The student is required to combine three algebraic fractions with different denominators using factoring to find a common denominator and then simplify the expression.
Explanation:The question entails a topic in algebra, specifically with respect to combining expressions with different denominators, which requires finding a common denominator, and working with signs and exponents. The problem presents three fractions that should be combined: (x+6)/(x^2+8x+15), (3x)/(x+5), and (x-3)/(x+3).
Firstly, note that the denominator x^2+8x+15 can be factored into (x+3)(x+5). This will allow us to identify a common denominator for all three fractions, which is (x+3)(x+5). We rewrite the fractions so that each has the common denominator:
(x+6)/((x+3)(x+5))(3x)/(x+5) will be rewritten as (3x)(x+3)/((x+3)(x+5))(x-3)/(x+3) will remain as (x-3)/(x+3) because it already has part of the common denominatorNow we simply combine these three fractions over the common denominator:
((x+6) + (3x)(x+3) - (x-3)) / ((x+3)(x+5))After the combination, the terms must be simplified and ordered in descending powers of x, which is the final answer.
Descending powers of x is [tex]\[ \frac{2x^2 + 8x + 15}{(x + 3)(x + 5)} \][/tex].
To combine the given expressions, we need to find a common denominator and then combine the numerators. Here are the expressions given:
[tex]\[ \frac{x}{x^2 + 8x + 15} + \frac{3x}{x + 5} - \frac{x - 3}{x + 3} \][/tex]
First, let's factor the quadratic denominator in the first term if possible and identify the common denominator:
The quadratic [tex]\( x^2 + 8x + 15 \)[/tex] can be factored into [tex]\( (x + 3)(x + 5) \)[/tex], since 3 and 5 are factors of 15 that add up to 8.
Now we have:
[tex]\[ \frac{x}{(x + 3)(x + 5)} + \frac{3x}{x + 5} - \frac{x - 3}{x + 3} \][/tex]
The common denominator will be [tex]\( (x + 3)(x + 5) \)[/tex].
Now, let's rewrite each fraction with the common denominator:
The second term already has [tex]\( x + 5 \)[/tex] in the denominator, so we multiply the numerator and denominator by [tex]\( x + 3 \)[/tex]to have the common denominator.
[tex]\[ \frac{3x}{x + 5} \rightarrow \frac{3x(x + 3)}{(x + 5)(x + 3)} \][/tex]
The third term has [tex]\( x + 3 \)[/tex] in the denominator, so we multiply the numerator and denominator by \( x + 5 \) to have the common denominator.
[tex]\[ \frac{x - 3}{x + 3} \rightarrow \frac{(x - 3)(x + 5)}{(x + 3)(x + 5)} \][/tex]
Now all terms have a common denominator, and we can combine them as follows:
[tex]\[ \frac{x}{(x + 3)(x + 5)} + \frac{3x(x + 3)}{(x + 3)(x + 5)} - \frac{(x - 3)(x + 5)}{(x + 3)(x + 5)} \][/tex]
Combine the numerators while keeping the denominator the same:
[tex]\[ \frac{x + 3x(x + 3) - (x - 3)(x + 5)}{(x + 3)(x + 5)} \][/tex]
Now, let's expand and simplify the numerator:
[tex]\[ x + 3x^2 + 9x - (x^2 + 2x - 15) \][/tex]
[tex]\[ x + 3x^2 + 9x - x^2 - 2x + 15 \][/tex]
Combine like terms:
[tex]\[ 3x^2 - x^2 + x + 9x - 2x + 15 \][/tex]
[tex]\[ 2x^2 + 8x + 15 \][/tex]
Now, let's put it all over the common denominator:
[tex]\[ \frac{2x^2 + 8x + 15}{(x + 3)(x + 5)} \][/tex]
This is the simplified expression in descending powers of \( x \). Since the numerator is already in descending powers of \( x \), this is the final answer. There is no further simplification possible because the numerator and the denominator do not have common factors other than 1.
I have no idea what to do with this!
Given the function g(x) = 4x - 5, compare and contrast g(2) and g(-4). Choose the statement that is true concerning these two values.
Answer
The value of g(2) is smaller than the value of g(-4).
The values of g(2) and g(-4) cannot be compared.
The value of g(2) is larger than the value of g(-4).
The value of g(2) is the same as the value of g(-4). ...?
By substituting the x-values into the linear function g(x) = 4x - 5, we find g(2) = 3 and g(-4) = -21. Therefore, the value of g(2) is larger than the value of g(-4).
Explanation:To determine the values of g(2) and g(-4) for the function g(x) = 4x - 5, we simply substitute the x-values into the function.
For g(2):
g(2) = 4(2) - 5g(2) = 8 - 5g(2) = 3For g(-4):
g(-4) = 4(-4) - 5g(-4) = -16 - 5g(-4) = -21Now, we can compare the two values:
The value of g(2), which is 3, is larger than the value of g(-4), which is -21.
The statement that is true concerning [tex]\( g(2) \)[/tex] and [tex]\( g(-4) \)[/tex] is option a) : The value of [tex]\( g(2) \)[/tex] is larger than the value of [tex]\( g(-4) \)[/tex].
To compare and contrast [tex]\( g(2) \)[/tex] and [tex]\( g(-4) \)[/tex] for the function [tex]\( g(x) = 4x - 5 \)[/tex]:
1. Calculate [tex]\( g(2) \)[/tex] :
[tex]\[ g(2) = 4 \cdot 2 - 5 = 8 - 5 = 3 \][/tex]
2. Calculate [tex]\( g(-4) \)[/tex] :
[tex]\[ g(-4) = 4 \cdot (-4) - 5 = -16 - 5 = -21 \][/tex]
Now, let's analyze the values:
- [tex]\( g(2) = 3 \)[/tex]
- [tex]\( g(-4) = -21 \)[/tex]
Comparing these values:
- [tex]\( g(2) \)[/tex] is positive (3).
- [tex]\( g(-4) \)[/tex] is negative (-21).
Therefore, [tex]\( g(2) \)[/tex] is greater than [tex]\( g(-4) \)[/tex]. This means:
[tex]\[ \text{The value of } g(2) \text{ is larger than the value of } g(-4). \][/tex]
Explanation:
The function [tex]\( g(x) = 4x - 5 \)[/tex] calculates a linear relationship where [tex]\( g(2) \)[/tex] evaluates to 3 and [tex]\( g(-4) \)[/tex] evaluates to -21. This comparison clearly shows that [tex]\( g(2) \)[/tex] yields a larger value than [tex]\( g(-4) \)[/tex]. This result stems from the fact that when x = 2 , [tex]\( g(x) \)[/tex] outputs a positive value, while for x = -4 , [tex]\( g(x) \)[/tex]outputs a larger negative value. Therefore, option:
Complete question : Given the function g(x) = 4x −5, compare and contrast g(2) and g(−4). Choose the statement that is true concerning these two values.
a The value of g(2) is larger than the value of g(−4).
b The value of g(2) is smaller than the value of g(−4).
c The value of g(2) is the same as the value of g(−4).
d The values of g(2) and g(−4) cannot be compared.
graph this: f(x)= IxI-1 ...?
The graph resembles a "V" shape with its vertex at the origin.
The function f(x) = |x| - 1 is a piecewise function that represents the absolute value of x minus 1.
The absolute value function |x| returns the distance of x from the origin on the number line.
When x is positive or zero, |x| = x , and when x is negative, |x| = -x.
Thus, f(x) = |x| - 1behaves differently for positive and negative values of x .
For [tex]\( x \geq 0 \)[/tex], the function is f(x) = x - 1, and for x < 0, the function is f(x) = -x - 1.
The graph of f(x) consists of two straight lines intersecting at the point (0, -1), with a slope of 1 for [tex]\( x \geq 0 \)[/tex] and a slope of -1 for x < 0.
The graph resembles a "V" shape with its vertex at the origin.
Dividing polynomials...
(16x^2-25y^2)divided by (4x+5y)...Please explain your answer... ...?
What is directly in between 0.5 and 1.0?
How do you sketch this graph? Sketch the graph of y=xlnx
Jaron paid 2.70 for 6 juice boxes how much should jaron expect to pay for 18 juice boxes
Answer:
8.10
Step-by-step explanation:
What is the perimeter of a triangle with vertices located at (1,4), (2,7), (0,5)
Marty made a $220 bank deposit using $10 bills and $5 bills. She gave the teller a total of 38 bills, how many $5 bills were in the deposit? A) 6 five-dollar bills B) 28 five-dollar bills C) 32 five-dollar bills D) 34 five-dollar bills
George sold 18, 22, 26, 12, 25, 20, and 19 cars per month over the past seven months. He followed the steps below to determine the number of cars he needs to sell in the next month to have a mean number of sales per month of 24.
Final answer:
To find the number of cars George needs to sell in the next month to have a mean of 24, calculate the total number of cars he has sold in the past seven months. Subtract that total from 24 multiplied by 8 (including the past seven months and the next month). Divide the result by 1 to get the number of cars George needs to sell in the next month.
Explanation:
To find the number of cars George needs to sell in the next month to have a mean (average) number of sales per month of 24, we need to calculate the total number of cars he has sold in the past seven months. Then, we can subtract that total from 24 multiplied by 8 (as George wants a mean of 24 sales per month for a total of 8 months, including the past seven months and the next month). Finally, we divide the result by 1 to get the number of cars George needs to sell in the next month.
In this case, the total number of cars George has sold in the past seven months is 18 + 22 + 26 + 12 + 25 + 20 + 19 = 142.
So, to determine the number of cars George needs to sell in the next month to have a mean of 24, we perform the following calculation:
(24 * 8) - 142 = 192 - 142 = 50.
Therefore, George needs to sell 50 cars in the next month to have a mean number of sales per month of 24.
F = 9/5 C + 32 : is the formula used to convert degrees Celsius to degrees Fahrenheit. Convert 35°C to degrees Fahrenheit.
A) 36°F
B) 63°F
C) 67°F
D) 95°F
Converting 35°C to degrees Fahrenheit is equal to 95°F.
Option D is the correct answer.
What is an expression?An expression contains one or more terms with addition, subtraction, multiplication, and division.
We always combine the like terms in an expression when we simplify.
We also keep all the like terms on one side of the expression if we are dealing with two sides of an expression.
Example:
1 + 3x + 4y = 7 is an expression.com
3 + 4 is an expression.
2 x 4 + 6 x 7 – 9 is an expression.
33 + 77 – 88 is an expression.
We have,
Using the formula F = (9/5)C + 32, where C is the temperature in Celsius and F is the temperature in Fahrenheit.
Substituting C = 35°C into the formula:
F = (9/5) × 35 + 32
F = 63 + 32
F = 95°F
Therefore,
35°C is equal to 95°F.
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"Which of the following are remote interior angles of 5? Check all that apply.
A.1
B.2
C.3
D.4
E.5
F. 6"
Answer:
A. 1
B. 2
Step-by-step explanation:
We have been given an image of a triangle. We are asked to find remote interior angles of angle 5 for our given triangle.
We can see that angle 5 is an exterior angle. We know that remote interior angles are those, which are inside the triangle and opposite to exterior angle.
We can see that angle 2 and angle 1 are are opposite angles of angle 5 and they are inside our given triangle.
Therefore, option A and B are correct choices.
Final answer:
Remote interior angles are non-adjacent angles within the same triangle. Relative to angle 5, angle 1 and angle 2 are remote interior angles because they are inside the triangle and do not share a vertex with angle 5.
Explanation:
In this mathematics problem, we are asked to find the remote interior angles of a given angle, which are angles that are inside the triangle and non-adjacent to the exterior angle being considered. For example, if we have a triangle with angles labeled 1, 2, 3, 4, 5, and 6, and we want to find the remote interior angles relative to angle 5, we would look for the angles inside the triangle that do not share a vertex with angle 5.
From the available options:
Angle 1 and angle 2 qualify as remote interior angles relative to angle 5 because they are non-adjacent to angle 5 and are within the same triangle.Angle 3 is adjacent to angle 5, so it is not a remote interior angle.Angles 4, 5, and 6 are not pertinent in this case because the question is specifically about the remote interior angles of angle 5, making angles 4 and 6 external to the triangle, whereas angle 5 is the angle in question itself.Therefore, options A (1) and B (2) are the correct answers to this problem.
Count the left-handed and right-handed spirals on the cacti in the photograph. You should get two consecutive terms of the Fibonacci sequence. What are the numbers? A. 34 and 55 B. 5 and 8 C. 8 and 13 D. 21 and 34 E. 13 and 21
The student's question involves counting the number of spirals in a cactus, which corresponds to two consecutive numbers in the Fibonacci sequence. The exact answer would depend on the actual count, which cannot be determined without the photograph.
Explanation:The question asks about the phenomenon related to the Fibonacci sequence observed in natural patterns, specifically in plant architecture such as cacti spirals. The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, usually starting with 0 and 1. It is often observable in nature, for example, in the arrangement of leaves or the spirals on sunflower heads and cacti.
In order to answer the question, one would need to physically count the number of left-handed and right-handed spirals on the cactus. The Fibonacci sequence relevant to this context begins with 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and so on. Since the given answer choices are consecutive terms from this sequence, the correct numbers would be identified based on the actual spiral counts which are expected to be consecutive Fibonacci numbers.
Without the photograph, it is impossible to provide the specific numeric answer, but we can understand that the two numbers representing the left-handed and right-handed spirals on the cactus would be consecutive Fibonacci numbers as indicated by the options provided: A) 34 and 55, B) 5 and 8, C) 8 and 13, D) 21 and 34, e) 13 and 21.
factor 3x^2-14x+8 ...?
A rancher with 750 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle.
1] Find a function that models the total area of the four pens. (Let w be the width of the rectangular area and A(w) be the area.)
2]Find the largest possible total area of the four pens. (Round your answer to one decimal place.)
The function modeling the total area of the four pens is [tex]A(w) = w * ((750 - 3w) / 2)[/tex]. The maximum area can be found by taking the derivative of this function, setting it to zero, and solving for w.
Explanation:The subject in question is Mathematics, specifically calculus and problem-solving involving areas. The rancher with 750 ft of fencing intends to make a rectangular area divided into four pens. Let's define w to be the width and l to be the length of the rectangle.
1] Since the total length of the fence is 750 ft and it will be divided into 3 parts for the width and 2 parts for the length, the function for calculating length will be [tex]l = (750 - 3w) / 2[/tex]. Since the area of a rectangle is given by A=w*l, we can substitute our equation for l into our Area function. Therefore, our function will be [tex]A(w) = w * ((750 - 3w) / 2).[/tex]
2] To find the maximum area of the rectangles, you would need to find the derivative of the Area function and solve for w. After that, plug the value of w into the area function to get maximum area. The exact calculation requires calculus skills and could be taught in a high school Calculus course.
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John has six bills of paper money in the following denominations $1, $5, $10, $20, $50, and $100. If he selects 3 bills at a time what is the probability of selecting a group that has an average value of at least $26?
The required probability is 0.55 while selecting a group that has an average value of at least $26.
What is probability?Probability is defined as the possibility of an event being equal to the ratio of the number of favorable outcomes and the total number of outcomes.
John has six bills of paper money in the following denominations $1, $5, $10, $20, $50, and $100.
The total number of possible combinations = [tex]^6c_3[/tex] = 6!/313! = 20
Combinations for which the average value is at least $26:
(1.5.100), (1,10,100), (1.20,100). (1.50,100), (5,10,100), (5,20,100), (5,50,100), (10,20,100), (10,20,50), (10,50,100), (20,50,100)
Number of favorable combinations = 11
So, required probability = 11/20 = 0.55
Thus, the required probability is 0.55 while selecting a group that has an average value of at least $26.
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Pears cost $1.25 per pound. What is the cost of 4.5 pounds of pears
3x-y=11
2x+5y=-4
systems of linear equations using substitution
find X and Y
Amy wants to run the 26 miles of the marathon in 4.5 hours. at what rate will she have to run to reach thisbgoal??
Store A is advertising a sale that will reduce prices on all merchandise by 15%. Store B is advertising a sale that will reduce prices on all merchandise by one over five. Which store is reducing its prices more, and by how much?
A
Store A is reducing prices by 5% more than Store B.
B
Store B is reducing prices by 10% more than Store A.
C
Store B is reducing prices by 5% more than Store A.
D
Both stores are reducing prices by the same amoun
Answer:
The correct option is C) Store B is reducing prices by 5% more than Store A.
Step-by-step explanation:
Consider the provided information.
Store B is advertising a sale that will reduce prices on all merchandise by one over five.
Convert one over five in percentage as shown below:
[tex]\frac{1}{5}=\frac{1}{5}\times{\frac{20}{20}}[/tex]
[tex]\frac{1}{5}=\frac{20}{100}[/tex]
[tex]\frac{1}{5}=20\%[/tex]
Thus, the store B reduce the prices by 20%.
Store A reduces the prices on all merchandise by 15%.
From the above calculation, it is clear that store B reduces more prices by 5%.
Therefore, the correct option is C) Store B is reducing prices by 5% more than Store A.
A farmer wants to fence an area of 24 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. What should the lengths of the sides of the rectangular field be so as to minimize the cost of the fence?
smaller value? and larger value?
A box contains four 40-W bulbs, five 60-W
bulbs, and six 75-W bulbs. If bulbs are se-
lected one by one in random order, what is
the probability that at least two bulbs must be
selected to obtain one that is rated 75W?
Answer: 0.99483
Step-by-step explanation:
Given : A box contains four 40-W bulbs, five 60-W bulbs, and six 75-W bulbs.
Total bulbs : 4+5+6=15
The probability of selecting a 75-W bulb :[tex]p=\dfrac{6}{15}=0.4[/tex]
Using the binomial probability :-
[tex]P(X)= ^nC_xp^x(1-p)^{n-x}[/tex], where P(x) is the probability of getting success in x trials , p is the probability of getting success in each trial and n is the total number of trials.
We have,
The probability that at least two bulbs must be selected to obtain one that is rated 75W :-
[tex]P(x\geq2)=1-(P(x<2))\\\\=1-(P(x=0)+P(x=1))\\\\=1-(^{15}C_0(0.4)^0(0.6)^{15}+^{15}C_{1}(0.4)^1(0.6)^{14})\\\\=1-((1)(0.6)^{15}+(15)(0.4)(0.6)^{14})\ \ [\because\ ^nC_0=1\ \&\ ^nC_1=n]\\\\\approx1-(0.00047+0.00470)\\\\=1-0.00517=0.99483[/tex]
Hence, the required probability = 0.99483
Taylor writes an expression with 5 terms. all 5 terms are like terms. how many terms are in the equivalent expression with the least number of terms?
Answer:
Step-by-step explanation: Taking into account that they are all like terms they can be combined, therefore there would only be one term. For example:
5x-3x+6x-x+2x=
Combine the like terms
2x+6x-x+2x=
It can be simplified further
8x-x+2x=
7x+2x=
9x
Greg went to the walmart to buy pencils and pens. each pencils costs $2 and each pen costs $3. he bought five items. he spent $12. how many pens and pencils did he buy?
Triangles can have some congruent corresponding parts, but not be congruent. true or false.
Answer:
True
Step-by-step explanation:
Two or more figures are congruent if all of their sides and angles are equal.
You can draw two not congruent triangles with equal congruent parts.
For example, you know that the sum of the inside angles of a triangle is 180º.
You could find two triangles with 90º 45º and 45º angles but with different-length sides.
Those would be congruent angles but not congruent triangles.
A linear equation in the form y = mx + b is written in slope-intercept form. What does the 'm' represent?
A. x-intercept
B. y-intercept
C. z-intercept
D. slope
2. A linear equation in the form y = mx + b is written in slope-intercept form. What does the 'b' represent?
A. x-intercept
B. y-intercept
C. z-intercept
D. slope
3. What is slope of equation y = 3x + 5?
A. 3
B. -5
C. 5
D. -3
4. What is y-intercept of equation y = 3x + 5?
A. 3
B. 5
C. -5
D. -3
If an object is moving at the speed of 36 kilometers per hour, how many meters does it travel in one second?
Answer:
10 meters per second
Step-by-step explanation:
write the equation of a line passing through (-2,1) with a slope of 3. explain how you arrived at your answer.
Kendra owns a restaurant. She charges $3.00 for 2 eggs and one piece of toast, and $1.80 for one egg and one piece of toast. How much does Kendra charge for an egg? A piece of toast?
The daily dose of ampicillin for the treatment of an ear infection is 115mg/kg of body weight. what is the daily dose, in mg, for a 34-lb child? ...?
What is the first four terms of the sequence n squared + 5
The first four terms of the sequence defined by n squared plus 5 are 6, 9, 14, and 21.
The first four terms of the sequence defined by the expression n squared plus 5, which can be represented mathematically as n^2 + 5.
To find the terms, we simply substitute the first four positive integers (1, 2, 3, 4) for n and calculate:
For n = 1: 1^2 + 5 = 1 + 5 = 6
For n = 2: 2^2 + 5 = 4 + 5 = 9
For n = 3: 3^2 + 5 = 9 + 5 = 14
For n = 4: 4^2 + 5 = 16 + 5 = 21