = (225 + 729 - 441) / (810)
cos(Z) = 513 / 810
cos(Z) = 0.6333
The next step is to find the angle whose cosine is 0.6333. We can use a calculator to find the inverse cosine:
Z = cos⁻¹(0.6333)
Hence, the measure of angle Z to the nearest degree is approximately:
Z ≈ 51°
The measure of angle Z is approximately 70 degrees (rounded to the nearest degree).
To find the measure of angle Z in triangle XYZ, you can use the Law of Cosines. The formula is:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(C) \][/tex]
where:
- c is the side opposite the angle you want to find (in this case, side XZ),
- a and b are the other two sides (XY and YZ),
- C is the angle opposite side c.
In this case, let C be the angle Z, and a = XY = 15, b = YZ = 21, and c = XZ = 27.
[tex]\[ 27^2 = 15^2 + 21^2 - 2(15)(21) \cos(Z) \][/tex]
Now, solve for cos(Z):
[tex]\[ 729 = 225 + 441 - 630 \cos(Z) \][/tex]
[tex]\[ 630 \cos(Z) = 441 - 225 \][/tex]
[tex]\[ 630 \cos(Z) = 216 \][/tex]
[tex]\[ \cos(Z) = \frac{216}{630} \][/tex]
[tex]\[ \cos(Z) \approx 0.343 \][/tex]
Now, find the angle Z:
[tex]\[ Z = \cos^{-1}(0.343) \][/tex]
[tex]\[ Z \approx 70.18 \][/tex]
So, the measure of angle Z is approximately 70 degrees (rounded to the nearest degree).
The summation expression in the following series has an absolute value in it. Expand and evaluate the summation notation. What is the sum of the series?
Answers
We have that [tex]3<\dfrac{10}3<4[/tex], which means [tex]|3i-10|[/tex] reduces to [tex]3i-10[/tex] when [tex]i\ge4[/tex], or reduces to [tex]10-3i[/tex] when [tex]i\le3[/tex].
So we can expand the summation as
[tex]\displaystyle\sum_{i=1}^6|3i-10|=\sum_{i=1}^3(10-3i)+\sum_{i=4}^6(3i-10)[/tex]
Notice that the 10 contributes a total of 30 from the first sum, and -30 from the second sum, so those terms cancel, leaving us with
[tex]\displaystyle3\left(\sum_{i=4}^6i-\sum_{i=1}^3i\right)=3((4+5+6)-(1+2+3))=3(15-6)=3(9)=27[/tex]
To expand and evaluate the summation notation with an absolute value, we can use the binomial theorem. The expanded form of the series will depend on whether the sum inside the absolute value is positive or negative. Substituting the given values and simplifying the expression will give the sum of the series.
Explanation:A binomial expansion is a way of expressing an algebraic quantity as a sum of an infinite series of terms. In this case, we have an absolute value in the summation expression.
To expand and evaluate the summation notation, we can use the binomial theorem.
The binomial theorem states that (a + b)^n = C(n,0)a^n + C(n,1)a^(n-1)b + C(n,2)a^(n-2)b^2 + ... + C(n,n)b^n, where C(n,k) represents the binomial coefficient.
In the given case, we have an expression of the form |a + b|.
To expand this, we can consider two cases: a + b ≥ 0 and a + b < 0. If a + b ≥ 0, then |a + b| = a + b. If a + b < 0, then |a + b| = -(a + b).
Thus, the expanded form of the series will be a + b when a + b ≥ 0, and -(a + b) when a + b < 0.
To evaluate the series, substitute the given values for a and b into the expanded form and simplify the expression.