A right triangular prism is shown.

A right triangular prism is shown. The right triangles have sides with lengths 15 and 8. The hypotenuse has a length of 17. The height of the prism is 15.

What is the volume of the prism?

900 cubic units
1,020 cubic units
1,800 cubic units
2,312 cubic units

The derivative of the function y = xeˣ when calculated is xeˣ + eˣ

Given that

y = xeˣ

To find the derivative of the function y = xeˣ, we can use the product rule and the chain rule of differentiation.

First, we use the product rule:

d/dx = x * d/dx eˣ + d/dx x * eˣ

Next, we use the chain rule to complete the derivative

d/dx = x * eˣ + 1 * eˣ

Substituting this into the product rule equation gives:

d/dx = xeˣ + eˣ

Therefore, the derivative of y = xeˣ is  xeˣ + eˣ

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In response to the stated question, we may state that As a result, there equation are 55 Democrats in the Senate.

An equation in mathematics is a statement that states the equality of two expressions. An equation is made up of two sides that are separated by an algebraic equation (=). For example, the argument "2x + 3 = 9" asserts that the phrase "2x + 3" equals the number "9". The purpose of equation solving is to determine the value or values of the variable(s) that will allow the equation to be true. Equations can be simple or complicated, regular or nonlinear, and include one or more elements. In the equation "x2 + 2x - 3 = 0," for example, the variable x is raised to the second power. Lines are utilised in many different areas of mathematics, such as algebra, calculus, and geometry.

Let's call "x" the number of Republicans in the Senate.

According to the problem, Democrats outnumber Republicans by ten. Therefore "x + 10" represents the number of Democrats in the Senate.

Because the Senate has a total of 100 senators, we may formulate an equation:

x + (x + 10) = 100

To simplify this equation:

2x + 10 = 100

Taking 10 off both sides:

2x = 90

Divide all sides by two:

x = 45

As a result, there are 45 Republicans in the Senate.

To get the number of Democrats, we may insert x = 45 into the following expression:

x + 10 = 45 + 10 = 55

As a result, there are 55 Democrats in the Senate.

Therefore:

Democrats have 55 seats.

Republicans have 45 seats.

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Simple interest just accounts for the beginning sum when calculating interest. After 15 years and under the specified circumstances, the interest earned on the given sum is $3,637.50.

How much does simple interest cost?

Simple interest is calculated as follows: I = (PxRxT)/100 if the beginning amount (also known as the principal amount) is P, the annual interest rate is R%, and the amount is left for T years.

In this instance, we are informed that:

P is equal to $5,000, R is 4.85%, and T is 15 years.

Hence, by using the method above to simple interest, we get at:

I = ($5,000x4.85x15)/100 = $3,637.50

As a result, the interest earned on the given sum with the specified circumstances after 15 years is provided by: $3,637.50

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a.There is a 63.21% chance that the light bulb will have to be replaced within 500 hours.

The probability that the light bulb will have to be replaced within 500 hours can be calculated by finding the area under the exponential probability density function (PDF) from 0 to 500. Using the formula for the exponential PDF with a mean of 500, we get:

P(X ≤ 500) = 1 - e^(-500/500) ≈ 0.6321

Therefore, there is a 63.21% chance that the light bulb will have to be replaced within 500 hours.

b. There is a 39.35% chance that the light bulb will last between 200 and 800 hours.

The probability that the light bulb will last more than 1,000 hours can be calculated by finding the area under the exponential PDF from 1000 to infinity. Using the same formula, we get:

P(X > 1000) = e^(-1000/500) ≈ 0.1353

Therefore, there is a 13.53% chance that the light bulb will last more than 1,000 hours.

c. The probability that the light bulb will last between 200 and 800 hours is0.3935.

It can be calculated by finding the area under the exponential PDF from 200 to 800. Again, using the same formula, we get:

P(200 < X < 800) = e^(-200/500) - e^(-800/500) ≈ 0.3935

Therefore, there is a 39.35% chance that the light bulb will last between 200 and 800 hours.

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The list above represents a polynomial of degree 4.The following polyeval( coefs,x ) function returns the value of the polynomial evaluated at x:1.

HTML representation of the polyeval( coefs,x ) function:```def polyeval(coefs, x):    poly_sum = 0    for i in range(len(coefs)):        poly_sum += coefs[i] * x**i    return poly_sum```Explanation:A polynomial can be represented in the form of a list. In Python, the representation of a polynomial as a list [1, 2, -3, 0, 2] means that the ith index corresponds to the coefficient of the term of degree i. Therefore, the list above represents a polynomial of degree 4.The following polyeval( coefs,x ) function returns the value of the polynomial evaluated at x:1. The function accepts two parameters: a list of polynomial coefficients from lowest to highest order and a value x at which to evaluate the polynomial.2. The function returns a float corresponding to the value of the polynomial evaluated at x.

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Answer:

In the picture.

34 ft.

The rate of change of the area of a square with respect to the length z, the diagonal of the square is  dA/dz = 2z; rate = 6. The correct answer is C.

We know that the area A of a square is given by A = s², where s is the length of the sides of the square. Also, we know that the diagonal of the square (z) is related to the sides by the Pythagorean theorem: s² + s² = z² or 2s² = z² or s² = z²/2.

Taking the derivative of both sides of the equation s² = z²/2 with respect to z, we get:

2s ds/dz = 2z/2

s ds/dz = z

Now, since the area A is given by A = s², we can take the derivative of both sides of this equation with respect to z:

dA/dz = d/dz (s²) = 2s ds/dz

Substituting the value of s ds/dz obtained earlier, we get:

dA/dz = 2s (z/s) = 2z

Therefore, the correct option is (c) dA/dz = 2z, and the rate of change of the area of the square with respect to the length z is 2z. When z = 3, the rate of change is 2(3) = 6. So, the answer is (c) dA/dz = 2z; rate = 6.

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It will take approximately 27.7 minutes for the coffee to cool from 54.8°C to 44.9°C when following exponential decay model.

A quantity declines over time proportionate to its existing value through a process known as exponential decay. An exponential function of the form f(t) = ab raised to t, where an is the beginning value, b is the decay factor (a number between 0 and 1), and t represents time, mathematically describes this.

Several real-world circumstances, like population increase, radioactive decay, and the loss of electrical charge in a capacitor, exhibit exponential decay.

Given that the situation follows a exponential decay model.

The exponential decay is given as:

[tex]T(t) = T0 * e^{(-rt)}[/tex]

Substituting the values T0 = 54.8, r = 0.02, and T(t) = 44.9.

[tex]44.9 = 54.8 * e^{(-0.02t)}\\0..8208 = e^{(-0.02t)}\\ln(0.8208) = -0.02t\\t = ln(0.8208)/(-0.02) = 27.7 minutes[/tex]

Hence, it will take approximately 27.7 minutes for the coffee to cool from 54.8°C to 44.9°C.

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Answer:

C = 21,309.4

Step-by-step explanation:

Diameter of moon is miles is,

d = 2159.8 miles

We have,

The circumference of the moon is, 6783 miles

Since, We know that,

the circumference of circle is,

C = 2πr

Substitute given values,

6783 miles = 2 × 3.14 × r

6783 = 6.28 × r

r = 6783 / 6.28

r = 1079.9 miles

Therefore, Diameter of moon is miles is,

d = 2 x r

d = 2 x 1079.9

d = 2159.8 miles

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Step-by-step explanation:

f''(x) = 12x² + 6x - 4

f'(x) = 4x³ + 3x² - 4x + c

f(x) = x⁴ + x³ - 2x² + cx + d

f(0) = 5 = 0⁴ + 0³ - 2×0² + c×0 + d

d = 5

f(1) = 4 = 1⁴ + 1³ - 2×1² + c×1 + 5

4 = 1 + 1 - 2 + c + 5

4 = c + 5

c = -1

f(x) = x⁴ + x³ - 2x² - x + 5

Answer:

m = -3/4

Step-by-step explanation:

Slope = rise/run or (y2 - y1) / (x2 - x1)

Pick 2 points (0, -2) (-3,2)

We see the y increase by 4 and the x decrease by 3, so the slope is

m = -3/4

The amount of energy stored in plant matter each second is 1.08 × 10^12 Joules.

Step 1: Calculation of joules of energy stored in plant matter each secondGiven that, 1.08 terawatts of energy gets stored in plant matter for photosynthesis in a second.Therefore, 1.08 × 1012 watts of energy gets stored in plant matter each second. Also, the energy stored in plants matter in Joules = Watts × seconds (Joule is the unit of energy)1.08 × 1012 watts of energy stored in plant matter in a second.1 watt-second = 1 JouleEnergy stored in plant matter in a second = 1.08 × 1012 watts × 1 second = 1.08 × 1012 Joules Answer: The amount of energy stored in plant matter each second is 1.08 × 10^12 Joules.

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When the initial population increases by 10% in 10 years, it is growing at a rate of approximately 78.04 persons per year at t = 60.

To find the population in 60 years, we need to use the formula:

P(t) = P0rt

P0 is the initial population

r is the rate of growth

t is the time in years.

So, given that the initial population of 500 increases by 10% in 10 years, we can find r as follows:

10% increase in 10 years means that the population has grown to (100% + 10%) = 110% of its original size in 10 years.

Therefore, we have:

P(10) = 500(1 + 0.10)

= 550

Now we can use these values to solve for:

r: 550 = 500er

⇒ er = 550/500

⇒ r = ln(550/500)/10

= 0.04879 (rounded to 5 decimal places)

Therefore, the population in 60 years is:

P(60) = 500e0.04879 × 60 ≈ 1599 (rounded to the nearest person)

The population is growing at a rate of:

P'(t) = rP(t),

so at t = 60, we have:

P'(60) = 0.04879 × 1599 ≈ 78.04 (rounded to two decimal places)

Therefore, the population is growing at a rate of approximately 78.04 persons per year at t = 60.

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Hence, 28 toy soldiers are the correct answer.

A group in mathematics is created by combining a set with a binary operation. For instance, a group is formed by a set of integers with an arithmetic operation and a group is also formed by a set of real numbers with a differential operator.

Let's refer to the quantity of toy soldiers as "x".

We are aware that x is within the range of 27 and 54 thanks to the problem.

x can be divided by 4 without any remainders.

The residual is 6 when x is divided by 7.

The leftover after dividing x by five is three.

These criteria allow us to construct an equation system and find x.

Firstly, we are aware that x can be divided by 4 without any residual. As a result, x needs to have a multiple of 4. We can phrase this as:

x = 4k, where k is some integer.

Secondly, we understand that the remaining is 6 when x is divided by 7. This can be stated as follows:

x ≡ 6 (mod 7)

This indicates that x is a multiple of 7 that is 6 more than. We can solve this problem by substituting x = 4k:

4k ≡ 6 (mod 7)

We can attempt several values of k until we discover one that makes sense for this equation in order to solve for k. We can enter k in to equation starting using k = 1, as follows:

4(1) ≡ 6 (mod 7)

4 ≡ 6 (mod 7)

It is not true; thus we need to attempt a next value for k. This procedure can be carried out repeatedly until the equation is satisfied for all values of k.

k = 2:

4(2) ≡ 6 (mod 7)

1 ≡ 6 (mod 7)

k = 3:

4(3) ≡ 6 (mod 7)

5 ≡ 6 (mod 7)

k = 4:

4(4) ≡ 6 (mod 7)

2 ≡ 6 (mod 7)

k = 5:

4(5) ≡ 6 (mod 7)

6 ≡ 6 (mod 7)

k = 6:

4(6) ≡ 6 (mod 7)

3 ≡ 6 (mod 7)

k = 7:

4(7) ≡ 6 (mod 7)

0 ≡ 6 (mod 7)

We have discovered that the equation 4k 6 (mod 7) is fulfilled when k = 7. Thus, we can change k = 7 to x = 4k to determine that:

x = 4(7) = 28

This indicates that there are 28 toy troops. Yet we also understand that the leftover is 3 when x is divided by 5. We don't need to take into account any other values of x because x = 28 satisfies this requirement.

28 toy soldiers are the correct response.

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The expression 4 triangles to 16 squares when simplified is 1 triangle to 4 squares

Given that

4 triangles to 16 squares

When expressed as ratio, we have

Triangle : Square = 4 : 16

To simplify the ratio Triangle : Square = 4 : 16, we can divide both the numerator and denominator by their greatest common factor, which is 4.

So, we have

Triangle : Square = 4 : 16

Divide both sides by 4:

Triangle/4 : Square/4 = 1 : 4

So the simplified ratio is 1 : 4, which means for every one triangle, there are four squares.

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Confidence Interval: The confidence interval for the true difference between the mean home prices in the two areas at a 95% confidence level is given as [tex]16,322 < \mu_1 - \mu_2 < 31,338.[/tex]

Given: Population 1: [tex]n_1 = 40, \mu_1 = 183,100, \sigma_1 = 21,845[/tex]

Population 2: [tex]n_2 = 33, \mu_2 = 161,500, \sigma2 = 24,820[/tex]

To construct a 95% confidence interval for the true difference between the mean home prices in the two areas, we need to calculate the sample mean difference between the two populations, as well as the standard error.

The sample mean difference is given by: [tex]\bar{x}_1 - \bar{x}_2 = 183,100 - 161,500 = 21,600[/tex]

The standard error is given by: [tex]s = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \\s= \sqrt{\frac{21,845^2}{40} + \frac{24,820^2}{33}} = 8,205[/tex]

Using the standard normal distribution table, with a confidence level of 95%, we obtain a z-score of ±1.96. Using these values, the 95% confidence interval for the true difference between the mean home prices in the two areas is given by: 21,600 - (1.96 * 8,205) < μ1 - μ2 < 21,600 + (1.96 * 8,205)

Solving for the inequality, we get 16,322 < μ1 - μ2 < 31,338

Therefore, the confidence interval for the true difference between the mean home prices in the two areas at a 95% confidence level is given as $16,322 < μ1 - μ2 < $31,338.

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Answer:

B & D

Step-by-step explanation:

For future reference, you can just a site called desmos. It has a graphing tool where you can just write the function and then check where the lines meet.

0.0166 correct to two significant figures is 0.016.

To write 0.0166 to two significant figures, we need to look at the first two significant digits of the number, which are 1 and 6.

Since the digit after 6 is less than 5, we round down the last significant digit (6) to get:

0.016

Therefore, 0.0166 correct to two significant figures is 0.016.

Significant figures (also known as the significant digits, precision or resolution) of a number in positional notation are digits in the number that are reliable and necessary to indicate the quantity of something.

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Answer:

A.

Step-by-step explanation:

Answer:

Step-by-step explanation:

For this problem, we set up an equation

You have 40 cents a pound and 149 cents a pound

You need to make 209 pounds of 85 cents a pounded mix.

Our equation will be:

40(209 - b) + 149b = 209 x 85

b represents the number of pounds in the second mix

209-b represents the number of pounds left in the first mix

Simplifying the equation will leave us to:

8,360 - 40b + 149b = 17,765

8,360 + 109b = 17,765

109b = 9,405

b = 86.284404

109 - b = 22.715596

Answer:

Step-by-step explanation:

Here goes:

So, we want to write out an expression for this situation. I'm not exactly sure what you've been taught in class, but personally, I would start out with a let statement looking something like this:

Let x = the total cost Maura spent canoeing and fishing.

From there, we know you have 4 canoeing and 3 fishing trips. From here, we just plug in numbers for what we know. So, we get an equation that looks like this:

4(8) + 3(5) = x

Now, this is an equation, however, if you just had an expression like the question asked for, there wouldn't be anything to evaluate. Plus, it's always kind of satisfying to get a number answer. So, with a little bit of math, we get 32+15 = x, and a grand total of 47 = x.

Forgot what x was? Thank goodness you wrote a let statement. Look back up if you need the refresher.

Finally, if you need any other clarification, feel free to reach out with another question.

Answer:

F(x) decreases by 6

Step-by-step explanation:

The two points are (-4,10) and (-1,4) and the line is going down as the x value increases. therefore subtract 4 from 10 to get a decrease of -6.

Based on what Mia recorded, we calculate that 152 people are expected to go to the Youth wing on Sunday.

We solve this problem using simple arithmetic methods. According to the data collected by mia,

The number of people who went to the "Youth wing" = 470

The number of people who went to "Social issues" = 413

The number of people who went to "Fiction and Literature" = 350

So, the total number of visitors that the library had on Saturday was,

470+413+350 = 1233

Hence, the proportion of people who went to the "Youth wing", "Social issues" and "Fiction and Literature" are (470/1233), (413/1233), (and 350/1233) respectively.

So, on Sunday the expected number of people who may go to the "Youth wing" should be,

(470/1233)×400

= 152.47 ≈ 152

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By answering the presented question, we may conclude that Therefore, the perimeter of the classroom is 96√2 + 42 feet (approx. 155.27 feet).

In Euclidean geometry, a rectangle is a parallelogram with four small angles. It may also be defined as a hexagon that is fundamental rule, or one in which all of the angles are equal. Another alternative for the parallelogram is a straight angle. Four of the vertices of a square are the same length. A quadrilateral with four 90° angle vertices and equal parallel sides has a rectangle-shaped cross section. As a result, it is also known as a "equirectangular rectangle." A rectangle is sometimes referred to as a parallelogram due to the equal and parallel dimensions of its two sides.

The perimeter-

d = √[(x2 - x1)² + (y2 - y1)²]

So, the perimeter of the classroom is:

d1 = √[(-12 - (-12))² + (15 - (-9))²] = √(24² + 24²) = √(2² × 24²) = 48√2

d2 = √[(-12 - 9)² + (15 - 15)²] = √(21²) = 21

d3 = √[(9 - 9)² + (15 - (-9))²] = √(24² + 24²) = √(2² × 24²) = 48√2

d4 = √[(9 - (-12))² + (-9 - 15)²] = √(21²) = 21

perimeter = d1 + d2 + d3 + d4

= 48√2 + 21 + 48√2 + 21

= 96√2 + 42

Therefore, the perimeter of the classroom is 96√2 + 42 feet (approx. 155.27 feet).

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Answer: 95%

Step-by-step explanation: 38 / 40 = 0.95

Step-by-step explanation:

hey, you just changed the angles in the question.

this following answer was about the angles of depression of 15° and 30°.

you cannot change the problem, when the answers are already given for the original problem.

so, I will add a copy with the adapted numbers for 45° and 30° after my original answer.

this creates 2 right-angled triangles.

the right angle is in both cases the angle where house meets the ground.

they also share one leg : the height of the house (10 m).

the second legs are the ground distances of the cars from the house.

the 2 Hypotenuses are the line of sight from the roof to the corresponding car.

remember, the sum of all angles in a triangle is always 180°.

again, we know one angle : the 90° angle.

but we also know a second angle based on the angles of depression (the "downward looking angles").

the triangle internal angle at the rooftop is the complementary angle (the difference to 90°) of the angle of depression.

so, this is 90-15 = 75° and 90-30 = 60°.

the angles at the cars on the ground are then

angle car 1 = 180 - 90 - 75 = 15°

angle car 2 = 180 - 90 - 60 = 30°

now, remember the trigonometric triangle inscribed in a circle.

imagine that the vertex at the car is the center of the corresponding circle around the trigonometric triangle.

the height of the house is then sine of the angle at the car multiplied by the Hypotenuse (= the line is sight from the rooftop to the car), which is the angle creating radius of the circle.

and the ground distance is the cosine of that same angle multiplied by the Hypotenuse.

so, we need to get the ratio of the height of the house / sin(car angle) to get the length of the Hypotenuse (line of sight). with that we can then calculate the ground distance as cosine of the angle multiplied by the same Hypotenuse.

for car 1 we have

10m/sin(15) = 38.63703305... m line of sight

that means ground distance of car 1 is

cos(15)×38.63703305... = 37.32050808... m

for car 2 we have

10m/sin(30) = 20 m line of sight

that means ground distance of car 2 is

cos(30)×20 = 17.32050808... m

since both cars are driving on the same side of the house in the same direction, the distance between both cars is purely the difference between their distances from the house :

37.32050808... - 17.32050808... = 20 m

the cars are 20 m apart.

and now for the angles of depression of 45° and 30° :

the triangle internal angle at the rooftop is the complementary angle (the difference to 90°) of the angle of depression.

so, this is 90-45 = 45° and 90-30 = 60°.

the angles at the cars on the ground are then

angle car 1 = 180 - 90 - 45 = 45°

angle car 2 = 180 - 90 - 60 = 30°

now, remember the trigonometric triangle inscribed in a circle.

imagine that the vertex at the car is the center of the corresponding circle around the trigonometric triangle.

the height of the house is then sine of the angle at the car multiplied by the Hypotenuse (= the line is sight from the rooftop to the car), which is the angle creating radius of the circle.

and the ground distance is the cosine of that same angle multiplied by the Hypotenuse.

so, we need to get the ratio of the height of the house / sin(car angle) to get the length of the Hypotenuse (line of sight). with that we can then calculate the ground distance as cosine of the angle multiplied by the same Hypotenuse.

for car 1 we have

10m/sin(45) = 14.14213562... m line of sight

that means ground distance of car 1 is

cos(45)×14.14213562... = 10 m

logically, as for 45° sine and cosine are equal.

for car 2 we have

10m/sin(30) = 20 m line of sight

that means ground distance of car 2 is

cos(30)×20 = 17.32050808... m

since both cars are driving on the same side of the house in the same direction, the distance between both cars is purely the difference between their distances from the house :

17.32050808... - 10 = 7.32050808... m

≈ 7.32 m

the cars are about 7.32 m apart.

Answer: 4x(x - 4)

Step-by-step explanation:

4x² - 16x = 4x(x - 4)

Now we can see that the expression inside the parentheses can also be factored:

x - 4 = (x - 4)

So the fully factorized expression is:

4x² - 16x = 4x(x - 4) = 4x(x - 4)

Answer:

ㅤ

- 4x( x + 4 )

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Step-by-step explanation:

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[tex]\large{\pmb{\sf{ - 4x^{2} - 16x}}}[/tex]

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[tex]\large{\underline{\underline{\sf{Taking \: Out \: {\green{4}} \: As \: Common:-}}}}[/tex]

ㅤ

[tex]\large{\pmb{\sf{\leadsto{- 4(x^{2} + 4x)}}}}[/tex]

ㅤ

[tex]\large{\underline{\underline{\sf{Taking \: Out \: {\green{x}} \: As \: Common:-}}}}[/tex]

ㅤ

[tex]\large{\purple{\boxed{\pmb{\sf{\leadsto{- 4x(x + 4)}}}}}}[/tex]

━━━━━━━━━━━━━━━━━━━━━━

[tex]\star \: {\large{\underline{\underline{\pink{\mathfrak{More:-}}}}}} \: \star[/tex]

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[tex]\large{\dashrightarrow}[/tex] Two positive always makes positive sign when multiplied.

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[tex]\large{\dashrightarrow}[/tex] Two negatives always makes positive sign when multiplied.

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[tex]\large{\dashrightarrow}[/tex] A positive and a negative always makes negative sign when multiplied.

ㅤ

[tex]\large{\dashrightarrow}[/tex] The sum of two positives is always positive with a positive sign.

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[tex]\large{\dashrightarrow}[/tex] The sum of two negatives is always positive with a negative sign.

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[tex]\large{\dashrightarrow}[/tex] The sum of a positive and a negative is always negative with the sign of whose number is greater.