Squaring both sides of an inequality can be valid if you know both sides are non-negative, but it can introduce extraneous solutions. Therefore, it's important to check any solutions against the original inequality to ensure their validity, especially in cases involving negative numbers or functions with restricted domains.
When you're working with inequalities, you have to be careful when performing operations like squaring both sides. Unlike equalities, where multiplication or division by the same number on both sides does not change equality, with inequalities, the effect can be more complex due to the direction of the inequality sign and the possibility of dealing with negative numbers.
For instance, squaring both sides of an inequality is not always a valid operation because if one or both sides of the inequality are negative, squaring could lead to incorrect results. When you square a negative number, it becomes positive, which could potentially reverse the inequality's direction. However, if you know that both sides of the inequality are non-negative, then squaring both sides is permissible. This concept is similar to solving quadratic constraints without introducing square roots, using identities like |(1 + ix)²|² = ([1 + ix|²)².
To avoid introducing solutions that were not there originally (extraneous solutions), it is important to check the solutions obtained after squaring against the original inequality. An example where squaring both sides might be questioned is when solving trigonometric inequalities, where a common mistake is to square both sides without considering the domain of the original function.
Which expression can be used to determine the average rate of change in f(x) over the interval [2,9]
a) f(9-2)
b) f(9)-f(2)
c) f(9-2)/9-2
c) f(9)-f(2)/9-2
Answer:
D. f(9-2)/9-2 on Edg2020
Step-by-step explanation:
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Carlos graphs the equations y = 0.5x^2 + 3 and y = –4x^2 + 24x – 35 and generates the graph below. Which conclusion is supported by the graph?
A. The equation 0.5x2 + 3 = –4x2 + 24x – 35 has no solutions.
B. The equation 0.5x2 + 3 = –4x2 + 24x – 35 has one solution.
C. The equation 0.5x2 + 3 = –4x2 + 24x – 35 has two solutions.
D. The equation 0.5x2 + 3 = –4x2 + 24x – 35 has three solutions.
Answer: The correct option is A. The equation 0.5x2 + 3 = –4x2 + 24x – 35 has no solutions.
Explanation:
The given equations are,
[tex]y=0.5x^2+3[/tex]
[tex]y=-4x^2+24x-35[/tex]
Since the above equations are quadratic equations, therefore they area parabola.
The coefficient of first equation is a positive therefore it is an upward parabola. The coefficient of second equation is a negative therefore it is an downward parabola.
From given figure it is noticed that the vertex of first parabola is (0,3) and the vertex of second parabola is (3,1).
The vertex is the extreme point of the parabola. So for first equation [tex]y\geq 3[/tex] and for second equation [tex]y\leq 1[/tex]. Therefore the parabolas will never intersect each other.
Since there is no intersection between parabolas therefore the equation
[tex]0.5x^2+3=-4x^2+24x-35[/tex] has no solution and the correct option is A.
what rotation will map the figure onto itself?
The correct answer is 180° about the center.
To map the given figure onto itself, we need to find the rotational symmetry. Let’s analyze the “X” shape:
A 90° rotation would not work because it would overlap the arms of the “X.”
Similarly, a 270° rotation would also result in overlapping arms.
A 45° rotation would not preserve the original shape due to the asymmetry of the “X.”
However, a 180° rotation about its center will bring each arm of the “X” back to a position occupied by another arm, effectively mapping it onto itself. An “X” has rotational symmetry at 180°; every half turn gives us an identical view.
The correct answer is 180° about the center. This specific degree of rotation is required due to the symmetrical nature of an “X” shape where each arm aligns with another after being rotated by this angle.
help will mark brainliest
What is the following product
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(06.01)Which point on the scatter plot is an outlier?
Point H
Point I
Point J
Point K
You invest a total of $5800 in two investments earning 3.5% and 5.5% simple interest. Your goal is to have a total annual interest income of $283. Write a system of linear equations that represents this situation where x represents the amount invested in the 3.5% fund and y represents the amount invested in the 5.5% fund. Solve this system to determine the smallest amount that you can invest at 5.5% in order to meet your objective.
If g is the variable, which mathematical sentence expresses the information below? "The number of gallons of gas in the tank plus 7 more adds up to 12 gallons."
Answer: [tex]g+7=12[/tex]
Step-by-step explanation:
The given statement : The number of gallons of gas in the tank plus 7 more adds up to 12 gallons.
Given variable : 'g'
if g represents the number of gallons of gas in the tank, then for the given statement we have the following expression:-
[tex]g+7=12[/tex]
Hence, the mathematical sentence expresses the given information :-
[tex]g+7=12[/tex]
Let u = <-9,4>, v = <8, -5>. Find u - v.
To find the difference between vectors u and v, subtract corresponding components.
Given vectors u = <-9,4> and v = <8, -5>, to find the difference u - v, subtract corresponding components:
Subtract the x-components: -9 - 8 = -17
Subtract the y-components: 4 - (-5) = 9
Therefore, u - v = <-17, 9>.
how many cookies will tony have if he bakes 5 more batches y=90+ 14 (x)
replace x with 5
y=90+14(5)
14*5 =70
90+70 =160
y=160
he will have 160 cookies
Answer:
Given the equation:
[tex]y = 90+14x[/tex] .....[1]
where,
y represents the total number of cookies and
x represents the number of batches bake.
From the statement:
we have;
x = 5
Substitute in [1] we have;
[tex]y = 90+14 \cdot 5 = 90+70 = 160[/tex]
Therefore, 160 cookies will tony have if he bakes 5 more batches
Calculate the upper and lower limit for a 95% confidence interval about this mean.
A family needs a new car, but isn't sure they can fit the payment into their budget. A sample of 36 months of grocery bills yields a mean of $94 with a standard deviation of $10. If the upper limit of a 95% confidence level is below $100, the family can afford to buy the car.
Standard error = (standard deviation)/(square root of sample size)
Upper limit (dollars and cents)
Lower limit (dollars and cents)
Answer:Upper limit ⇒ 94+(1.96)(10÷√36) = 94+(1.96)(10/6) = $97.27
Lower limit ⇒ 94 - (1.96)(10÷√36) = 94 - (1.96)(10/6) = $90.73
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A store is selling two mixtures of nuts in 20-ounce bags. The first mixture has 15 ounces of peanuts combined with five ounces of cashews, and costs $4.25. The second mixture has five ounces of peanuts and 15 ounces of cashews, and costs $6.75. How much does one ounce of peanuts and one ounce of cashews cost?
Select one:
a. $0.40 for peanuts and $0.15 for cashews
b. $0.15 for peanuts and $0.40 for cashews
c. $0.12 for peanuts and $0.49 for cashews
d. There is no solution.
The cost of one ounce of peanuts is $ 0.15 and the cost of one ounce of cashews is $ 0.40. Then the correct option is B.
What is the linear system?A Linear system is a system in which the degree of the variable in the equation is one. It may contain one, two, or more than two variables.
A store is selling two mixtures of nuts in 20-ounce bags.
The first mixture has 15 ounces of peanuts combined with five ounces of cashews and costs $4.25.
The second mixture has five ounces of peanuts and 15 ounces of cashews and costs $6.75.
Let the cost of one ounce of peanuts be x and one ounce of cashews cost be y. Then we have
15x + 5y = 4.25 ...1
5x + 15y = 6.75 ...2
By solving equations 1 and 2, Then we have
x = $ 0.15 and y = $ 0.40
Then the cost of one ounce of peanuts is $ 0.15 and the cost of one ounce of cashews is $ 0.40.
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Assuming that n 1 , rewrite the compound interest equation B t 900 1.185 t in the general form.
The table and the graph below each show a different relationship between the same two variables, x and y:
How much more would the value of y be on the graph than its value in the table when x = 12?
Please just help me understand, do not give me the answer! thank you
The football team has 30 more members than the basketball teams. If each group had 10 more members, the ratio of their membership would be 3.2. How many members are in each group? Write your answer as football x basketball y
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three consecutive even numbers have a sum where one half of the sum is between 84 and 96
86 *2 = 172
172/3 = 57.33
54 + 56 +58 =168
168/2 =84
56 + 58 +60 = 174
174/2 =87
58 +60 +62 = 180
180/2 = 90
60 +62 +64 = 186
186/2 = 93
62 +64+66 = 192
192/2 = 96
there are 3 sets of numbers that fall between 84 & 96
1 set equals 84 and 1 set equals 96
so if you are counting those there are 5 sets
Sammy borrowed $10,000 to purchase a new car at an annual interest rate of 11%. She is to pay it back in equal monthly payments over a 5-year period. How much total interest will be paid over the period of the loan? Round to the nearest dollar.
Write logbx + logby - logbz as a single logarithm.
What is the sum of the two vectors (2,-1) and (3,3)?
One of the UK lottery systems consists of selecting six numbers from forty-nine for a one-pound stake. The winning numbers are drawn at random and the order is not important. Determine the probability that a randomly selected set of six numbers will win the lottery.
4.167 × 10-6
8.63 × 10-7
5.467 × 10-6
7.15 × 10-8
The answers is D.)7.15 × 10-8
Answer:
7.15 × 10-8 0 or 7.15 * 10^-8
Step-by-step explanation:
(6!(49 - 6)!)/49!
720/10068347520
= 7.15 * 10^-8
I hope this answer gives you an idea of the process used to find the answer.
Enter the degree of the polynomial below
7x^7+10x^4+4x^3-5x^11-10x^6-6x^7
A. 11
B. 10
C. 6
D. 7
What is the equation of the quadratic graph with a focus of (1, 1) and a directrix of y = −1?
f(x) = − one fourth (x − 1)2 + 1
f(x) = − one fourth (x − 1)2
f(x) = one fourth (x − 1)2 + 1
f(x) = one fourth (x − 1)2
Solve and check the inequality
|8m+4|<12
A regular octagon rotates 360° about its center. How many times does the image of the octagon coincide with the preimage during the rotation?
In your lab, a substance's temperature has been observed to follow the function T(x) = (x + 5)3 + 7. The turning point of the graph is where the substance changes from a liquid to a solid. Explain to your fellow scientists how to find the turning point of this function?
The curve y = x/(1 + x2) is called a serpentine. find an equation of the tangent line to this curve at the point (2, 0.40). (round the slope and y-intercept to two decimal places.) y =
The equation of the tangent line to the curve y = x/(1 + x2) at the point (2, 0.40) is given by y = -0.24x + 0.88, rounded to two decimal places.
Explanation:The subject of the question refers to calculus, specifically the concept of finding the equation of a tangent line to a curve. This process typically involves finding the derivative of the function (which gives the slope of the tangent), and then using the point-slope form of the line equation.
The derivative of y = x/(1 + x2) can be found using the quotient rule for differentiation and is given by y' = (1 - x2)/(1 + x2)2. Evaluating this at x = 2 gives a slope of about -0.24 (rounded to two decimal places).
The equation of the line is usually written in the slope-intercept form, y = mx + b, where m represents the slope and b represents the y-intercept. Since we know that the line passes through the point (2, 0.40), we can substitute these values into the equation and solve for b. This yields a y-intercept of about 0.88 (rounded to two decimal places).
Therefore, the equation of the tangent line to the curve y = x/(1 + x2) at the point (2, 0.40) is approximately y = -0.24x + 0.88.
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If x and y are both negative when is x-y positive
A plane has 360 total seats, which are divided into economy class and business class. For every 13 seats in economy class, there are 5 seats in business class.
13 +5 =18
360/18=20
13*20 = 260
5*20 =100
260 economy & 100 business
Do as follows:
13 +5 =18
360/18=20
13*20 = 260
5*20 =100
260 economy & 100 business
HELP! Algebra!
The following two-way table shows the number of students of a school who have a cell phone and/or have a part-time job:
Based on the table, how many students have both a cell phone and a part-time job?
Answers:
A- 10
B- 40
C- 50
D- 100
The number of students have both a cell phone and a part-time job is 40.
The correct option is (B)
What is algebra?
Algebra is a branch of mathematics that deals with symbols or variables and uses arithmetic operations (+, –, ×, ÷) to find the unknown quantities represented by these variables.
Given table shows that,
The students having whether having cell phones or not and whether the students are having part time job or not.
As we can see from the Column I shows students having cellphones and the first row shows students having part time job. which is 40.
Hence, the students have both a cell phone and a part-time job is 40.
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