Given f of x is equal to 2 divided by the quantity x squared plus 3x minus 10 end quantity, which of the following is true?
f(x) is positive for all x < –5
f(x) is negative for all x < –5
f(x) is positive for all x < 2
f(x) is negative for all x > 2
Answers
Answer:
(a) positive for x < -5
Step-by-step explanation:
The sign of a rational function can be determined from the signs of its factors. The sign of the function will change when either the sign of the numerator or the sign of the denominator changes. The sign will change on either side of the zero of a linear factor with odd multiplicity.
__
sign changesThe factorization of the given rational function is ....
\(f(x)=\dfrac{2}{x^2+3x-10}=\dfrac{2}{(x+5)(x-2)}\)
Denominator zeros are ...
x +5 = 0 ⇒ x = -5
x -2 = 0 ⇒ x = 2
The sign of the function will change at x=-5 and at x=2.
__
signThe denominator factors will have different signs in the interval -5 < x < 2. That is the only interval in which the function value is negative. (Eliminates choices B and D.)
The denominator factors will have the same sign for x < -5 (both negative) or x > 2 (both positive). The function value will be positive in those intervals. (Eliminates choice C.)
f(x) is positive for all x < -5.
_____
Additional comment
If a zero has even multiplicity, the sign of the function on either side of it is the same. The product of an even number of negatives has the same sign as the product of an even number of positives — positive.
The function f(x) is negative for all x < 2": true, as the function is negative in the interval (-5, 2).
To determine the intervals where the function f(x) = 2 / (x² + 3x - 10) is positive or negative, we need to find the critical points and analyze the sign of the function in different intervals.
First, let's factor the denominator:
x² + 3x - 10 = (x + 5)(x - 2)
The critical points occur where the denominator is equal to zero:
x + 5 = 0
=> x = -5
x - 2 = 0
=> x = 2
Now, we have three intervals to consider: (-∞, -5), (-5, 2), and (2, ∞).
In the interval (-∞, -5):
Pick a test point, e.g., x = -6.
Calculate f(-6) = 2 / ((-6)² + 3*(-6) - 10)
f(6) = 2 / (36 - 18 - 10)
f(6) = -2 / 8
f(6) = -1/4.
Since the function is negative in this interval, the statement "f(x) is positive for all x < -5" is not true.
In the interval (-5, 2):
Pick a test point, e.g., x = 0.
Calculate f(0) = 2 / (0² + 3*0 - 10)
f(0) = 2 / (-10)
f(0) = -1/5.
Again, the function is negative in this interval, so the statement "f(x) is positive for all x < 2" is not true.
In the interval (2, ∞):
Pick a test point, e.g., x = 3.
Calculate f(3) = 2 / (3²+ 3*3 - 10)
f(3) = 2 / (9 + 9 - 10)
f(3) = 2 / 8
f(3) = 1/4.
In this interval, the function is positive, so the statement "f(x) is negative for all x > 2" is not true.
The correct answer is: f(x) is negative for all x < 2.
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Related Questions
ill give 50 points and brailets if u ansewr fast and ergent
Answers
The linear equation that models the cost as a function of the number of hours is:
y = 40*x + 20
How to write the equation?We know that a repair that takes two hours costs $100 and a repair that takes 6 hours costs $260.
Let's assume this is a linear equation, then:
A general linear equation is written as:
y = a*x + b
Where a is the slope and b is the y-intercept.
If the line passes through two points (x1, y1) and (x2, y2), then the slope is:
slope = (y2 - y1)/(x2 - x1)
Here we have the two points (2, 100) and (6, 260), then the slope is:
a = (260 - 100)/(6 - 2) = 160/4 = 40
So our line is something like:
y = 40*x +b
To find the value of b we use the fact that our line passes trhough (2, 100), then:
100 = 40*2 + b
100 - 80 = b
20 = b
Then the linear equation is:
y = 40*x + b
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46. Machine A produces 500 springs a day. The number of defective springs produced by this machine each day is recorded for 60 days. Based on the distribution given below, what is the expected value of the number of defective springs produced by Machine A in any single day?
F. 0.00
G. 0.45
H. 0.70
J. 1.00
K. 1.50
Answers
Answer:
G. 0.45
Step-by-step explanation:
To find expected value, you simply multiply the value of each outcome (the numbers in the left column) by its probability
(the numbers in the right column) and then add them all together.
0(0.7) + 1(0.2) + 2(0.05) + 3(0.05)
0 + 0.2 + 0.1 + 0.15 = 0.3 + 0.15 = 0.45
The expected value is 0.45. Thus, the correct option is G.
What is the expected value?In parameter estimation, the expected value is an application of the weighted sum. Informally, the expected value is the simple average of a considerable number of individually determined outcomes of a randomly picked variable.
The expected value is given below.
E(x) = np
Where n is the number of samples and p is the probability.
The expected value is calculated as,
E(x) = 0 x 0.70 + 1 x 0.20 + 2 x 0.05 + 3 x 0.05
E(x) = 0 + 0.20 + 0.10 + 0.15
E(x) = 0.45
Thus, the correct option is G.
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Find the area of the shape shown below.
Answers
Answer:
The answer is 32
Step-by-step explanation:
The formula for finding the area of a trapezoid is:
((base 1 + base 2)/ 2 )* h
Now all you have to do is substitute the numbers in.
Note: bases will always be the ones like 12 and 4 in this case. We have just named then 1 and 2.
Answer:
32 square units
Step-by-step explanation:
\(\displaystyle A=\frac{1}{2}(b_1+b_2)h=\frac{1}{2}(12+4)(4)=\frac{1}{2}(16)(4)=\frac{1}{2}(64)=32\)
Note that \(b_1\) and \(b_2\) are the lengths of each base of the trapezoid, so it doesn't matter which is which.
Given that the measure of ∠x is 120°, and the measure of ∠y is 60°, find the measure of ∠z.
Answers
The measure of ∠x is 120°, and the measure of ∠y is 60°, find the measure of , ∠z = 0°
What about the triangle angle?A triangle's internal angles always add up to 180°, although its exterior angles are equal to the sum of the two interior angles that are not next to it. Another method for calculating a triangle's exterior angle is to subtract the angle of the vertex of interest from 180°.
Given that,
For triangle,
∠x is 120°
∠y is 60°
now , ∠z = [180° - (120°+60°)]
∠z = 0°
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what are the zeros of the polynomial y=3(x+4)(x+1)(x-3) a) -4
Answers
The zeroes of the given Polynomial equation; y=3 (x+4) (x+1) (x-3) are; -4, -1 and 3.
What are the zeroes of the given Polynomial equation?It follows from the task content that the zeroes of the given polynomial equation are to be determined.
Recall that the zeroes of a polynomial equation can be determined by equating all of the factors of the polynomial to zero.
Since the given Polynomial is; y=3(x+4)(x+1)(x-3).
The zeroes are as follows;
x + 4 = 0; x = -4.x + 1 = 0; x = -1.x - 3 = 0; x = 3.On this note, it can be concluded that the zeroes of the polynomial as required are; -4, -1 and 3.
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what is equivalent to 2/8
Answers
Answer:
1/4,
Step-by-step explanation:
Answer:
1/4
Step-by-step explanation:
A roofer leaned a 16-foot ladder against a house. If the base of the ladder is 5 feet from the house, how high up the house does the ladder reach?
Answers
Answer:
15.20feet
Step-by-step explanation:
Length of the ladder = 16foot (hypotenuse)
BAse of the ladder from the house = 5feet (adjacent)
height of the house = x (opposite)
Using the pythagoras theorem
hyp^2 = opp^2 + adj^2
16^2 = 5^2 + opp^2
256 = 25 + opp^2
opp^2 = 256-25
opp^2 = 231
opp =√231
opp = 15.20
Hence the house is 15.20feet high up the house
Determine the total cost of the items if they were NOT to be sold as a combo if both they were sold at R100 and save R37
Answers
If the items were not sold as a combo and were sold separately at R100 each, the total cost of the items would be R100 + R100 = R200.
What is Algebraic expression ?
Algebraic expression can be defined as combination of variables and constants.
If the items were sold separately for R100 each, and the combo price is R63 for both items (which saves R37), we can set up the following equations:
2x = 200 (total cost of the two items if sold separately)
x + y = 63 (combo price)
We can solve for x and y by subtracting the second equation from twice the first equation:
2x - (x + y) = 137
x - y = 37
Now we have two equations:
x + y = 63
x - y = 37
Adding these two equations gives:
2x = 100
x = 50
Substituting this value of x into either of the equations above gives:
y = 13
Therefore, if the items were not sold as a combo and were sold separately at R100 each, the total cost of the items would be R100 + R100 = R200.
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A regional survey found that 60% of all families who indicated an intention to buy a new car bought a new car within 3 months, that 10% of families who did not indicate an intention to buy a new car bought one within 3 months, and that 26% indicated an intention to buy a new car. If a family chosen at random bought a car, find the probability (as a percent) that the family had not previously indicated an intention to buy a car. (Round your answer to two decimal places.)
Answers
Let A be the event that a family chose at random indicated an intention to buy a new car and let B be the event that a family chose at random bought a new car within 3 months.
We are asked to find P(A'), which is the probability that a family chosen at random had not previously indicated an intention to buy a car.
We can use the given information to write the following conditional probabilities:
P(B | A) = 60%
P(B | A') = 10%
P(A) = 26%
We can use these probabilities to apply Bayes' theorem to find P(A' | B), which is:
P(A' | B) = P(B | A') * P(A') / P(B)
Substituting the values from the problem, we have:
P(A' | B) = (10%) * (100% - 26%) / (60% * 26% + 10% * (100% - 26%))
= (10%) * (74%) / (60% * 26% + 10% * 74%)
= (10%) * (74%) / (15.4% + 7.4%)
= (10%) * (74%) / (22.8%)
= 43.86%
Therefore, the probability that a family chosen at random had not previously indicated an intention to buy a car is 43.86%.
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Which of these statements is true?
Answers
Option B is correct, WX is perpendicular to ZY and WX is parallel to AB.
What is Coordinate Geometry?Coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space.
To find the solution of this problem.
Firstly let us understand the parallel and perpendicular lines.
Parallel lines are the lines that do not intersect or meet each other at any point in a plane.
Two geometric objects are perpendicular if they intersect at a right angle. The condition of perpendicularity may be represented graphically using the perpendicular symbol, ⟂.
As we check all the options, WX is perpendicular to ZY and WX is parallel to AB.
Hence, Option B is correct, WX is perpendicular to ZY and WX is parallel to AB.
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A restaurant with 25 tables has 4 tables by the window. Parties are seated randomly at the tables as they come in.
What is the probability that the first 3 parties of the night are all seated at window seats?
Answers
Answer:
50/50
Step-by-step explanation:
It's random
A three-way intersection is sometimes called a
T-intersection
Y-intersection
Both a and b
Neither a nor b
Answers
Answer:
Both A and B
Step-by-step explanation:
Both a and b. A three-way intersection can be referred to as a T-intersection or a Y-intersection, depending on the shape of the intersection. In a T-intersection, one road intersects another road perpendicularly, forming a T-shape. In a Y-intersection, one road splits into two branches, forming a Y-shape.
I do not understand why the answer is a) for this equation: y'=2y+x. I assumed that the answer is c) or a), because numbers in the equation are positive, but I'm not sure this is the correct method here
Answers
The slope field for the differential equation would be D . Graph D .
What are slope fields ?A slope field provides a pictorial representation of differential equations that displays the magnitude and direction of the derivative or slope for solution curves at various points in the plane .
The length of line segments represents the magnitude, while the direction indicates the sign of the slope.
The equation given is y = 2 y + x which means that the slope is positive. This is why we can tell that Graph D has the correct slope field as it goes up for positive.
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6. A company car purchased for $39,600 depreciates at 12% per annum. What is the car
worth after 3 years?
Answers
Answer:
$26,986.29
Step-by-step explanation:
We can use the formula for calculating the depreciation of an asset over time:
wor
\(\bold{D = P(1 - \frac{r}{100} )^t}\)
where:
D= the current value of the asset
P = the initial purchase price of the asset
r = the annual depreciation rate as a decimal
t = the number of years the asset has been in use
In this case, we have:
P = $39,600
r = 12% = 0.12
t = 3 years
Substituting these values into the formula, we get:
\(D= 39,600(1 - \frac{12}{100})^3\\D= 39,600(1 - 0.12)^3\\D= 39,600*0.88^3\\D= 39,600*0.681472\\D=26986.2912\)
Therefore, the car is worth approximately $26,986.29 after 3 years of depreciation at a rate of 12% per annum.
Answer:
$26,986.29
Step-by-step explanation:
As the car's value depreciates at a constant rate of 12% per annum, we can use the exponential decay formula to create a function for the value of the car f(t) after t years.
Exponential Decay formula\(\boxed{f(t)=a(1-r)^t}\)
where:
f(t) is the value of the car (in dollars) after t years.a is the initial value of the car.r is the depreciation rate (as a decimal).t is the time period (number of years after purchase).In this case, the initial value is $39,600, and the rate of depreciation is 12% per year. Therefore, the function that models the value of the car after t years is:
\(f(t)=39600(1-0.12)^t\)
\(f(t)=39600(0.88)^t\)
To calculate the value of the car after 3 years, substitute t = 3 into the function:
\(\begin{aligned} f(3)&=39600(0.88)^3\\&=39600(0.681472)\\&=26986.2912\\&=26986.29\;(\sf 2\;d.p.)\end{aligned}\)
Therefore, the car is worth $26,986.29 after 3 years.
Find the quotient of z₁ by z2. Express your answer in
trigonometric
form.
² - 3 (0 (4) + (*))
Z₁ cos
+/sin
Z₂
²2 = 7 (cos(377)+
COS
8
O A. 7 (cos (577) + i sin (5/77))
8
B.
21(cos(577)+isin (577))
8
OC. 21 cos
21(cos(-7)+ i sin(-77))
O D. 7 (cos(-7) + + sin(-7))
i
+/sin
37T
8
Answers
The quotient of z₁ by z₂ in trigonometric form is:
7/21 * (cos(584°) + i sin(584°))
To find the quotient of z₁ by z₂ in trigonometric form, we'll express both complex numbers in trigonometric form and then divide them.
Let's represent z₁ in trigonometric form as z₁ = r₁(cosθ₁ + isinθ₁), where r₁ is the magnitude of z₁ and θ₁ is the argument of z₁.
We have:
z₁ = 7(cos(577°) + i sin(577°))
Now, let's represent z₂ in trigonometric form as z₂ = r₂(cosθ₂ + isinθ₂), where r₂ is the magnitude of z₂ and θ₂ is the argument of z₂.
From the given information, we have:
z₂ = 21(cos(-7°) + i sin(-77°))
To find the quotient, we divide z₁ by z₂:
z₁ / z₂ = (r₁/r₂) * [cos(θ₁ - θ₂) + i sin(θ₁ - θ₂)]
Substituting the given values, we have:
z₁ / z₂ = (7/21) * [cos(577° - (-7°)) + i sin(577° - (-7°))]
= (7/21) * [cos(584°) + i sin(584°)]
The quotient of z₁ by z₂ in trigonometric form is:
7/21 * (cos(584°) + i sin(584°))
Option C, 21(cos(-7°) + i sin(-77°)), is not the correct answer as it does not represent the quotient of z₁ by z₂.
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Find the slope intercept form of the equation of the line through the points(2,-4) parallel to the line 8x+2y+2=0
Answers
Answer:
y = -4x + 4
Step-by-step explanation:
8x+2y+2=0
turn that into slope intercept form:
y = -4x - 1
a parallel line has the same slope, but different y-intercept
y = -4x + b
put (2,-4) into the equation
-4 = -4(2) + b
you get b=4
final product:
y = -4x + 4
A line passes through the points (-2, 1) and (x2, y2), and has a slope of 3/4. If point (x2, y2) is located in quadrant I, find x2.
A. 2
B. -1
C. -4
D. -3
Answers
Answer:
the answer is x=2
Step-by-step explanation:
You can actually solve this by inspection. For Quadrant I in the Cartesian plane the x-axis and y-axis must be positive. Hence x2 = 2 is the only positive answer in the option
NEED HELP FAST ASAP!!
Answers
Answer:
The answer is C. 7 and -7
Answer:
c
Step-by-step explanation:
I hope you get it right
Can someone help me?
Answers
Y=-x+3 is the slope-intercept form equation for the line.
What is meant by slope?The slope or gradient of a line is a numerical representation of a line's steepness and direction in mathematics. The reason for this usage is unknown, although it first appears in English in O'Brien (1844), who wrote "y = mx + b," and in Todhunter (1888), who wrote "y = mx + c." The letter m is widely used to signify slope.
The slope can be calculated by comparing the "vertical change" to "horizontal change" between any two distinct points on a line. In cases when the ratio is expressed as a quotient ("rise over run").
From the above graph,
The line is passing through the points (-1, 4) and (3, 0)
Let the slope intercept form be
y= mx +c
Slope m=(0-4/3+1)
=-4/4
= -1
y= -x+c
The equation is passing through the point (3, 0)
-3+c=0
c=3
y= -x+3
Therefore, y=-x+3 is the slope-intercept form equation for the line.
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5 tons of rocks cost $9,900.00. What is the price per pound?
Answers
Answer:
$0.99 per pound of rock
Step-by-step explanation:
We take
$9,900 / 5 = $1980 per ton of rock
1 ton = 2000 pound
What is the price per pound?
We take
1980 / 2000 = $0.99 per pound of rock
So, the price per pound of rock is $0.99
Please answer the question in the picture
Answers
Answer:
(y-8) (y-8)Step-by-step explanation:
→ y² -16y +64
→ y²-8y-8y+64
→ y(y-8) -8(y-8)
→ (y-8) (y-8)
P is inversely proportional to the square of Q
when q = 5 and p = 3.4
find the value of P when q = 2
show working
Answers
The numerical value of p is 21.25 when q is equal to 2, owing to the fact that p is inversely proportional to the square of q.
How to solve proportionality problems?Proportional relationships are relationships between two variables where their ratios are equivalent.
The inverse variation equation is expressed as; p∝ 1/q², then p = k/q².
Where k is the proportionality constant.
Given the data in the question;
q = 5p = 3.4k = ?First, we determine the proportionality constant.
p = k/q²
3.4 = k/(5)²
3.4 = k/25
k = 3.4 × 25
k = 85
Now, we find the value of p when q = 2.
p = k/q²
p = 85/(2)²
p = 85/4
p = 21.25
The numerical value of p is 21.25 when q is equal to 2, owing to the fact that p is inversely proportional to the square of q.
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Only 3 more questions to go
Answers
What is the equation of the line through (2,3) and (-1,-12)
Answers
Answer:
slope intercept: y=5x-7
Step-by-step explanation:
first you need slope: y2-y1/x2-x1
-12-3/-1-2=
-15/-3=
5
then plug into this formula: y-y1=m(x-x1)
y-3=5(x-2)
y-3=5x-10
y=5x-7
Amy's car holds at most 18 gallons of gas. Now it has 6 gallons. Use pencil and paper. Explain how to find the amount of gas she needs, in liters, to fill the gas tank.
Answers
Answer:
10 gallons
Amy's car's maximum capacity to hold gas = 19 gallons
present amount of gas = 9 gallons
As the car's capacity is 19 gallons and present amount in tank is 9 gallons, so we can fill it as much as so that the amount in tank becomes 19 gallons.
let assume she fills x gallons
then x gallons and 9 gallons present earlier should be equal to 19 gallon, as the tank can not hold more than that.
lets write it mathematically
x + 9 = 19
subtracting 9 from both side
x + 9 - 9 = 19 - 9
=> x = 10
The amount of gas she needs, is 12 liters, to fill the gas tank.
What is the unitary method?The unitary method is a method for solving a problem by the first value of a single unit and then finding the value by multiplying the single value.
Given that Amy's car's maximum capacity to hold gas is 18 gallons
The present amount of gas is 6 gallons
Thus we can fill it as much as so that the amount in the tank becomes 18 gallons.
Now assume she fills x gallons
Then x gallons and 6 gallons present earlier should be 18 gallons, as the tank can not hold more than that.
write it mathematically
x + 6 = 18
Now subtracting 6 from both sides
x + 6 - 6= 18 - 6
x = 12
Hence, the amount of gas she needs, is 12 liters, to fill the gas tank.
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A softball player hits a pitched ball when it is 4 feet above the ground. The initial velocity is 75 feet per second. Use the formula h=-16t^2+vt+s. How long will it take for the ball to hit the ground?
Answers
If the initial velocity is 75 feet per second, it will take approximately 5.125 seconds for the ball to hit the ground.
The given formula h= -16t²+vt+s represents the height (h) of an object thrown vertically in the air at time (t), with initial velocity (v) and initial height (s). In this case, we are given that the initial height of the softball is 4 feet and the initial velocity is 75 feet per second.
We want to find out how long it will take for the ball to hit the ground, which means we want to find the time (t) when the height (h) is 0.
Substituting the given values into the formula, we get:
0 = -16t² + 75t + 4
This is a quadratic equation in standard form, which we can solve using the quadratic formula:
t = (-b ± √(b² - 4ac)) / 2a
Where a=-16, b=75, and c=4. Substituting these values into the formula, we get:
t = (-75 ± √(75² - 4(-16)(4))) / 2(-16)
t = (-75 ± √(5625 + 256)) / (-32)
t = (-75 ± √(5881)) / (-32)
We can simplify the expression under the square root as follows:
√(5881) = √(49121) = 711 = 77
So we have:
t = (-75 ± 77) / (-32)
Simplifying further, we get two possible solutions:
t = 0.5 seconds or t = 5.125 seconds
Since the softball player hits the ball when it is 4 feet above the ground, we can disregard the solution t=0.5 seconds (which corresponds to when the ball is at its maximum height) and conclude that it will take approximately 5.125 seconds for the ball to hit the ground.
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what is the answer for 6x + 3(x-2)
Answers
Answer:
9x - 6
Hope this helps! <3
(If you need a step by step lmk)
Answer:
6x+3(x-2)
6x+3x-6
9x-6
please help
Select the correct answer. Which graph represents the solution to this inequality?
Answers
ANSWER i think its c Step-by-step explanation:
Help me out Ik you love Algebra as much as I do
Answers
Answer:
pop
yes no total
country yes 11 38 49
no 43 8 51
total 54 46 100
Hope you understand this haha
The answer is in the file. Hope it helps.
Graciela begins the month with $300 and is spending $20 per
week. Her brother Sergio begins the month with $150 and is saving $10 per week. Sergio wants to know when they will have the same amount of money.
W
Let represent the number of weeks since the beginning of the
month. Which equation represents this situation?
Answers
Answer:
300-20w=?
150+10w=?
Step-by-step explanation:
Answer:
Step-by-step explanation:
320