A social media survey found that 75% of parents are "friends" with their children on a certain online networking site. A random sample of 160 parents was selected.
a. Calculate the standard error of the proportion.
b. What is the probability that 125 or more parents from this sample are "friends" with their children on this online networking site?
c. What is the probability that between 120 and 128 patents

The volume of the parallelepiped determined by the vectors u, v, and w is 20.

To calculate the volume of the parallelepiped determined by the vectors u, v, and w, we need to use the scalar triple product. The scalar triple product of three vectors is defined as the dot product of one vector with the cross product of the other two vectors. In this case, the volume of the parallelepiped is given by:

Volume = |u . (v x w)|

Where "." represents the dot product and "x" represents the cross product.

First, let's calculate the cross product of v and w:

v x w = (2i + k) x (4j) = 8k - 4i

Next, let's take the dot product of u and the result of the cross product:

u . (v x w) = (i - 2j + 3k) . (8k - 4i) = -4 + 24 = 20

Finally, we take the absolute value of the result to get the volume of the parallelepiped:

Volume = |20| = 20

Therefore, the volume of the parallelepiped determined by the vectors u, v, and w is 20.

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Answer:

Now, This number is neither a perfect square nor a perfect cube

So, 34 can be evaluated as, 34 = (30 + 4) (15 + 15 + 4)

Here is the evaluated form of 34.

Step-by-step explanation:

is of goggle btw so just change some of the words

It's not a. If I expanded "a" I would get = (9x - 64y)(9x - 64y) hence the square root. If I did FOIL I would not get the correct answer. "A" is essentially "b" written differently.

Answer:

(3x + 8y) • (3x - 8y)

Step-by-step explanation:

An expression that shows the associative property has been applied to (6+8)+4 is 6+(8+4).

The associative property is a mathematical rule that states that the way numbers are grouped within an expression does not affect the final result. In other words, you can add or multiply numbers in any order, and the result will be the same.

This property is represented as (a+b)+c=a+(b+c) or (a*b)*c=a*(b*c).

In the given expression, (6+8)+4, the associative property can be applied by changing the grouping of the numbers. This can be done by moving the parentheses from the first two numbers to the last two numbers. The new expression would be 6+(8+4).

Therefore, an expression that shows the associative property has been applied to (6+8)+4 is 6+(8+4).

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Answer:

Step-by-step explanation:

There is no graph attached.  

Convert the English phrases into expressions:

1.  "A number (Let's call it x) is equal to the sum of half a second number (y) and 3"

2.  "The first number (x) is also equal to the sum of one-quarter of the second number (y) and 5"

Let's rewrite these in standard form:

x = (1/2)y+3

2x = y + 6

y = 2x - 6

and

x = (1/4)y + 5

4x = y + 20

y = 4x - 20

A plot of these two lines is attached.  Match them with the graph.

The concession stand made 18 fewer cups of coffee than cups of hot chocolate.

To find out how many fewer cups of coffee they made than cups of hot chocolate, we need to add together the number of cups of hot chocolate with marshmallows and the number of cups without marshmallows:

20 cups + 15 cups = 35 cups of hot chocolate

Now, we can subtract the number of cups of coffee from the number of cups of hot chocolate to find out how many fewer cups of coffee they made:

35 cups - 17 cups = 18 cups

So, the concession stand made 18 fewer cups of coffee than cups of hot chocolate.

In conclusion, the concession stand made 18 fewer cups of coffee than cups of hot chocolate.

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Answer:

3/4

Step-by-step explanation:

1. The values of x, y, and z are x = 15.5, y = 9.54, and z = 0. 2. The values of x and y are x = 1.4375 and y = 8. 3. The values of x and y are x = 6 and y = 52.5. 4. X can have any value and the triangles will still be similar

1. We are given that ΔPRS is congruent to ΔCFH.

From ΔPRS, we know that:

∠P = 180 - 28 - ∠R

∠P = 152 - 13y

From ΔCFH, we know that CH is the hypotenuse and CF is one of the legs. So, using the Pythagorean Theorem, we have:

CH^2 = CF^2 + FH^2

39^2 = 24^2 + FH^2

FH^2 = 39^2 - 24^2

FH = sqrt(39^2 - 24^2) = 30

Since ΔPRS is congruent to ΔCFH, their corresponding sides are equal. Therefore:

PS = CH = 39

2x - 7 = CF = 24

Solving for x and y:

2x - 7 = 24

2x = 31

x = 15.5

39 = 2x - 7

46 = 2x

x = 23

∠P = 152 - 13y

28 = 152 - 13y

124 = 13y

y = 9.54

Solving for z:

PS = 2x - 7

39 = 2(15.5) - 7

39 = 31

z = 0

Therefore, the values of x, y, and z are x = 15.5, y = 9.54, and z = 0.

2. We are given that ΔABC is similar to ΔDEF. Therefore, the corresponding sides are proportional:

AB/DE = BC/EF = AC/DF

Substituting the given values:

8/(y-6) = 19/(4x-1) = 14/DF

We can solve for x and y using any two of the three ratios.

Let's first solve for x and y using the first two ratios:

8/(y-6) = 19/(4x-1)

Cross-multiplying, we get:

8(4x-1) = 19(y-6)

Expanding the brackets, we get:

32x - 8 = 19y - 114

32x - 19y = -106

Now let's use the third ratio:

14/DF = 8/(y-6)

Cross-multiplying, we get:

14(y-6) = 8DF

Simplifying, we get:

y = (4/7)DF + 6

Substituting this into the equation we got earlier:

32x - 19y = -106

32x - 19[(4/7)DF + 6] = -106

32x - (76/7)DF - 114 = -106

32x - (76/7)DF = 8

Multiplying both sides by 7, we get:

224x - 76DF = 56

Using the equation we got from the third ratio:

14(y-6) = 8DF

14y - 84 = 8DF

14y = 8DF + 84

y = (4/7)DF + 6

Substituting this into the equation we just got:

14[(4/7)DF + 6] = 8DF + 84

8DF + 84 = (56/7)DF + 84

8DF = (56/7)DF

DF = 7

Substituting DF = 7 into the third ratio:

14/DF = 8/(y-6)

14/7 = 8/(y-6)

2 = y-6

y = 8

Now we can substitute y = 8 into the equation we got earlier:

32x - 19y = -106

32x - 19(8) = -106

32x - 152 = -106

32x = 46

x = 1.4375

Therefore, the values of x and y are x = 1.4375 and y = 8.

3. Since ΔZMK ≈ ΔAPY, we know that the corresponding angles are congruent:

m∠M = m∠A

m∠K = m∠Y

Therefore, we can write two equations:

m∠M = 2y + 7

m∠K = 41°

Also, we know that:

m∠M + m∠K + (13x - 37)° = 180°

Substituting the values we have:

112° + 41° + (13x - 37)° = 180°

13x + 116 = 180

13x = 64

x = 4.9231

Substituting x into the third equation:

112° + 41° + (13x - 37)° = 180°

13x + 116 = 180

13(4.9231) + 116 + m∠K = 180

m∠K = 41°

Substituting m∠K = 41° into the second equation:

m∠K = m∠Y

13x - 37 = 41

13x = 78

x = 6

Substituting x into the first equation:

m∠M = 2y + 7

112 = 2y + 7

105 = 2y

y = 52.5

Therefore, the values of x and y are x = 6 and y = 52.5.

4. Since ΔBTS ≈ ΔGHD, we know that the corresponding angles are congruent:

m∠S = m∠H

m∠B = m∠G

Therefore, we can write two equations:

m∠S = 7y + 5

m∠B = m∠G = 21°

Also, we know that:

m∠B + m∠T + m∠S = 180°

Substituting the values we have:

21° + m∠T + 56° = 180°

m∠T = 103°

Now we can use the fact that the sum of the angles in a triangle is 180° to find m∠G:

m∠B + m∠T + m∠G = 180°

21° + 103° + m∠G = 180°

m∠G = 56°

Since we have a pair of similar triangles, we can use their side lengths to set up a proportion:

BS/BT = GD/GH

Substituting the given values:

25/31 = (4x-11)/GH

Solving for GH:

GH = (31/25)(4x-11)

Now we can use the fact that the sum of the angles in a triangle is 180° to find m∠H:

m∠G + m∠H + m∠D = 180°

56° + m∠H + 90° = 180°

m∠H = 34°

Substituting the values we have:

m∠S = 7y + 5

56 = 7y + 5

51 = 7y

y = 7.2857

Substituting y into the first equation:

m∠S = 7y + 5

m∠S = 7(7.2857) + 5

m∠S = 59

Now we can use the fact that the sum of the angles in a triangle is 180° to find m∠T:

m∠B + m∠T + m∠S = 180°

21° + m∠T + 59° = 180°

m∠T = 100°

Now we can use the fact that we have a pair of similar triangles to find x:

BS/BT = GD/GH

25/31 = (4x-11)/GH

25/31 = (4x-11)/((31/25)(4x-11))

Simplifying:

25/31 = 25/31

Therefore, x can have any value and the triangles will still be similar.

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The resulting polynomial in descending order is 11x7 + 5x6

To combine like terms and write the resulting polynomial in descending order, we need to first identify the like terms and then add or subtract their coefficients. Like terms are terms that have the same variable and the same exponent.

In this case, the like terms are 2x^(7) and 9x^(7).

We can combine these like terms by adding their coefficients:

2x^(7) + 9x^(7) = 11x^(7)

Now we have:

11x^(7) + 5x^(6)

Since the exponents are different, we cannot combine these terms.

Therefore, the resulting polynomial in descending order is 11x7 + 5x6

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Answer:

380

Step-by-step explanation:

To find out what number is 600% of another number, we need to divide that number by 600% (or 6).

So:

x = 2,280 ÷ 6

x = 380

Therefore, 600% of 380 is equal to 2,280.

Hope this helped !

The identity \[ \frac{1}{\tan x(1+\cos 2 x)}=\csc 2 x \] is proved.

To prove the identity \[ \frac{1}{\tan x(1+\cos 2 x)}=\csc 2 x \], we can use the Double Angle Formula for Cosines and the Pythagorean Identity.

Using the Double Angle Formula for Cosines, we get:
$\cos2x = 2\cos^2 x - 1$

We can then substitute this into the original identity and simplify:
$\frac{1}{\tan x(1+\cos 2 x)}=\frac{1}{\tan x(1+2\cos^2 x - 1)}$

Using the Pythagorean Identity, $\cos^2 x + \sin^2 x = 1$, we get:
$\frac{1}{\tan x(1+\cos 2 x)}=\frac{1}{\tan x(\sin^2 x)}$

Using the inverse tangent function, $\tan^{-1}x = \frac{\pi}{2}-\sin^{-1}x$, and since $\sin 2x = 2 \sin x \cos x$, we can rewrite this as:
$\frac{1}{\tan x(1+\cos 2 x)}=\frac{1}{2 \sin x \cos x}$

Finally, using the definition of cosecant, $\csc x = \frac{1}{\sin x}$, we get:
$\frac{1}{\tan x(1+\cos 2 x)}=\csc 2 x$

Therefore, the identity \[ \frac{1}{\tan x(1+\cos 2 x)}=\csc 2 x \] is proved.

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The quadrilateral formed from the points P(7, -3), Q(0,3), R(-9,5), and S(-2,-1)  is a rhombus.

To find the desired slopes and lengths, we can use the slope formula and the distance formula. The slope formula is (y₂ - y₁)/(x₂ - x₁) and the distance formula is √((x₂ - x₁)² + (y₂ - y₁)²).

Slope of PQ = (3 - (-3))/(0 - 7) = 6/(-7) = -6/7
Slope of QR = (5 - 3)/(-9 - 0) = 2/(-9) = -2/9
Slope of RS = (-1 - 5)/(-2 - (-9)) = (-6)/(7) = -6/7
Slope of SP = ((-3) - (-1))/(7 - (-2)) = (-2)/(9) = -2/9

Length of PQ = √((0 - 7)² + (3 - (-3))²) = √(49 + 36) = √85
Length of QR = √((-9 - 0)² + (5 - 3)²) = √(81 + 4) = √85
Length of RS = √((-2 - (-9))² + ((-1) - 5)²) = √(49 + 36) = √85
Length of SP = √((7 - (-2))² + ((-3) - (-1))²) = √(81 + 4) = √85

Since the opposite slopes are equal and all of the lengths are equal, the quadrilateral is a rhombus.

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In quadrilateral EFGH, sides FG and EH are congruent because they each have a length of 7.07

The area of quadrilateral EFGH is closest to 30 square units.

From the question, we have the following parameters that can be used in our computation:

The quadrilateral EFGH

This quadrilateral is a rectangle

From the figure, we can see that the following side lengths

EF = 3√2 = 4.24

EH = 5√2 = 7.07

So, we have

Area = 3√2 * 5√2

Evaluate

Area = 30

Hence, the area is 30 square units

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The areas of the triangle with length of two side are 3cm and 4cm and the angle included between these sides is 30° is equals to the (3√3/2) cm².

Area is defined as a measure the space inside a two-dimensional shapes, like square, triangle, etc. Area is denoted by square units, like cm², m², etc. Area of triangle is equals to the (1/2)× base length × height. We have a triangle with side lengths. Let triangle be named as ABC. Let

Base length of triangle, BC = 3 cm

Other side of triangle ABC, AC = 4 cm

Angle included between these sides is

= 30°

Now, the above assumption results a right angled triangle ABC, with base 3 cm and hypothenuse 4 cm. We have to calculate the area of triangle ABC. First we determine the height of ∆ABC. Using the Trigonometric functions,

tan 30° = AB/BC

=> tan 30° = AB/3

=> AB = 3 tan 30°

=> AB = 3(1/√3) = √3 cm

Now, area of triangle ABC = (1/2)× base length × height.

= (1/2) × √3 cm × 3 cm

= 3√3/2 cm²

Hence, the required area is 3√3/2 cm².

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Answer:     7 .

Step-by-step explanation:

The middle term is, +7x its coefficient is 7 . Step-2 : Find two factors of -8 whose sum equals the coefficient of the middle term, which is

The equation (bd)/(b-d) = c can be used to determine how many hours a day the remaining women must work in order to produce the same number of articles each day, when some of the women have been released.

An equation is an expression used to relate two or more quantities, usually by using mathematical operators such as addition, subtraction, multiplication and division. An equation typically has an equal sign (=) as its central element, with the left and right sides of the equation representing the two sides of the equation.

The answer to this question can be determined by using the following equation: (bd)/(b-d) = c. This equation is a simple ratio that takes into account the number of hours the women work per day (b), the number of women released (d), and the number of articles produced each day (c).

For example, if a factory had 10 women working 8 hours a day and produced 100 articles each day, and 5 of those women were released, then the equation (8 x 5) / (8-5) = 100 would determine that the remaining women must work 12.5 hours per day in order to produce the same number of articles.

In conclusion, the equation (bd)/(b-d) = c can be used to determine how many hours a day the remaining women must work in order to produce the same number of articles each day, when some of the women have been released.

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Answer:

In order to pay Jimmy's credit card debt off in the next 12 months, then the total minimum credit card payment will be $411.25.

Step-by-step explanation:

Answer:

y - 8 ≤ -27

Step-by-step explanation:

The difference of y and 8 is less than or equal to -27

y - 8 ≤ -27

Answer:

= 42m²

Step-by-step explanation:

The ratio of two similar hexagons is 3:7

where the smaller correspond to 3

And

the bigger correspond to 7

where the area of the smaller haxagon is 18

so if 3 = 18m²

then 7 = ?

therefore

[tex] \frac{7}{3} \times 18 \\ = 42 {m}^{2} [/tex]

therefore the area of the bigger hexagon is 42m²

the real solutions to this equation are x = 2√2 and x = -2√2.

Solution:

(a) -n + √(6n + 19) = 2

To find the solution algebraically, we need to isolate the radical on one side of the equation and then square both sides to get rid of the radical.

-n = 2 - √(6n + 19)

(-n - 2)^2 = (2 - √(6n + 19))^2

n^2 + 4n + 4 = 4 - 4√(6n + 19) + 6n + 19

n^2 - 2n - 19 = -4√(6n + 19)

(n^2 - 2n - 19)^2 = (-4√(6n + 19))^2

n^4 - 4n^3 - 18n^2 + 76n + 361 = 96n + 304

n^4 - 4n^3 - 114n^2 + 76n + 57 = 0

This is a quartic equation that can be solved using the Rational Root Theorem or by factoring. However, this equation does not have any real solutions. Therefore, there are no real solutions to this equation.

(b) x^4 - 20x^2 + 64 = 0

This equation can be factored as:

(x^2 - 8)^2 = 0

x^2 - 8 = 0

x^2 = 8

x = ±√8

x = ±2√2

Therefore, the real solutions to this equation are x = 2√2 and x = -2√2.

Remember to check your solutions by plugging them back into the original equation and seeing if they make the equation true.

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Answer:

Step-by-step explanation:

1) Cannot be determined.

2) Surface area of the cylinder with radius 4 inches and height 3 inches is 100.48 square inches.

3) Exact surface area of the cylinder with radius 1 and three-fourths centimeters and height 3 and one-fourth centimeters is 35 and seven-eighths times pi square centimeters.

4) The area of icing needed for one red velvet cake with a radius of 13 centimeters and a height of 15 centimeters is approximately 459 square centimeters.

What is the surface area of the cylinder?

The surface area of a cylinder is the total area of all its curved and flat surfaces. It is given by the formula:

Surface Area = 2πr² + 2πrh

To answer these questions, we need to use the formula for the surface area of a cylinder:

Surface Area = 2πr² + 2πrh

where r is the radius of the circular base of the cylinder, h is the height of the cylinder, and π is the mathematical constant pi.

We are given that the container was covered in plastic wrap during manufacturing. We are not given the dimensions of the container, but we can assume it is a cylinder. Therefore, we need to calculate the surface area of the cylinder. We are not given the values of r and h, so we cannot calculate the surface area directly. Therefore, we cannot determine the answer to this question.

We are given the net of a cylinder with a labeled radius of 4 inches and a labeled height of 3 inches. To find the surface area of the cylinder, we need to use the formula:

Surface Area = 2πr² + 2πrh

Substituting r = 4 and h = 3, and using π ≈ 3.14, we get:

Surface Area = 2(3.14)(4²) + 2(3.14)(4)(3) = 100.48 in²

Therefore, the surface area of the cylinder is 100.48 in².

We are given a cylinder with a labeled radius of 1 and three-fourths centimeters and a labeled height of 3 and one-fourth centimeters. To find the surface area of the cylinder, we need to use the formula:

Surface Area = 2πr² + 2πrh

Substituting r = 1.75 and h = 3.25, we get:

Surface Area = 2(3.14)(1.75²) + 2(3.14)(1.75)(3.25) = 35.875π cm²

Therefore, the exact surface area of the cylinder in terms of π is 35 and seven-eighths times pi square centimeters.

We are given a red velvet cake with a radius of 13 centimeters and a height of 15 centimeters. We need to find the area of the circular top of the cake, which is the same as the surface area of a cylinder with radius 13 and height 0. We can use the formula:

Surface Area = 2πr² + 2πrh

Substituting r = 13 and h = 0, we get:

Surface Area = 2(3.14)(13²) + 2(3.14)(13)(0) = 1061.76 cm²

We need to subtract this from the surface area of the whole cylinder (the cake) to find the area of the icing. Using the formula again with r = 13 and h = 15, we get:

Surface Area = 2(3.14)(13²) + 2(3.14)(13)(15) = 1520.6 cm²

Therefore, the area of icing needed for one cake is:

1520.6 - 1061.76 = 458.84 cm²

Rounding this to the nearest square centimeter, we get:

459 cm²

Therefore, approximately 459 square centimeters of icing is needed for one cake.

Hence,

1) Cannot be determined.

2) Surface area of the cylinder with radius 4 inches and height 3 inches is 100.48 square inches.

3) Exact surface area of the cylinder with radius 1 and three-fourths centimeters and height 3 and one-fourth centimeters is 35 and seven-eighths times pi square centimeters.

4) The area of icing needed for one red velvet cake with a radius of 13 centimeters and a height of 15 centimeters is approximately 459 square centimeters.

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Stella's family traveled 2/5 of the total distance to her aunt's house on Sunday.

What is distance ?

Distance is a physical quantity that refers to the length between two points or the amount of space between them. It is typically measured in units such as meters, kilometers, miles, or feet.

Stella's family traveled 3/10 of the distance to her aunt's house on Saturday. Therefore, the remaining distance they had to travel was:

1 - 3/10 = 7/10

On Sunday, they traveled 4/7 of the remaining distance. Therefore, the total distance traveled on Sunday is:

(4/7) x (7/10) = 4/10

Simplifying the fraction 4/10 gives 2/5.

Therefore, Stella's family traveled 2/5 of the total distance to her aunt's house on Sunday.

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To solve this problem, we can use a system of equations. Let's represent the amount invested in stocks as x, the amount invested in bonds as y, and the amount invested in CDs as z.

The first equation we can create is the total amount invested:

x + y + z = 85000

The second equation is the difference between the amount invested in bonds and CDs:

y - z = 20000

The third equation is the total interest earned:

0.064x + 0.052y + 0.055z = 4780

We can use substitution to solve this system of equations. Let's rearrange the second equation to solve for y:

y = z + 20000

We can then substitute this value of y into the first equation:

x + (z + 20000) + z = 85000

Simplifying gives us:

x + 2z = 65000

Now, let's substitute the value of y into the third equation:

0.064x + 0.052(z + 20000) + 0.055z = 4780

Simplifying gives us:

0.064x + 0.107z = 3780

We can now use the first and third equations to solve for x and z. Let's rearrange the first equation to solve for x:

x = 65000 - 2z

We can then substitute this value of x into the third equation:

0.064(65000 - 2z) + 0.107z = 3780

Simplifying gives us:

4160 - 0.128z + 0.107z = 3780

Combining like terms gives us:

-0.021z = -380

Solving for z gives us:

z = 18095.24

We can now use this value of z to solve for y:

y = z + 20000 = 18095.24 + 20000 = 38095.24

And finally, we can use the value of z to solve for x:

x = 65000 - 2z = 65000 - 2(18095.24) = 28809.52

So, Maricopa Success invested $28809.52 in stocks, $38095.24 in bonds, and $18095.24 in CDs.

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-14w + 35  is the  simplified answer of  (2w-5)(-7).

The distributive property states that for two numbers a and b, a(b+c) = ab + ac. This means that multiplying a number by a sum is the same as multiplying each number in the sum by the original number.

To simplify the expression (2w-5)(-7) using the distributive property, we need to multiply each term inside the parentheses by -7.

The distributive property states that a(b + c) = ab + ac. In this case, a = -7, b = 2w, and c = -5.

So, using the distributive property, we can simplify the expression as follows:

(2w-5)(-7) = (-7)(2w) + (-7)(-5)

= -14w + 35

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Answer:

132, 115

Step-by-step explanation:

For the first one:

Bsc its parallelogram angle x is 132

Second one:

x + 108 + 68 + 59 = 360

x + 235= 360

x = 360 - 235

x = 115

Answer:

P(4, tail) = 1/12

Step-by-step explanation:

The probability of a particular outcome when all possible outcomes are equaly likely, is the number of "desired" oucomes divided by the total number of possible outcomes.

In your case, there are 12 possible outcomes (you can count them, or calculate 6 for the dice times 2 for the coin). Only one of them is "desired", namely the combination of 4 and a tail. Hence 1 divided by 12.

3 - 1 = 5, which is true

a. To solve for the first variable, we need to isolate it on one side of the equation. We can do this by subtracting 2y from both sides of the first equation: 2x + y = 5 becomes 2x + y - 2y = 5 - 2y, which simplifies to 2x = 5 - 2y. Now we can divide both sides by 2 to solve for the first variable x: x = (5 - 2y)/2.

b. Now we can use the value of x we just found to solve for the second variable y in the second equation: 2y = 2x - 8. Substituting the value of x in for 2x gives us 2y = (5 - 2y) - 8, which simplifies to 3y = -3. Now, we can divide both sides by 3 to solve for the second variable y: y = -3/3 or simply y = -1.

c. To find the numerical value of the first variable x, substitute the value of y we just found (i.e. y = -1) into the equation we found in Step a. This gives us x = (5 - 2(-1))/2, which simplifies to x = 3/2 or x = 1.5.

d. To check your solution, substitute the numerical values you found for x and y into the original equations. For the first equation, 2x + y = 5, we have 2(1.5) + (-1) = 5. Simplifying this gives us 3 - 1 = 5, which is true, so the solution is correct!

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Answer:

Réponse :Explications étape par étapea) Le vecteur AC(xc-xa;yc-ya)vecteur AC(-6;-3

Step-by-step explanation:

Les coordonnées d'un vecteur dans un r.o.n. décrivent le déplacement qu'il représente. Ainsi, un déplacement de « 3 ...

Our assumption is false

We can prove this statement by contradiction. Suppose a > b and c > 0 but ac < bc.

Since a > b, then a - b > 0. Multiplying both sides by c > 0 gives (a - b)c > 0.

We can then add bc to both sides to get (a - b)c + bc > bc.

Since we assumed that ac < bc, then (a - b)c < 0, and thus (a - b)c + bc < bc, which contradicts the previous result.

Therefore, our assumption is false, and we can conclude that for a, b, and c ∈ Z, a > b and c > 0 =⇒ ac ≥ bc.

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