[tex](\stackrel{x_1}{3}~,~\stackrel{y_1}{4})\hspace{10em} \stackrel{slope}{m} ~=~ - 8 \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{4}=\stackrel{m}{- 8}(x-\stackrel{x_1}{3}) \\\\\\ y-4=-8x+24\implies {\Large \begin{array}{llll} y=-8x+28 \end{array}}[/tex]
Heather is six years younger than her husband Ryan. Thesum of their ages is 52. How old is Ryan23293146
Let's begin by listing out the information given to us:
Ryan (r) = r
Heather (h) = r - 6
[tex]\begin{gathered} r+h=52 \\ h=r-6 \\ r+r-6=52 \\ 2r-6=52 \\ 2r=52+6 \\ 2r=58 \\ r=\frac{58}{2}=29 \\ r=29 \\ \\ \therefore Ryan^{\prime}s\text{ age is 29 years old} \end{gathered}[/tex]Therefore, Ryan is 29 years old
Answer:74
Step-by-step explanation:
If h(x) = 2x and k(x) =2x -k, what is h(k(x))?
If h(x) = 2x and k(x) =2x -k, then h(k(x)) would be:
h(k(x))= 2*(2x-k)
h(k(x))= 4x - 2k Using distributive property
Final answer is: h(k(x))= 4x - 2k
What is the area of a rectangle with side lengths of 6/8 meter and 4/10 meter? In fraction in simplest form
SOLUTION
We want to find the area of the rectangle with the given side lengths as indicated in the question.
The area of a rectangle is given as
[tex]\text{Area = leghth }\times\text{ width}[/tex]So, this means to get the area, we will multiply the side lengths, we have
[tex]\begin{gathered} \text{Area = leghth }\times\text{ width} \\ \text{Area = }\frac{6}{8}\times\frac{4}{10} \end{gathered}[/tex]Solving the fraction, we have
[tex]\begin{gathered} \text{Area = }\frac{6}{8}\times\frac{4}{10} \\ 4\text{ divide itself is 1, }8\text{ divide 4 is 2, we have } \\ \text{Area = }\frac{6}{2}\times\frac{1}{10} \\ \text{now, 2 divide itself is 1, 6 divide 2 is 3, we have } \\ \text{Area = }\frac{3}{1}\times\frac{1}{10} \\ \text{Area = }\frac{3}{10} \end{gathered}[/tex]Hence the area in fraction in the simplest form is
[tex]\frac{3}{10}\text{ meter}[/tex]Type in the value ol 29 С A 21 20 B sin A= COSA = tanA= sinC = cosC =
Given,
The measure of side AC is 29.
The measure of side AB is 20.
The measure of side BC is 21.
The angle ABC is right angle.
The expression of sin in trigonometric ratio is,
[tex]\begin{gathered} \sin \text{ A=}\frac{\text{side opposite to angle A}}{\text{Hypotenuse}} \\ \sin \text{ A=}\frac{21}{29} \\ \sin \text{ C=}\frac{\text{side opposite to angle C}}{\text{Hypotenuse}} \\ \sin \text{ C=}\frac{20}{29} \end{gathered}[/tex]The expression of cos in trigonometric ratio is,
[tex]\begin{gathered} \cos \text{ A=}\frac{\text{side adjacent to angle A}}{\text{Hypotenuse}} \\ \cos \text{ A=}\frac{20}{29} \\ \cos \text{ C=}\frac{\text{side adjacent to angle C}}{\text{Hypotenuse}} \\ \cos \text{ C=}\frac{21}{29} \end{gathered}[/tex]The expression of tan in trigonometric ratio is,
[tex]\begin{gathered} \tan \text{ A=}\frac{\text{side opposite to angle A}}{\text{side adjacent to angle A}} \\ \tan \text{ A=}\frac{21}{20} \\ \tan \text{ C=}\frac{\text{side opposite to angle C}}{\text{side adjacent to angle C}} \\ \tan \text{ C=}\frac{20}{21} \end{gathered}[/tex]Hence, the value of sin A is 21/29, cos A is 20/29 , tan A is 21/20 and the value of
Brad stands 30 feet from a tree. He estimates the angle of elevation from a point onthe ground 30 feet from the tree to the top of the tree to be 60° as shown below.60°30 feetWhich of the following is closest to the height of the tree?
To determine the height of tree using trigonometric ratio,
Find the coordinate point for that would make ABCD a rhombus. Answer Choices: (3,5), (5,1), (1,1), (3,4)
A rhomus has te following shape:
therefore, the top part is missing, if we analyze, it should be 5 on the y axis and 3 on the x axis.
Answer: (3,5)
The table and the graph represent the rate at which two machines arebottling milk in gallons per second.
Given:
There are given the information about the two machines, one is in table form and another is in the graph.
Explanation:
We need to find the rate of change from both of the machines.
So,
For machine 1:
Choose two-point and find the slope by using the slope formula:
So,
The points are;
[tex](1,0.6),and,(2,1.2)[/tex]From the formula to find the rate of change:
[tex]r=\frac{y_2-y_1}{x_2-x_1}[/tex]Then,
[tex]\begin{gathered} r=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\ r=\frac{1.2-0.6}{2-1} \\ r=\frac{0.6}{1} \\ r=0.6 \end{gathered}[/tex]Now,
For machine 2:
We need to choose two points from the graph.
So,
Two points are:
[tex](8,6),and,(16,12)[/tex]Then,
From the formula:
[tex]\begin{gathered} r=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\ r=\frac{12-6}{16-8} \\ r=\frac{6}{8} \\ r=0.75 \end{gathered}[/tex]Final answer:
Hence, the machine 2 is faster at botting milk because the value of the rate of machine 2 is greater than the value of rate of machine 1.
Perform the indicated operation. 22 6/11 x 1 1/2
Multiplying the two fractions gives us the result 372/11
Here, we are given an expression- 22 6/11 x 1 1/2
To multiply these fractions, we first need to first convert the mixed fraction into an improper fraction. This can be done as follows-
22 6/11 = (22 * 11 + 6)/ 11
= (242 + 6)/ 11
= 248/ 11
similarly we have,
1 1/2 = (1*2 + 1)/2
= (2 + 1)/2
= 3/2
Thus, the given expression becomes-
248/ 11 x 3/ 2
= (248 * 3)/ (11 * 2)
= 744/ 22
= 372/ 11
We cannot simplify this fraction further.
Thus, multiplying the two fractions gives us the result 372/11
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A parabola has a directrix of y=1 and a focus of (1,12). Which statement is true?A)The parabola opens upward.B)The parabola opens downward.C)The parabola opens to the left.D)The parabola opens to the right.
1) Since the directrix is y=1 and the focus (1,12) We can write that, using the distance formula:
[tex]\begin{gathered} \sqrt[]{(y-1)^2}=\sqrt[]{(x-1)^2+(y-12)^2} \\ (y-1)^2=(x-1)^2+(y-12)^2 \\ y^2-2y+1=x^2-2x+1+y^2-24y+144 \\ -2y+1=x^2-2x+1-24y+144 \\ -2y+24y=x^2-2x+145 \\ -22y=x^2-2x+145 \\ y=\frac{x^2-2x+145}{22} \\ \\ y=\frac{x^2-2x+145}{22} \end{gathered}[/tex]2) Then we can state that the correct option is A the parabola opens upward.
The average American student is in class 330 minutes/day. How many seconds/day is this?
The given time duration is
330 minutes/day
So converting it into seconds, we know that 60 seconds is 1 minute, therefore,
[tex]330\frac{\text{ minutes}}{\text{day}}=330\times\frac{60\sec onds}{day}=19800\text{ seconds/day}[/tex]Thus the answer is 19800 seconds/day.
Find the perimeter of the triangle whose vertices are
(-2, 4), (-2, 1), (-4,-2).
The sum of the length of the sides is the perimeter of the triangle.
The perimeter of the triangle is 5[tex]\sqrt{10}[/tex] + [tex]\sqrt{13}[/tex].
How to find the perimeter of a triangle?The sum of the length of the sides is the perimeter of the triangle.
d = [tex]\sqrt{(x^2 - x^1)^2 + (y^2 - y^1)^2}[/tex]
Lets take (-2, 4) and (-2, 1).
[tex]d = \sqrt{(-2+2)^2 + (1 - 4)^2}\\\\d = \sqrt{0 + -3^2} \\\\d = \sqrt{9} \\[/tex]
d = 3
Now, let's take the points (-2, 1) and (-4,-2).
d = [tex]\sqrt{(x^2 - x^1)^2 + (y^2 - y^1)^2}[/tex]
[tex]d = \sqrt{(-4+2)^2 + (-2-1)^2}\\\\d = \sqrt{-2^2 + -3^2} \\\\d = \sqrt{4 + 9} \\[/tex]
d = [tex]\sqrt{13}[/tex]
Now, let's take the points (-2, 4) and (-4,-2).
d = [tex]\sqrt{(x^2 - x^1)^2 + (y^2 - y^1)^2}[/tex]
[tex]d = \sqrt{(-4+2)^2 + (-2-4)^2}\\\\d = \sqrt{-2^2 + -6^2} \\\\d = \sqrt{4 + 36} \\[/tex]
d = 40 = 2[tex]\sqrt{10}[/tex]
The sum of the length of the sides is the perimeter of the triangle d= 3 + [tex]\sqrt{13}[/tex] + 2[tex]\sqrt{10}[/tex]
The perimeter of the triangle is 5[tex]\sqrt{10}[/tex] + [tex]\sqrt{13}[/tex].
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Patsy’s pizza sells one 20-inch cheese pizza or two 12-inch cheese pizzas for 11.99. Determine which size gives more pizza (A= 3.14r^2)
Since 20-inch pizza, means it is a circle with a diameter of 20 inches
Since the diameter is twice the radius, then
The raduis = 20/2 = 10 inches
Then the area of it is
[tex]\begin{gathered} A_1=\pi(r)^2 \\ A_1=3.14(10)^2 \\ A_1=3.14(100) \\ A_1=314 \end{gathered}[/tex]The area of the first pizza is 314 square inches
Since the diameter of the other area is 12 inches, then
The radius = 12/2 = 6 inches
Then the area of it is
[tex]\begin{gathered} A_2=3.14(6)^2 \\ A_2=3.14(36) \\ A_2=113.04 \end{gathered}[/tex]Then the area of the second pizza is 113.04 x 2 = 226.08 square inches
Since the area of the first offer is greater than the area of the second offer, then
The first size gives more pizza
I need help with part "C" and "D"
3x²cos( x³ ) and 3sin²( x ) cos( x ) are the derivatives of the composite functions f(x) = sin(x³) and f(x) = sin³(x) respectively.
What are the derivative of f(x) = sin(x³) and f(x) = sin³(x)?Chain rule simply shows how to find the derivative of a composite function. It states that;
d/dx[f(g(x))] = f'(g(x))g'(x)
Given the data in the question;
f(x) = sin(x³) = ?f(x) = sin³(x) = ?First, we find the derivate of the composite function f(x) = sin(x³) using chain rule.
d/dx[f(g(x))] = f'(g(x))g'(x)
f(x) = sin(x)
g(x) = x³
Apply chain rule, set u as x³
d/du[ sin( u )] d/dx[ x³ ]
cos( u ) d/dx[ x³ ]
cos( x³ ) d/dx[ x³ ]
Now, differentiate using power rule.
d/dx[ xⁿ ] is nxⁿ⁻¹
cos( x³ ) d/dx[ x³ ]
In our case, n = 3
cos( x³ ) ( 3x² )
Reorder the factors
3x²cos( x³ )
Next, we find the derivative of f(x) = sin³(x)
d/dx[f(g(x))] = f'(g(x))g'(x)
f( x ) = x³
g( x ) = sin( x )
Apply chain rule, set u as sin( x )
d/du[ u³ ] d/dx[ sin( x )]
Now, differentiate using power rule.
d/dx[ xⁿ ] is nxⁿ⁻¹
d/du[ u³ ] d/dx[ sin( x )]
3u² d/dx[ sin( x )]
Replace the u with sin( x )
3sin²(x) d/dx[ sin( x )]
Derivative of sin x with respect to x is cos (x)
3sin²( x ) cos( x )
Therefore, the derivatives of the functions are 3x²cos( x³ ) and 3sin²( x ) cos( x ).
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Solve each of these systems of equations. You may use either the substitution or elimination method.2x+y=15x+6y =4
Answer:
x=2/7 and y=3/7
Explanation:
Given the systems of equations:
[tex]\begin{gathered} 2x+y=1 \\ 5x+6y=4 \end{gathered}[/tex]We use the substitution method to solve.
Step 1: Make y the subject in the first equation.
[tex]\begin{gathered} 2x+y=1 \\ y=1-2x \end{gathered}[/tex]Step 2: Substitute y into the second equation.
[tex]\begin{gathered} 5x+6y=4 \\ 5x+6(1-2x)=4 \\ 5x+6-12x=4 \\ 5x-12x=4-6 \\ -7x=-2 \\ x=-\frac{2}{-7} \\ x=\frac{2}{7} \end{gathered}[/tex]Step 3: Solve for y
[tex]\begin{gathered} y=1-2x \\ =1-2(\frac{2}{7}) \\ =1-\frac{4}{7} \\ y=\frac{3}{7} \end{gathered}[/tex]Therefore, x=2/7 and y=3/7.
Represent the following sentence as an algebraic expression, where "a number" is the letter x. You do not need to simplify.2 minus twice a number.
Given the statement:
2 minus twice a number.
Let's represent the sentence as an algebraic expression where the number is the letter x.
To represent the statement as an algebraic expression, we have:
A number is the letter x ==> x
• 2 minus twice the number ==> 2 - 2x.
Therefore, the algebraic expression that represents the sentence is:
2 - 2x
ANSWER:
2 - 2x
I need help with my algebra
N 1
we have the ordered pairs
(1,1) and (4,2)
m=(2-1)/(4-1)
m=1/3
the slope is 1/3The accompanying table shows the median household income (in dollars) for 25 randomly selected regions. Complete parts (a) through (g) below.B!: Click the icon to view the table of data.(a) Construct a frequency distribution. Use a first class having a lower class limit of 35,000 and a class width of 5000.IncomeFrequency
Step 1:
Draw a table with class intervals to construct the frequency distribution table.
Step 2:
Choose a lower class and a class interval.
Lower class limit = 35,000
Class interval = 5000
Step 3:
Get the smallest and largest values from the table
Smallest value = 39712
Largest value = 67729
Step 4:
Draw the table using the class interval.
a) Frequency distribution table
FINANCE Yetunde is purchasing a refrigerator, which costs $950. The store has a finance option: pay 10% now, and pay the remaining balance
in one year, with a 5% annual simple interest rate applied to the remaining balance.
How much will she pay up front if she uses the finance option? $
How much interest will she pay at the end of one year? $
What is the total cost of the refrigerator after using the one-year finance option? $
What is the total cost of the refrigerator if she pays the balance and interest at 6 months? $
S
1. The amount that Yetunde will pay upfront if she uses the finance option is $95.
2. The interest she will pay at the end of one year is $42.75.
3. The total refrigerator cost after using the one-year finance option is $992.75.
4. The total cost of the refrigerator if she pays the balance and interest at 6 months is $971.375.
What is a finance charge?A finance charge is the interest cost added to the amount borrowed or used for credit purchases.
Cost of the refrigerator = $950
Finance Option: Initial deposit = 10% = $95 ($950 x 10%)
Balance to be financed = $855 ($950 - $95)
Simple Interest rate = 5%
Simple interest amount = $42.75 ($855 x 5%)
The total cost with finance option is $992.75 ($950 + $42.75)
Simple interest amount at 6 months = $21.375 ($42.75/2)
The total cost at 6 months = $971.375 ($950 + $21.375)
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please answer this question
It's important to know that similar figures have equal angles, for example, if two triangles are similar, then their corresponding angles are congruent.
Having said that, D is true because all rectangles have equal angles.
Which statement about the following equation is true?
2x2 – 9x + 2 = –1
The discriminant is less than 0, so there are two real roots.
The discriminant is less than 0, so there are two complex roots.
The discriminant is greater than 0, so there are two real roots.
The discriminant is greater than 0, so there are two complex roots.
Answer:
The discriminant is greater than 0, so there are two real roots.
Step-by-step explanation:
[tex]\boxed{\begin{minipage}{6.2 cm}\underline{Discriminant}\\\\$b^2-4ac$ \quad when $ax^2+bx+c=0$\\\\If $b^2-4ac > 0 \implies$ two real roots\\If $b^2-4ac=0 \implies$ one real root\\If $b^2-4ac < 0 \implies$ no real roots\\\end{minipage}}[/tex]
Given equation:
[tex]2x^2-9x+2=-1[/tex]
Add 1 to both sides of the equation so that the equation equals zero:
[tex]\implies 2x^2-9x+2+1=-1+1[/tex]
[tex]\implies 2x^2-9x+3=0[/tex]
Compare the equation with ax²+bx+c=0:
a = 2b = -9c = 3Substitute the values of a, b and c into the discriminant formula and solve:
[tex]\begin{aligned}\implies b^2-4ac&=(-9)^2-4(2)(3)\\&=81-4(2)(3)\\&=81-8(3)\\&=81-24\\&=57\end{aligned}[/tex]
As 57 > 0, the discriminant is greater than zero, so there are two real roots.
Answer:
B on Edge 23
Step-by-step explanation:
trust me bro
A motorcycle can be purchased for $9000 or leased for a down payment of $400 and $290 per month. Find a function that describes how the cost of the lease depends on time. Assuming that the monthly payments are made, how long can the motorcycle be leased before more than the purchase price has been paid?Question content area bottomPart 1The function that models the situation is p= enter your response here, where p is the amount paid on the lease in dollars and t is the time in months.(Simplify your answer.)
Let's find the function p that describes that situation, at the initial time we must pay 400, then
400 - 0
After one month we must pay 290, it will sum with the previous 400 that we have paid
400 - 0
690 - 1 month
In the third month, we must add 400, the previous 290 and 290 again, two times.
400 - 0
690 - 1 month
980 - 2 months
If we repeat that we will see that we will have 290*3 plus 400, in other words, more 290 per month. We can write an expression to do it, it's
[tex]p=400+290\cdot t[/tex]Where t is the time in months, verify that our expression is coherent with our previous logic, now we have our expression let's go to the next step. The lease will be equal to the purchase price when p = 9000, which means that the person paid 9000 on the lease, if we use it in our equation we can solve it for t
[tex]9000=400+290\cdot t[/tex]Let's solve it for t
[tex]\begin{gathered} 9000=400+290t \\ \\ 290t=9000-400 \\ \\ 290t=8600 \\ \\ t=\frac{8600}{290} \\ \\ t=29.65 \end{gathered}[/tex]Then, if we lease the motorcycle for 30 months, we will pay $9100, more than the purchase price, if we lease it for 29 months, we will pay $8810.
Final answer:
Expression:
[tex]p=400+290t[/tex]How long you can lease it before it costs more than the purchase price:
[tex]t=29[/tex]Solve 2x2 + x - 4=0.X2+be += 0DONEIntro
Use the following figure and information to complete the proof. Given: m∠4=m∠2m∠5=m∠3m∠DBE=180∘ Prove: m∠1+m∠2+m∠3=180∘ Parallel lines m & n. Triangle A B C with A & C on n & B on m. A is to the left of C with angle 2 at A & angle 3 is at C. B is above A & C with angles 4 3 & 5 at C from left to right.© 2019 StrongMind. Created using GeoGebra. Match each numbered statement in the proof to its correct reason.
Explanation:
The given information is
m∠4 = m∠2
m∠5 = m∠3
m∠DBE = 180
The angle addition postulate says that the angle that the sum of the angles 4, 1, and 5 is equal to the angle DBE, so we can write the following equation
m∠4 + m∠1 + m∠5 = m∠DBE
But, we know that m∠DBE = 180, so by transitivity, we get:
m∠4 + m∠1 + m∠5 = 180
Then, we also know that m∠4 = m∠2 and m∠5 = m∠3, so we can substitute the angles to get
m∠2 + m∠1 + m∠3 = 180
Therefore, we can change the order by the commutative property to get
m∠1 + m∠2 + m∠3 = 180
Answer:
So, the answer is
Solve for the variable: 1 - p = 7
Given,
Solve for the variable,
[tex]1-p=7[/tex]Solution:
Take the variable on one side of the equation and the constant on another side of the equation.
[tex]\begin{gathered} -p=7-1 \\ -p=6 \\ p=-6 \end{gathered}[/tex]Thus, the value of p is -6.
Josephine can correct her students’ test papers in 7 hours, but if her teacher’s assistant helps, it would take them 5 hours. How long, in hours and minutes, would it take the assistant to do it alone. Write in mixed units
Answer: it will take the assistant to do it alone 17 1/2 hours
Explanation:
Let x represent the number of hours it will take the assistant to do it alone. This means that the unit work rate per hour is 1/x
If Josephine can correct her students’ test papers in 7 hours, it means that the unit work rate per hour is 1/7
If both of them can do the job in 5 hours, it means that their combined unit rate is 1/5
Since the rates are independent, it means that
1/x + 1/7 = 1/5
The lowest common multiple of the denominators is 35x. Multiplying through by 35x, we have
35 + 5x = 7x
7x - 5x = 35
2x = 35
x = 35/2 = 17/2
it will take the assistant to do it alone 17 1/2 hours
Find the values of x that make mlln
Answer:
4x² = 100° (corresponding angles)
x² = 100÷4
= 25
x = √25
= 5°
3x+1x2+5x+6=ax+2+bx+3
Answer: If you have an equation of the form "ax2 + bx + c = 0".
x2 = 3x -1, x2 - 3x + 1 = 0, a=1, b=-3, c=1.
Step-by-step explanation:
Help please
Stem: 6, 7, 8, 9
Leaf: 8, 579, 02, 2667
Questions: What score represents the median of this data? What score represents the mode of this data? What is the mean of this data? If you use the mean, median, or mode to describe this data, which one would be the most misleading?
1. The score that represents the median is 81.
2. The score that represents the mode is 96.
3. The score that represents the mean is 84.2.
4. The mode will be the most misleading since it's far away from the mean and median.
What is a mean?The stem and leaf plot data are 68, 75, 77, 79, 80, 82, 92, 96, 96, and 97.
A mean simply means the average of a set of numbers. The mean will be:
= Sum of all data values / Number of values
= 842 / 10
= 84.2
The mode is the number that occurs most and this will be 96.
The median is the number in the middle. This will be:
= (80 + 82)/2
= 162/2
= 81
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Match each equation with its solution set.2(5 – 4x) = 8x + 2a. no solution8(x + 5) = 8x + 40b. all real numbers2(5 – 4x) = 2 – 8xc. 1/28(x + 5) = 8x + 5d. 2
To match the equation with its solution set we need to solve each of them. Let's do that
First equation:
[tex]\begin{gathered} 2(5-4x)=8x+2 \\ 10-8x=8x+2 \\ 8x+8x=10-2 \\ 16x=8 \\ x=\frac{8}{16} \\ x=\frac{1}{2} \end{gathered}[/tex]Second equation:
[tex]\begin{gathered} 8(x+5)=8x+40 \\ 8x+40=8x+40 \end{gathered}[/tex]Since both sides of the equation are exactly the same expression we conclude that this equation is satisfied by all real numbers.
Third equation:
[tex]\begin{gathered} 2(5-4x)=2-8x \\ 10-8x=2-8x \\ 8x-8x=2-10 \\ 0=-8 \end{gathered}[/tex]Since the last line is a contradiction we conclude that this equation has no solutions.
Fourth equation:
[tex]\begin{gathered} 8(x+5)=8x+5 \\ 8x+40=8x+5 \\ 8x-8x=5-40 \\ 0=-35 \end{gathered}[/tex]Once again we have a contradiction in the last line, then this equation has no solutions.
Then the correct match is:
First equation with c.
Second equation with b.
Third equation with a.
Fourth equation with a.
which number set(s) does -10 belong toirrational numberswhole numbersrational numbersintegersreal numberscounting or natural numbersNo number set describes this number.
Given:
The number is -10.
To describe the number:
Explanation:
Since 10 is a whole number and it has a minus sign.
So, -10 is an integer number.
Since it is an integer number.
So, it must be a rational number as well as the real number.
Therefore, -10 is a number that belongs to the sets
Integer numbers
Rational numbers
Real numbers
Final answer:
• Integer numbers
,• Rational numbers
,• Real numbers
