Where the graphs intersect or touch, the supplied statement is true, and that is the solution to any system.
What are the principles governing linear equations?One or two variables make up a linear equation. Neither the numerator nor the denominator of a fraction can be a variable in a linear equation raised to a power higher than 1. The lines that connect all the points on a coordinate grid when you identify the values that make a linear equation true are congruent.
A linear equation is what?An algebraic equation with simply a constant and a first-order (linear) term, such as y=mx+b, where m is the slope and b is the y-intercept, is known as a linear equation. The above is sometimes referred to as "linear equation of two variables," where y and x are the variables.
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A sandwich shop offers 4 types of bread, 5 meats, 7 cheeses, 4 condiment options, and you can pick veggies or no veggies. How many sandwiches are possible?
Answer:
Step-by-step explanation: OK so I don't know if I'm exactly right on this but what I would do is I would first read it more than 1 time and I would see that the sandwich shop offers 4 types of bread, that's 4 different types of bread. Then it says 5 meats, 7 cheeses, 4 condiment options, and you can pick veggies or no veggies. So what I did was I multiplied 4 by 5 and got 20, then I multiplied 7 by 4 and got 28, then I added 20 and 28 and got 48 sandwiches are possible. Then to make sure I was correct I did 4 times 5 times 7 times 4 by accident and I got 560. So I don't know which one is correct but I just thought that I could at least try to help you because even I have problems with problems like this one too. So I hope I helped you even a little bit.
What is the variability of the mean is 14 and the MAD is 3.2
Therefore, the estimated mean is roughly 0.154, but without more data, we are unable to calculate the SEM or group size.
Define mean.The total of all values divided by all of the values constitutes the result of a set, also called as the arithmetic mean. It is considered to be the most popular central trend indicator, and the word "mean" is commonly used to describe it. multiply the total amount of numbers in the library by the overall amount of values inside the gathering to get this result. Either original data or data that's been combined into frequency charts can be used for calculations. The average of a figure is known to as the average.
Here,
Given a normal distribution, the MAD and standard deviation (SD) are roughly inversely proportional to one another as follows:
=> MAD = 1.4826 × SD
As a result, we can calculate the standard deviation as follows:
=> SD = MAD / 1.4826
Since the MAD in this instance is provided as 3.2, we can calculate the standard deviation as follows:
=> SD ≈ 3.2 / 1.4826 ≈ 2.16
The standard error of the mean (SEM), which can be determined as follows, then provides the variability of the mean.
=> SEM ≈ SD / sqrt(n)
=> CV Equals SD/mean
To find the mean, we can change this equation as follows:
a) Mean Equals SD / CV
With the estimated numbers for SD and CV substituted, we obtain:
=> mean ≈ 2.16 / 14 ≈ 0.154
Therefore, the estimated mean is roughly 0.154, but without more data, we are unable to calculate the SEM or group size.
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The world's poorest people are ranked by their net worth.
O True
O False
Answer: I think its true
explanation: Correct me if im wrong ty!
Ethan and David are brothers. If Ethan is 3 years older than David and the sum of their ages is 17, how old is David?
Answer: 7 years old
Step-by-step explanation: The two quickest ways to solve these types of problems are by guess and check, and systems of equations. I will show you how to solve this problem with systems of equations since it is a systematic method.
We can replace the ages of Ethan and David with variables. Ethan's age is x and David's is y. Ethan is 3 years older than David and this can be represented by: y+3=x since 3 will have to be added to David's age, y to get Ethan's age, x. A second equation we can make is x+y=17 since the sum of their two ages is 17. Now we solve the system by first isolating a variable and then plugging that into the second equation.
y+3=x x is already isolated for us
x+y=17
(y+3)+y=17 We substitute x in the second equation for y+3
2y+3=17 Rearrange variables
2y=14
y=7 David's age, y, is 7.
Hope this helps!
I need help please answer 15 points !!!
The option that describes the correct order of steps for constructing an angle bisector of ∠DEF using only a straightedge and compass is option C
C.
Draw a circle with a radius less than [tex]\overline{DE}[/tex] and center E, and label the points of intersection G and H.Draw a line from point G to point HDraw a circle with radius [tex]\overline{GH}[/tex]and center G and another circle with radius [tex]\overline{GH}[/tex]and center H, and draw an equilateral triangle.Use straightedge to connect E to the rightmost vertex of the equilateral triangle. What is the angle bisector of an angle?An angle bisector is a segment that splits an angle to form two smaller congruent angles.
The steps for construct an angle bisector of the angle ∠DEF, by making use of a compass and a straightedge are;
With point E as the center, draw a circle with a radius less than the segment [tex]\overline{DE}[/tex], to intersect [tex]\overline{DE}[/tex] and [tex]\overline{EF}[/tex] at point G and HWith point G as the center, draw a circle with radius [tex]\overline{GH}[/tex], and with point E as center and with the radius [tex]\overline{GH}[/tex], draw another circleDraw an equilateral triangle to the points G and H from the point of intersection of the two circlesConnect point E to the rightmost vertex of the equilateral triangle, which is the point of intersection of the two circles to draw the angle bisector of ∠DEFLearn more on the geometric constructions here: https://brainly.com/question/29096545
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principal = $150
rate = 6%
time =?
simple interest = $54
Answer:
150 %6=es igual 1% dividimis 150 ÷100 para tener 150, se calcula el porcwntaje
Problem 5
Explain what you could do first to each side of the equation so that there would be no fractions.You do not have to solve the equations (unless you want more practice).
The first thing to do to each side of the equation so that there would be no fractions is to cross product.
The unknown number, a is -48
How to solve fractions?A fraction refers to a number which has a numerator (top value) and a denominator (bottom value).
2(a - 7) / 15 = (a + 4) / 6
cross product
6{2(a - 7)} = 15(a + 4)
open parenthesis
6(2a - 14) = 15a + 60
12a - 84 = 15a + 60
combine like terms
12a - 15a = 60 + 84
-3a = 144
divide both sides by -3
a = 144/-3
a = -48
In conclusion, the solution to the fraction is a = -48
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Please help as fast as possible! I know that’s what everyone asks but seriously I’ll give you points
Answer:
Number 2 and number 1
Step-by-step explanation:
youre all good
If f(x) = x2-9/x+5, find f(0), f(-1), f(1/k)
Step-by-step explanation:
f(0):
f(0) = (0^2 - 9) / (0 + 5)
= -9/5
f(-1):
f(-1) = ((-1)^2 - 9) / (-1 + 5)
= (-8) / 4
= -2
f(1/k):
f(1/k) = ((1/k)^2 - 9) / (1/k + 5)
= (1/k^2 - 9) / (1/k) + 5
= (1 - 9k^2) / (k^2 + 5k)
Therefore, f(0) = -9/5, f(-1) = -2, and f(1/k) = (1 - 9k^2) / (k^2 + 5k).
Answer:
Step-by-step explanation:
To find the values of f(x) at the given points, we substitute the values of x into the expression for f(x) and simplify:
f(x) = (x^2 - 9)/(x + 5)
a) To find f(0), we substitute 0 for x:
f(0) = (0^2 - 9)/(0 + 5) = -9/5
Therefore, f(0) = -1.8.
b) To find f(-1), we substitute -1 for x:
f(-1) = (-1^2 - 9)/(-1 + 5) = (-10)/4
We can simplify this fraction by dividing both the numerator and denominator by their greatest common factor of 2:
f(-1) = -5/2
Therefore, f(-1) = -2.5.
c) To find f(1/k), we substitute 1/k for x:
f(1/k) = ((1/k)^2 - 9)/(1/k + 5)
We can simplify the numerator by multiplying out the square:
f(1/k) = (1/k^2 - 9)/(1/k + 5)
We can simplify the denominator by multiplying by k/k:
f(1/k) = (1 - 9k^2)/(1 + 5k)
Therefore, f(1/k) = (1 - 9k^2)/(1 + 5k).
In the diagram below, triangle PQR is congruent with triangle STR. Complete the statement PQ is congruent with
A. RP
B. QT
C. ST
D. RQ
Answer:
Step-by-step explanation:
10 + 10 = 20
☁️Answer☁️ I think D.
️Step-by-step explanation️ hope this helps
Rectangle JKLM undergoes a sequence of transformations, resulting in rectangle J'K'L'M'.
The length of side K'L' is 6 units. The coordinates of vertex K' are (-3,2), and the coordinates of vertex M' are (3,-2).
Describe a sequence of transformations to rectangle JKLM that would result in rectangle J'K'L'M.
Show your work.
A sequence of transformations to rectangle JKLM that will result in rectangle J'K'L'M' is a translation by 3 units and -4 units, a reflection about the y-axis, and a rotation of 90 degrees counterclockwise about the origin.
What is a rectangle?A rectangle is a quadrilateral with four right angles and four sides of equal length. It is one of the most common shapes in nature and is found in everything from architecture to floor patterns. A rectangle is a two-dimensional shape that has four sides and four angles, all of which are right angles, and two pairs of opposite sides which are parallel.
In order to transform rectangle JKLM into J'K'L'M', a series of transformations must be applied. The first transformation is a translation of the rectangle along the x-axis by 3 units and along the y-axis by -4 units. This will move the original rectangle from its original position, JKLM, to the new position, J'K'L'M'. Next, a reflection about the y-axis needs to be applied, which will flip the rectangle around the y-axis. This will result in the length of side K'L' being 6 units. Finally, a rotation about the origin needs to be applied. The rotation should be 90 degrees counterclockwise in order for the coordinates of vertex K' to be (-3,2) and the coordinates of vertex M' to be (3,-2). Therefore, a sequence of transformations to rectangle JKLM that will result in rectangle J'K'L'M' is a translation by 3 units and -4 units, a reflection about the y-axis, and a rotation of 90 degrees counterclockwise about the origin.
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Solve this system of equations for x using the addition method. (Must show steps of addition method to get credit. Do not solve using the substitution method.)
The solution for the inequality expression given is x > -2.154.
What is Linear Inequalities?Linear inequalities are defined as those expressions which are connected by inequality signs like >, <, ≤, ≥ and ≠ and the value of the exponent of the variable is 1.
Given linear inequality is,
-5/7 (21x + 35) < 1/3 (9 - 6x)
We have to solve the inequality.
Using the distributive property,
(-5/7 × 21x) + (-5/7 × 35) < (1/3 × 9) - (1/3 × 6x)
-15x - 25 < 3 - 2x
-15x + 2x < 3 + 25
-13x < 28
-x < 2.154
Multiplying throughout by negative 1.
x > -2.154
Hence the solution is x > -2.154.
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15. The solid shape below is made up four equilateral triangles and a square. What is the sum of all the edges in the solid shape?
The sum of all the edges in the solid shape is 24 unit.
What is length?Length is a measurement οf distance οr size. It can refer tο the physical extent οf an οbject, the size οf an area, οr the amοunt οf time sοmething takes. Length is measured in units such as centimeters, meters, feet, inches, yards, miles, and sο οn. Length is an impοrtant cοncept in mathematics and physics, used tο measure distances and calculate speed and οther quantities.
The sum οf all the edges in the sοlid shape is 24. This is calculated by adding the 4 edges οf the square (4 x 1 = 4) plus the 12 edges οf the 4 equilateral triangles (12 x 1 = 12).
Therefοre, 4 + 12 = 24.
Since all the sides οf the triangles and the square are equal in length, it can be said that all the edges in the sοlid shape are equal. This means that the length οf each edge is equal tο 24/24, οr 1. This means that the length οf each edge οf the sοlid shape is 1 unit.
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Complete question:
The solid shape below is made up four equilateral triangles and a square. What is the sum of all the edges in the solid shape?
[Given each side is 1 unit]
Which of the lines or segments below is secant to circle
The lines or segments which are secant to a circle are:
AK, TW, RN & UN
What is secant of the circle?A line that crosses a circle at two different locations is referred to as a "secant of a circle" in geometry. The Latin verb "secare," which meaning "to cut," is where the English term "secant" originates.
More technically, a secant of a circle is a line that crosses it twice, whereas a tangent of a circle is a line that crosses it only once. A circle's diameter is a chord that runs through its center, whereas a chord of a circle is a line segment connecting two locations on the circle.
Given that,
Two circle which contains various tangents, normal & secants
As we know from the definition of secant it intersect the circle at two points,
Line segments AK, TW, RN & UN are intersecting the circle at two points,
so all this segments are secant of the circle.
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The youth center has installed a swimming pool on level ground. The pool is a right circular cylinder with a diameter of 24 feet and a height of 6 feet. To the nearest cubic foot, what is the volume of water that will be in the pool when it is filled with?
Rounding to the nearest cubic foot, the volume of water that will be in the pool when it is filled is approximately 2713 feet³.
What is volume?
Each thing in three dimensions takes up some space. The volume of this area is what is being measured. The space occupied within an object's borders in three dimensions is referred to as its volume. It is sometimes referred to as the object's capacity.
The first step is to find the radius of the cylinder.
Since the diameter is given as 24 feet, the radius is half of the diameter or 12 feet.
Next, we can use the formula for the volume of a cylinder -
V = πr²h
where V is the volume, r is the radius, and h is the height.
Substituting the given values, we get -
V = π(12)²(6)
= π(144)(6)
= 2712.96
≈ 2713 cubic feet
Therefore, the volume value is obtained as 2713 cubic feet.
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If the two shakes below are similar, calculate the value of x.
X=
Answer:
try 1/2
Step-by-step explanation:
Henry claims that MN is a midsegment of AJKL. Explain why this is not true.
50 Points, Will Give Brainlist and Hearts if work is shown and correctly.
I bought my daughter 2 guppy fish for her birthday. Guppy populations grow at a rate of 1700% per month.
Write a formula that models the population of guppies and find the number of guppies I will have in 4 months.
Answer:
2x(1700%x4)+2=138
Step-by-step explanation:
2x2.00=4 2x17=34 34 per month plus the 2 that they already had. 34x4=136 there will be 136 plus 2 after the 4 months
You want to estimate the average class size of Basic Statistics classes at a certain college. You take a random sample of 50 students who are taking Basic Statistics this semester and ask them how large their class is. You average the results and get an average class size of 38. You then look at an Admissions and Records spreadsheet for this semester for the same college and get an average class size of Basic Statistics equal to 25. This is too large of a difference to attribute to random variation. Also, there's nothing wrong with your sample. What's wrong?
The sample size οf 50 might nοt be large οr randοm enοugh, and there might be sοme systematic bias οr errοr in the data sοurces, leading tο a large difference between the sample mean (38) and Admissiοns and Recοrds data (25) in estimating the average class size οf Basic Statistics classes at the cοllege.
What are the statistics?Statistics is a branch οf mathematics that deals with the cοllectiοn, analysis, interpretatiοn, presentatiοn, and οrganizatiοn οf data. It invοlves the use οf mathematical tοοls and techniques tο extract useful infοrmatiοn frοm data, and tο make decisiοns οr predictiοns based οn that infοrmatiοn.
In the given scenariο, it seems that there might be an issue with the methοd used tο cοllect the data, οr the data sοurces might be different. The sample size οf 50 students is nοt large enοugh tο represent the whοle pοpulatiοn οf Basic Statistics classes at the cοllege, especially if the cοllege has many sectiοns οf the class οr the class size varies widely between sectiοns. Mοreοver, the sample might nοt be truly randοm, as there cοuld be sοme bias in hοw the sample was selected, such as selecting οnly students frοm a particular sectiοn οr time slοt.
Alsο, the difference between the average class size οbtained frοm the sample (38) and the οne οbtained frοm Admissiοns and Recοrds (25) is tοο large tο attribute tο randοm variatiοn, which suggests that there might be sοme systematic bias οr errοr in either the sample οr the Admissiοns and Recοrds data.
Tο get a mοre accurate estimate οf the average class size οf Basic Statistics classes at the cοllege, a larger and mοre representative sample might be needed, οr the data cοllectiοn methοd might need tο be revised.
Hence, The sample size οf 50 might nοt be large οr randοm enοugh, and there might be sοme systematic bias οr errοr in the data sοurces, leading tο a large difference between the sample mean (38) and Admissiοns and Recοrds data (25) in estimating the average class size οf Basic Statistics classes at the cοllege.
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The area of each figure can be calculated using the given dimensions. Which of the
figures has an area formula that is a linear function?
The figure that has an area formula that is a linear function is the rectangle.
What is area?Area is the size or measurement of a two-dimensional space. It is usually expressed in square units such as feet, meters, or kilometers. Area is a fundamental concept in mathematics, and is used to measure the size of shapes and surfaces. Area is also used to calculate the amount of material or resources needed to cover a given space. In addition, it can be used to measure the cost or value of a piece of land.
The formula for the area of a rectangle is A=lw, where l is the length and w is the width. This formula is a linear equation because it is a product of two variables, where the relationship between the two variables is a constant. Thus, the area of a rectangle is a linear function of its dimensions.
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What value of x makes the equation 1+\sqrt{x-1=5} true?
The value of x is -1 and 3 for which x makes the equation.
What is factorization?
Writing a number or other mathematical object as the result of numerous factors—typically smaller or simpler objects of the same kind—is known as factorization or factoring in mathematics.
Here, we have
Given: 1 + (x-1)² = 5
We have to solve it for x and find the value of x.
1 + (x-1)² = 5
We solve this equation by factorization and we get
1 + x² - 2x + 1 = 5
x² - 2x + 2 -5 = 0
x² - 2x - 3 = 0
x² + x - 3x - 3 = 0
x(x+1) -3(x+1) = 0
(x+1)(x-3) = 0
x= -1, 3
Hence, the value of x is -1 and 3 for which x makes the equation.
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Given
f(x) = 4x2 − 1
and
g(x) = −4x + 8,
find the following.
f(2) − g(2)
This means that when function f(x) = 4x²- 1 and g(x) = 4x + 8, f(2) - g(2) = 15, and by using the equation as a substitute.
what is function ?
A function in mathematics is a relationship between a set of inputs (also referred to as the domain) and a set of outputs (also referred to as the range), where each input has a unique outcome. Typically, the variable x stands for the input values, the variable y for the output values, and the variable f for the function itself (x). Tables, graphs, equations, and verbal descriptions can all be used to represent functions. In a wide variety of disciplines, including science, engineering, economics, and social studies, they are employed to model relationships between quantities. Functions are frequently used to explain the relationship between two quantities.
given
We must first determine the values of f(2) and g(2) independently using the provided functions in order to evaluate f(2) - g(2):
f(2) = 4(2)² - 1 = 15
g(2) = -4(2) + 8 = 0
These values can now be added to the expression f(2) - g(2) as a substitute:
f(2) - g(2) = 15 - 0 = 15
This means that when f(x) = 4x²- 1 and g(x) = 4x + 8, f(2) - g(2) = 15, and by using the equation as a substitute.
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Question Which statements are true? Select each correct answer. Responses 24g4+18g2=6g2(4g2+3g) 24 g begin power 4 end power plus 18 g squared equals 6 g squared left parenthesis 4 g squared plus 3 g right parenthesis 4g2−g=g2(4−g) 4 g squared minus g equals g squared left parenthesis 4 minus g right parenthesis 9g3+12=3(3g3+4) 9 g cubed plus 12 equals 3 left parenthesis 3 g cubed plus 4 right parenthesis 35g5−25g2=5g2(7g3−5)
All four statements are true. The first statement is true because 24g4+18g2 can be factored into 6g2(4g2+3g). The second statement is true because 4g2−g can be factored into g2(4−g). The third statement is true because 9g3+12 can be factored into 3(3g3+4). The fourth statement is true because 35g5−25g2 can be factored into 5g2(7g3−5).
What are parenthesis?Parenthesis are punctuation marks that are used to set apart a clause or phrase within a sentence. They are also known as round brackets and are used to add additional information or asides to a sentence. Parenthesis are used to provide additional information, to emphasis a point, to give examples, to define terms, and to explain ideas.
The first statement is 24g4+18g2=6g2(4g2+3g). This translates to 24g to the fourth power plus 18g squared equals 6g squared left parenthesis 4g squared plus 3g right parenthesis. The second statement is 4g2−g=g2(4−g). This translates to 4g squared minus g equals g squared left parenthesis 4 minus g right parenthesis. The third statement is 9g3+12=3(3g3+4). This translates to 9g cubed plus 12 equals 3 left parenthesis 3g cubed plus 4 right parenthesis. The fourth statement is 35g5−25g2=5g2(7g3−5). This translates to 35g to the fifth power minus 25g squared equals 5g squared left parenthesis 7g cubed minus 5 right parenthesis.
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What are all possible values of x that satisfy the inequality x3−8≥1?
By answering the above question, we may state that Therefore, all possible values of x that satisfy the inequality [tex]x^3[/tex] - 8 ≥ 1 are x ≥ 2.08.
What is inequality?A connection between two expressions or values that is not equal in mathematics is referred to as an inequality. Hence, inequity results from imbalance. In mathematics, an inequality establishes the connection between two non-equal numbers. Egality and inequality are not the same. Use the not equal symbol most frequently when two values are not equal (). Values of any size can be contrasted using a variety of inequalities. By changing the two sides until just the variables are left, many straightforward inequalities may be solved. Nonetheless, a variety of factors support inequality: Both sides' negative values are split or added. Exchange the left and the right.
[tex]x^3[/tex] - 8 ≥ 1
[tex]x^3[/tex] ≥ 9
Taking the cube root of both sides, we get:
x ≥ ∛9
Since ∛9 ≈ 2.08, the solution to the inequality is:
x ≥ 2.08
Therefore, all possible values of x that satisfy the inequality [tex]x^3[/tex] - 8 ≥ 1 are x ≥ 2.08.
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A petri dish of bacteria grow continuously at a rate of 200% each day. If the petri dish began with 10 bacteria, how many bacteria are there after 5 days? Use the exponential growth function f(t) = aert, and give your answer to the nearest whole number.
Answer: 320
Step-by-step explanation:
Multiply each time by 2 so 10*2=20 then 40, 80, 160, and 320
I need help please! Im not sure how to solve
3x^2 + 4x - 3 = ( 3x -2 ) (x + 2) + 1
What is Remainder Theorem ?
Remainder Theorem is an approach of Euclidean division of polynomials. According to this theorem, if we divide a polynomial P(x) by a factor ( x – a); that isn't essentially an element of the polynomial; you will find a smaller polynomial along with a remainder.
Given:- p(x) = 3x^2 + 4x - 3 and g(x) = 3x - 2
p(x) ÷ g(x)
3x^2 + 4x - 3 ÷ ( 3x - 2)
[tex]\frac{3x^2+4x-3}{3x-2} =(x+2 )+ \frac{1}{3x-2}[/tex]
Quotient = x + 2
Remainder = 1
Hence, 3x^2 + 4x - 3 = ( 3x -2 ) (x + 2) + 1
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The library in Booneville collected $3,470 in overdue fees last year. This year, the library collected $347. What is the percent of decrease in the amount collected?
simplify:(-8+35)+|-16+4| a.15 b.31 c.39d.55
c. 39
Step-by-step explanation:
the answer is 39
the domain of discourse is set of male patients in a clinical study. define the following predicates: p(x): x was given the placebo
(a) One patient did not receive the medication.
(b) One patient did not get either the medicine or the placebo.
(c) None of the patients took the drug or experienced migraines (or both).
(d) One patient who received a placebo experienced no migraines.
what is variable?A variable in algebra is an alphabet that is used to symbolize the unknowable integer. It stands in for the value. An amount that can fluctuate depending on the mathematical issue is referred to as a variable. The generic letters x, y, and z are frequently employed in mathematical expressions and equations. A variable, then, is a symbol denoting a number whose value is unknown.
from the question:
(a)
Logical expression: ∀x D(x)
Negation: ¬(∀x D(x))
Applying De Morgan's law: ∃x ¬D(x)
One patient did not receive the medication.
(b)
Logical expression: ∀x (D(x) ∨ P(x))
Negation: ¬(∀x (D(x) ∨ P(x)))
Applying De Morgan's law: ∃x ¬(D(x) ∨ P(x))
Applying De Morgan's law: ∃x (¬D(x) ∧ ¬P(x))
One patient did not get either the medicine or the placebo.
(c)
Logical expression: ∃x (D(x) ∧ M(x))
Negation: ¬∃x (D(x) ∧ M(x))
Applying De Morgan's law: ∀x ¬(D(x) ∧ M(x))
Applying De Morgan's law: ∀x (¬D(x) ∨ ¬M(x))
None of the patients took the drug or experienced migraines (or both)
(d)
Logical expression: ∀x (P(x) → M(x))
Using the conditional identity: ∀x (¬P(x) ∨ M(x))
Negation: ¬∀x (¬P(x) ∨ M(x))
Applying De Morgan's law: ∃x ¬(¬P(x) ∨ M(x))
Applying De Morgan's law: ∃x (P(x) ∧ ¬M(x))
One patient who received a placebo experienced no migraines.
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complete question;
In the following question, the domain of discourse is a set of male patients in a clinical study. Define the following predicates:
P(x): x was given the placebo
D(x): x was given the medication
M(x): x had migraines
Translate each statement into a logical expression. Then negate the expression by adding a negation operation to the beginning of the expression. Apply De Morgan's law until each negation operation applies directly to a predicate and then translate the logical expression back into English.
Sample question: Some patient was given the placebo and the medication.
∃x (P(x) ∧ D(x))
Negation: ¬∃x (P(x) ∧ D(x))
Applying De Morgan's law: ∀x (¬P(x) ∨ ¬D(x))
English: Every patient was either not given the placebo or not given the medication (or both).
(a)Every patient was given the medication.
(b)Every patient was given the medication or the placebo or both.
(c)There is a patient who took the medication and had migraines.
(d)Every patient who took the placebo had migraines. (Hint: you will need to apply the conditional identity, p → q ≡ ¬p ∨ q.)
Which number can be placed in the blank to make this number sentence true? In other words, 316
times which number equals 1?
The number that can be placed in the blank to make the number sentence true is given as follows:
1/316.
How to solve the number sentence?The number sentence for this problem is given as follows:
"316 times which number equals 1".
The number is unknown, hence the variable that represents the number is given as follows:
x.
The times word means a multiplication operation, hence the expression is given as follows:
316x = 1.
Now we must simply solve the expression 316x = 1 for the variable x, applying the division, which is the inverse operation of the multiplication, hence:
x = 1/316.
Which is the number that satisfies the number sentence.
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