Firstly, let us define natural numbers.
Natural numbers are simply positive intergers, that is integers that are greater than zero e.g 1,2,3,4...
Secondly, when is a set said to be closed under a particular operation?
A set is said to be closed under an operation if and only if the operation on any two elements of the set produces another element of the same set. for example when you perform an operation (like addition) on a set of number (like natural numbers) it must give you a number from that same set.
E.g
let's check whether natural numbers are closed under the operation of adddition.
taking two numbers 3 and 5.
[tex]3+5=8[/tex]since the operation gives us 8, which is a natural number, it means natural numbers are closed under the operation of addition.
Back to the main question, which is to check whether natural numbers are closed under the operation of subtraction.
taking two cases.
case 1; taking numbers 3 and 5
[tex]3-5=-2[/tex]case 2: taking numbers 7 and 2
[tex]7-2=5[/tex]From the two cases, case 2 gave us a natural number while case 1 does not give us a natural number.
This means that Natural numbers are not Closed under subtraction.
Because subtraction operation on natural numbers does not necessarily produce another natural number.
On a general note;
Natural numbers are closed under Addition and Multiplication operations
while They are not closed under subtraction and division operation
Solve the following inequality. Graph the solution set and then write it in interval notation .
Given:
-2x ≥ 6
Solve for x
Divide both sides by -2
-2x/-2 ≤ 6/-2
x ≤ -3
Graph:
Interval notation (-∞, -3 ]
Rectangle ABCD has vertex coordinates
A(1, -2), B(4, -2), C(4, -4), and D(1,
-4). It is translated 1 unit to the left and 1 3 units up. What are the coordinates
of B?
A vertex is a point on a polygon where two rays or line segments meet, the sides, or the edges of the object come together. Vertex is the plural form of vertices.
Response: C
What is a graph's vertex?A node of a graph, or one of the points on which the graph is defined and which may be connected by graph edges, is referred to as a "vertex" in computing.
For instance, a rectangle's four sides result in its four vertices.
Response: C . The coordinates are obtained by first subtracting 1 from 4 to obtain 3 and then adding 3 to -2 to obtain 1. (3, 1)
The vertex is the collective endpoint. Vertex, on the other hand, refers to the common terminal point where two rays converge to make an angle. In a similar manner, we must understand an angle's arm. The term "arm of an angle" refers to the two rays that unite to make an angle.
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Solving Equations ** Reminder need to show ALL work **
solution
For this case we have the following equation:
[tex]\frac{3}{4}x-5=4[/tex]Then we can add 5 in both sides and we got:
[tex]\frac{3}{4}x=9[/tex]Then we can multiply both sides by 4/3 and we got:
[tex]x=9\cdot\frac{4}{3}=12[/tex]And the final solution for this case is x= 12
how can you obtain the points for the log below from its inverse?
Answer
Check Explanation
Explanation
The laws of logarithms will come in handy in helping to obtain the points for the log from its inverse.
Before we commence, the two laws of logarithms that we will be using is that
log₄ 4ᶜ = c (log₄ 4)
And
log₄ 4 = 1
So, no matter the value of inverse given, a simple mathematical evaluation will give the log points.
when x = 1
f(x) = log₄ x
f(1) = log₄ 1 = log₄ 4⁰ = 0 log₄ 4 = 0 × 1 = 0
when x = 4
f(x) = log₄ x
f(4) = log₄ 4 = log₄ 4¹ = 1 (log₄ 4) = 1 × 1 = 1
when x = 16
f(x) = log₄ x
f(16) = log₄ 16 = log₄ 4² = 2 (log₄ 4) = 2 × 1 = 2
when x = 64
f(x) = log₄ x
f(64) = log₄ 64 = log₄ 4³ = 3 (log₄ 4) = 3 × 1 = 3
Hope this Helps!!!
Slope =
y-intercept = (0,
Answer:
y intercept= (0,-3)
slope= 2/1 or simplified 2
Step-by-step explanation:
I need to write this equation that has a infinite number of solutions
2 (8x+4) -x =
Simplify the equation:
2(8x)+2(4) -x
16x +8 -x
15x +8
If both sides of the equation are equal o equivalent, there is an infinite number of solutions.
2 (8x+4) -x = 15x + 8
2. State two (2) values of θ (theta) to the nearest degree forsin θ = − 0. 966
To find a value of θ theta given a value of sin(θ) we must use the arcsin function, it receives a value of an sin as argument and returns the value of the angle θ. Then we must use a calculator and input
[tex]\begin{gathered} \theta=\arcsin\left(x\right) \\ \\ \theta=\arcsin(-0.966) \\ \\ \theta=−75 \end{gathered}[/tex]The result is already rounded to the nearest degree. Therefore, one value of θ that satisfies sin θ = −0.966 is θ= -75°
Now to find the other value we will look at the symmetry in the trigonometric circle:
Then, the other value of theta will be
[tex]\begin{gathered} \theta_2=-75°-30° \\ \\ \theta_2=105° \end{gathered}[/tex]Final answer:
[tex]\begin{gathered} \theta=-75° \\ \theta_2=-105° \end{gathered}[/tex]Use long division or a calculator to write 4/99 as a decimal. Then tell whether the decimal is terminating or repeating
0.0404
Repeating decimal (periodic)
1) Let's proceed with the long division of 4 by 99:
1.1) Since 4 is way smaller than 99 let's add one zero to the dividend and another for the quotient followed by a dot.
But 40 is still lesser than 99, so let's add another zero after the dot to make it 400. Now we can divide 400 by 99
1.2) Again to proceed with that division we'll need to write a zero at that 4 and another one in the quotient.
As and we can see already this a repeating decimal or periodic. This division will yield 0.0404040404.....
2) Hence, the answer is 0.0404
What is a feature of function g if g(x) = log (x-4) -8
The domain and range of the logarithmic function are
[tex]\begin{gathered} \text{domain}(\log x)=(0,\infty) \\ \text{range}(\log x)=(-\infty,\infty) \end{gathered}[/tex]Therefore, if
[tex]g(x)=\log (x-4)-8[/tex]We require that
[tex]\begin{gathered} x-4>0 \\ \Rightarrow x>4 \end{gathered}[/tex]Notice that the -8 term does not affect the range of function g(x); thus,
[tex]\begin{gathered} \text{domain}(g(x))=(4,\infty) \\ \text{range}(g(x))=(-\infty,\infty) \end{gathered}[/tex]Set g(x)=-8; then,
[tex]\begin{gathered} \Rightarrow\log (x-4)-8=-8 \\ \Rightarrow\log (x-4)=0 \\ \Rightarrow x=5 \end{gathered}[/tex]Therefore, y=-8 is not an asymptote of g(x), and, as shown above, the domain and range of g(x) are x>4, y->all real numbers.
Calculate the limit when x->4 as shown below,
[tex]\lim _{x\to4}g(x)=(\lim _{x\to4}\log (x-4))-8=(-\infty)-8=-\infty[/tex]Therefore, there is a vertical asymptote at x=4
Answer:
Hope this helps ;)
Step-by-step explanation:
A student at a junior college conducted a survey of 20 randomly selected full-time students to determine the relation between the number of hours of video game playing each week. X, and grade-point average, y. Shefound that a linear relation exists between the two variables. The least-squares regression line that describes this relation is ý = -0.0579x + 2.9408(Round to the nearest hundredth as needed.)(Part b) Interpret the slope.For each additional hour that a student spends playing video games in a week, the grade-point average will decrease by 0.0579 points, on average.(Part c) if appropriate, interpret the y-intercept.A. The grade-point average of a student who does not play video games is 2.9408.B. The average number of video games played in a week by students is 2.9408.C. It cannot be interpreted without more information.
exactly For sentence A "The grade-point average of a student who does not play video games is 2.9408" we can verify as follows:
That means this sentence "The grade-point average of a student who does not play video games is 2.9408" is true.
For sentence B "The average number of video games played in a week by students is 2.9408" is not exactelly correct because the average number of hours of video game played is not specified.
For sentence C "It cannot be interpreted without more information" let's look over an illustration.
We can see, we can interpret the y-intercept as the moment where x = 0 which means when the student does not play a video game. So this sentence is false.
I mean the correct sentence is sentence A.
13х-17y+16z= 73
-11x + 15y + 17z= 61
46x+10y-30z = -18
The solution of the linear system of three simultaneous equations is presented as follows; x = 2, y = 1 and z = 4
What is a set of simultaneous equation?Simultaneous system of equations consists of a finite set of equations for which a solution to the equation system is required.
The linear system of three equations can be presented as follows;
13•x - 17•y + 16•z = 73...(1)
-11•x + 15•y + 17•z = 61...(2)
46•x + 10•y - 30•z = -18...(3)
The above system of equations can be solved using common multiples of the coefficients as follows;
Multiply equation (2) by 2 and equation (3) by 3 to get;
2 × (-11•x + 15•y + 17•z) = 2 × 61 = 122
-22•x + 30•y + 34•z = 122...(4)3 × (46•x + 10•y - 30•z) = 3 × (-18) = -54
138•x + 30•y - 90•y = -54...(5)Subtracting equation (4) from equation (5) gives;
138•x + 30•y - 90•z - (-22•x + 30•y + 34•z) = -54 - 122 = -176
138•x - (-22•x) + 30•y - 30•y - 90•z - 34•z = -176
160•x - 124•z = -176
40•x - 31•z = 44
[tex] \displaystyle {z = \frac{(44 + 40\cdot x)}{31}}[/tex]
Plugging in the value of z in equation (1) and (2) gives;
1043•x - 527•y + 704 = 73 × 31 = 2236...(6)
Which gives;
[tex] \displaystyle {y = \frac{(1043\cdot x - 1559)}{527}}[/tex]
339•x + 465•y + 748 = 61 × 31 = 1891...(7)
Which gives; [tex] \displaystyle {y = \frac{(381 - 113\cdot x )}{155}}[/tex] which gives;
[tex] \displaystyle { \frac{(1043\cdot x - 1559)}{527}= \frac{(381 - 113\cdot x )}{155}}[/tex]
Therefore; 221216•x - 442432 = 0
x = 442432 ÷ 221216 = 2
x = 2
y = (1043×2 - 1559)÷527 = 1
y = 1
z = (44 + 40×2) ÷ 31 = 4
z = 4
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Which graph represents the solution of −2x≤4(x−6)?
Answer:
See attachments.
Step-by-step explanation:
Given inequality:
[tex]-2x\leq 4(x-6)[/tex]
Solve the inequality by first expanding the brackets:
[tex]\implies -2x\leq 4x-24[/tex]
Subtract 4x from both sides:
[tex]\implies -2x-4x\leq 4x-24-4x[/tex]
[tex]\implies -6x\leq -24[/tex]
Divide both sides by -6 (remembering to reverse the inequality sign as we are dividing by a negative number).
[tex]\implies \dfrac{-6x}{-6}\leq \dfrac{-24}{-6}[/tex]
[tex]\implies x\geq 4[/tex]
When graphing inequalities on a coordinate plane:
< or > : dashed line.≤ or ≥ : solid line.< or ≤ : shade under the line.> or ≥ : shade above the line.Therefore, to graph the given inequality on a coordinate plane:
Draw a solid line at x = 4.Shade above the line (i.e. shade to the right of the line).(See attachment 1).
When graphing inequalities on a number line:
< or > : open circle.≤ or ≥ : closed circle.< or ≤ : shade to the left of the circle.> or ≥ : shade to the right of the circle..Therefore, to graph the given inequality on a number line:
Place a closed circle at 4.Shade to the right of the circle.(See attachment 2).
james harmon pays 850.80 per year for his life insurance. if he where to the premiums quarterly, the payments would would be 221.21 what percentage more is mr hamrmon paying for the year using yhe quarterly rate
Percent is given by the expression:
[tex]\begin{gathered} \text{Total}\cdot\frac{\text{percent}}{100}=\text{Equivalent number to the percent} \\ 850.8\cdot\frac{x}{100}=221.21 \\ x=\frac{221.21\cdot100}{850.8} \\ x=26\text{ percent} \end{gathered}[/tex]So, he is paying 74% more using the quarterly rate
Joe bought 8 comic books for $36. How much does 1 comic book cost?
Answer:
1 comic book costs $4.5
Explanation:
Given that 8 comic books cost $36
To know how much 1 comic book costs, let x be the cost of 1 comic book, then, we have:
8 comic books = $36
1 comic book = x
Then we have the equation
8x = 36
where we can solve for x
Divide both sides by 8
8x/8 = 36/8
x = 4.5
Therefore, 1 comic book costs $4.5
Finding Slope
HELP ME PLS
The slope of the line that passes through the points (-5,6) and (-9,-6) is m = 3
The first point = (-5,6)
The second point = (-9,-6)
The slope of the line defined as the change in y coordinates with respect to the change in x coordinates.
The slope of the line m = [tex]\frac{y_2-y_1}{x_2-x_1}[/tex]
Where m is the slope of the line
[tex](x_1,y_1)[/tex] is the coordinates of the first point
[tex](x_2,y_2)[/tex] is the coordinates of the second point
Substitute the values in the equation
The slope of the line m = [tex]\frac{-6-6}{-9-(-5)}[/tex]
= -12/-4
= 3
Hence, the slope of the line that passes through the points (-5,6) and (-9,-6) is m = 3
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Given the equation y = x (x - 3)(2x + 7), find the rational roots. Complete theexplanation.The rational roots are x =, and. I found my answers bygraphing the equation, then finding where the equation crossed the (select) ▼4'
ANSWER
[tex]0,3,-\frac{7}{2}[/tex]EXPLANATION
The roots are the x values where the equation intercepts the x-axis.
Created by Cortin Lyons riables on Both Sides #3 2.12r – 14 = 8x + 16 12x – 14 = 6x + 16 as with Distributive Property
Problem N 2
we have
12x-14=8x+16
solve for x
subtract 8x both sides
12x-14-8x=8x+16-8x
4x-14=16
Adds 14 both sides
4x-14+14=16+14
4x=30
divide by 4 both sides
x=30/4
x=7.5
Problem N 4
we have
12x-14=6x+16
solve for x
subtract 6x both sides
12x-14-6x=6x+16-6x
6x-14=16
adds 14 both sides
6x=16+14
6x=30
divide by 6 both sides
x=30/6
x=5
When you are on the 2nd step of factoring this trinomial,you should be listing the factors of?
Given:
The trinomial equation is given as,
[tex]3v^2-4v-7[/tex]The objective is to choose the correct value for which the factors are to be obtained for factorization.
Explanation:
From the step 1, to perform the factorization the values of a and c has to be multiplied.
Then in step 2, the factors need to be calculated for the product of a and c.
Product of a and c :
The product of a and c will be,
[tex]a\times c=3\times7=21[/tex]Thus, factors are to be listed for the number 21.
Hence, option (1) is the correct answer.
#11 When you were born, your grandparents deposited $10,000 in a CD. for your college education. If theaccount earns 5% interest, compounded monthly, how much will be in the account for your collegeeducation?
Replacing with the values we already know, we have:
PV = 10,000
i = 5% or 0.05 but it is compounded monthly, then it is 0.05/12
n = 18 years or 216 months
10,000 = FV * 0.4073
FV = 10,000/0.4073
FV = $ 24,551.93
It is very close to option D, where the term "about" is included.
Draw a line connecting each sphere to its volume in terms of π and round it to the nearest tenth. (Not all of the values will be used.)
Remember that
The volume of a sphere is equal to
[tex]V=\frac{4}{3}\pi r^3[/tex]N 1
we have
D=9 units
r=9/2=4.5 units
substitute
[tex]\begin{gathered} V=\frac{4}{3}\pi(4.5)^3 \\ V=121.5\pi\text{ unit3} \\ V=381.5\text{ unit3} \end{gathered}[/tex]N 2
we have
r=2 units
[tex]\begin{gathered} V=\frac{4}{3}\pi2^3 \\ V=10.6\pi\text{ unit3} \\ V=33.5\text{ unit3} \end{gathered}[/tex]N 3
we have
D=14 units
r=14/2=7 units
[tex]\begin{gathered} V=\frac{4}{3}\pi7^3 \\ V=457.3\pi\text{ unit3} \\ V=1,436\text{ unit3} \end{gathered}[/tex]N 4
we have
r=9 units
[tex]\begin{gathered} V=\frac{4}{3}\pi9^3 \\ V=972\pi\text{ unit3} \\ V=3,052.1\text{ unit3} \end{gathered}[/tex]Simplify the expression by combing like terms.21v + 8 - 12v - 7 + 3t - t
We need to simplify the like terms.
"The like terms are whose with the same variable and exponent"
Therefore, the like terms are:
21v - 12v = 9v
8 - 7 = 1
3t - t = 2t
Now, the result is :
9v + 2t + 1
Hence, the correct answer is option D.
Classify the triangle with side lengths 8,13,20. a) Acute b) Right c) Obtuse
for right angles triangle,
hyposenuse square should be equal to sum of square of other two sides
it fails that law so its not right angled triangle
polynomials: classifying, simplifying adding and subtracting polynomials write in standard formplease do minimum steps
If joey walked east for 15 2/3 meters from home. Then, he walked west for 21 3/4 meters. How far was joey from home?
The distance between Joey and his home was such that joey walked east for 15 2/3 meters from home. Then, he walked west for 21 3/4 meters is 6 1/12 meters.
What is subtraction?To subtract in mathematics is to take something away from a group or a number of objects.
The group's total number of items decreases or becomes lower when we subtract from it.
It is known that East and West are the opposite of each other.
So, 15 2/3 towards the east let's take it positively.
And 21 3/4 towards left let's consider it negative.
So, the distance from the home
⇒ | ( 15 + 2/3) - (21 +3/4) |
⇒ | 15 -21 + 2/3 - 3/4 |
⇒ | -6 + (8 - 9)/12 |
⇒ | -6 - 1/12 |
⇒ | -(6 +1/12) |
⇒ 6 1/12
Hence "The distance between Joey and his home was such that joey walked east for 15 2/3 meters from home. Then, he walked west for 21 3/4 meters is 6 1/12 meters".
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find the value of the expression 4d ÷ c when c=3and d=6 simplify your answer
A company has developed a new deluxe AAA battery that is supposed to last longer than its regular AAA battery. However these new batteries are more expensive to produce. So the company would like to be convinced that they really do last longer. Based on years of experience, the company knows that its regular AAA batteries last lor 45 hours of continuous use. On average. The company selects an SRS Of 50 new batteries and uses them continuously until they are completely drained. The Sample mean lifetime is X =46.9 hours with a Standard deviation of S=4.6 hours A) Check for the conditions for the situation B) Calculate the test statistic for this situation C) What is the P value for this situation? D) What conclusion do you draw with 5% significance level? why? E) What type of error could you possibly make here? I
1 sample t-test
[tex]\mu=the\text{ true mean lifetime of new AAA batteries}[/tex][tex]\begin{gathered} H_0\colon\mu=45\text{ hours} \\ H_a\colon\mu>45\text{ hours} \end{gathered}[/tex][tex]n=50[/tex]B)
Calculating t test statistic
[tex]\begin{gathered} t=\frac{statistic-parameter}{s\tan dard\text{ deviation of statistic}} \\ t=\frac{\bar{x}-\mu_0}{\frac{s_x}{\sqrt[]{n}}} \end{gathered}[/tex]Plugging in the values, we have:
[tex]\begin{gathered} t=\frac{\bar{x}-\mu_0}{\frac{s_x}{\sqrt[]{n}}} \\ t=\frac{46.9-45}{\frac{4.6}{\sqrt[]{50}}} \\ t=\frac{1.9}{0.6505} \\ t=2.9207 \end{gathered}[/tex]C)t test statistic = 2.9207
degrees of freedom = n - 1 = 50 - 1 = 49
Using a calculator, we can calculate the p-value.
[tex]p-\text{value}=0.002633[/tex]D)Since p value is less than significance level (p value < alpha), then we will reject H_0 and take the alternate hypothesis.
Thus the test suggests that the new batteries do last more than 45 hours.
E)
We could've done Type I error here.
Type I error or α: Reject the null when it’s true.
Question 8 > Find the area of the trapezoid shown below 9 19 18 21 23 I Question Help ve
288 u²
1) Let's calculate the area of that trapezoid by plugging into the formula below the measures of the altitude, larger base, smaller one:
[tex]\begin{gathered} S=\frac{(B+b)h}{2} \\ S=\frac{(23+9)18}{2} \\ S=288 \end{gathered}[/tex]2) So that trapezoid has an area of 288 u²
Find the surface area of the triangular prism. 13 in. 5 in. 4 in. 12 in.
The first face is a triangle with height 5in and base 12in
Traingular face area = 1/2 x bh
=1/2 x 12 x 5
= 30 in^2
The area of the other triangular base = 30 in^2
Area of left side face = Length x breadth
= 5 x 4 = 20in^2
Area of the slant face = Length x breadth
= 13 x 4 = 52in^2
Area of the bottom face = Length x breadth
= 12 x 4 = 48in^2
Total surface area = 30 in^2 + 30 in^2 + 20in^2 + 52in^2 + 48in^2
=180in^2
suppose s is between r and t use the segment addition postulate to solve for each variable RS equals 2z plus 6 St equals 4z - 3 RT = 5z + 12
If s is between r and t, then:
RS + ST = RT
Where RS = 2z + 6
ST = 4z - 3
RT = 5z + 12
So, we get:
(2z + 6) + (4z - 3) = 5z + 12
Solving for z, we get:
2z + 6 + 4z - 3 = 5z + 12
6z + 3 = 5z + 12
6z + 3 - 5z = 12
z + 3 = 12
z = 12 - 3
z = 9
Answer: z = 9
Select the correct answer.Using long division, what is the quotient of 3r4 + 2023 + 1422 + 17= + 30 and I + 67
EXPLANATION
we are asked to use the long division method to solve the division
[tex]\frac{3x^4+20x^3+14x^2+17x+30}{x+6}[/tex]We will have