Taking attempt 2 of 2.
Question #1*
Find the measure of the indicated arc
67°
134°

Answer:

(fg)(x)= (x²+6)(x²-x+9)

multiply the terms:

(fg)(x)= x²(x²-x+9) +6(x²-x+9)

add the like terms:

(fg)(x)= (x⁴-x³+9x²)+(6x²-6x+54)

and you get your final answer:

(fg)(x)= x⁴-x³+15x²-6x+54

The width of the house on the plan is 0.615 meters.

To find the scale of the plan in the form 1:n, we can compare the measurements on the plan to the actual measurements of the house.

Length of the rectangle on the plan = 78.5 cm

Actual length of the house = 15.7 m

We need to convert the actual length of the house to the same unit as the length on the plan, which is centimeters.

1 meter = 100 centimeters

So, the actual length of the house in centimeters = 15.7 m [tex]\times[/tex] 100 cm/m = 1570 cm

Now, we can find the scale of the plan by dividing the length on the plan by the actual length of the house:

Scale = Length on the plan / Actual length of the house

= 78.5 cm / 1570 cm

Simplifying this fraction, we get:

Scale = 1/20

Therefore, the scale of the plan is 1:20.

To find the width of the house on the plan, we can use the same scale.

Width of the house in actual measurements = 12.3 m.

Width of the house on the plan = (Width of the house in actual measurements) / Scale

= 12.3 m / 20

= 0.615 m.

So, the width of the house on the plan is 0.615 meters.

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(a) Starting at 12 and making 1/2 revolution clockwise, the hand stops at 6.

(b) Starting at 2 and making 1/2 revolution clockwise, the hand stops at 8.

(c) Starting at 5 and making 1/4 revolution clockwise, the hand stops at 8.

(d) Starting at 5 and making 3/4 revolution clockwise, the hand stops at 11.

To determine where the hand of a clock will stop, we need to consider the fractions of a revolution made by the hand starting from different positions.

(a) If the hand starts at 12 and makes 1/2 of a revolution clockwise, it will stop at 6.

This is because a half revolution corresponds to the hand moving from 12 to 6 on the clock face.

(b) If the hand starts at 2 and makes 1/2 of a revolution clockwise, it will stop at 8.

Again, a half revolution corresponds to the hand moving from 2 to 8 on the clock face.

(c) If the hand starts at 5 and makes 1/4 of a revolution clockwise, it will stop at 8.

A quarter revolution corresponds to the hand moving from 5 to 8 on the clock face.

(d) If the hand starts at 5 and makes 3/4 of a revolution clockwise, it will stop at 11.

A three-quarter revolution corresponds to the hand moving from 5 to 11 on the clock face.

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The quotient of the given rational expressions, (x^2 - 16)/(x + 5) divided by (x^2 - 8x + 16)/(2x + 10), is (x - 4)/(x - 4), which simplifies to 2.

To divide rational expressions, we invert the second expression and multiply it with the first expression. So, we have:

[(x^2 - 16)/(x + 5)] / [(x^2 - 8x + 16)/(2x + 10)]

To simplify this expression, we can multiply by the reciprocal of the second rational expression:

[(x^2 - 16)/(x + 5)] * [(2x + 10)/(x^2 - 8x + 16)]

Next, let's factorize the numerators and denominators of both expressions:

[(x + 4)(x - 4)/(x + 5)] * [2(x + 5)/((x - 4)(x - 4))]

Now, we can cancel out the common factors:

[(x + 4) * 2(x + 5)] / [(x + 5) * (x - 4)(x - 4)]

The (x + 5) factors cancel out:

[(x + 4) * 2(x + 5)] / [(x - 4)(x - 4)]

Further simplification:

[2(x + 4)(x + 5)] / [(x - 4)(x - 4)]

Now, we observe that the factors (x - 4)(x - 4) are the same in the numerator and denominator. Therefore, they cancel out:

2(x + 4)(x + 5) / (x - 4)(x - 4) = 2(x + 4)(x + 5) / (x - 4)(x - 4) = 2

Therefore, the quotient of the given rational expressions is 2.

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The biconditional statement for the given conditional statement would be:

2x = 18 if and only if x = 9.

The given conditional statement "If 2x = 18, then x = 9" can be represented symbolically as p → q, where p represents the statement "2x = 18" and q represents the statement "x = 9".

To form the biconditional statement, we need to determine if the converse of the conditional statement is also true. The converse of the original statement is "If x = 9, then 2x = 18". Let's evaluate the converse statement.

If x = 9, then substituting this value into the equation 2x = 18 gives us 2(9) = 18, which is indeed true. Therefore, the converse of the original statement is true.

Based on this, we can write the biconditional statement:

2x = 18 if and only if x = 9.

The biconditional statement implies that if 2x is equal to 18, then x must be equal to 9, and conversely, if x is equal to 9, then 2x is equal to 18. The biconditional statement asserts the equivalence between the two statements, indicating that they always hold true together.

In summary, the biconditional statement is a concise way of expressing that 2x = 18 if and only if x = 9, capturing the mutual implication between the two statements.

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Answer:

If Tonia eats 1/8 of the cake, then the fraction of the cake left is:

1 - 1/8 = 7/8

If Trinny eats 1/6 of the cake, then the fraction of the cake left is:

1 - 1/6 = 5/6

Since Tonia and Trinny have identical cakes, the amount of cake left is the same for both of them. Therefore, the amount of cake left is:

(7/8 + 5/6) / 2 = 41/48

So there is 41/48 of the cake left.

Vinay buys "two thousand seven" fruits.

To find the number of fruits that Vinay buys, we need to determine the place value of 2 in the number 37,523 and add 7 to it.

In the number 37,523, the digit 2 is in the thousands place.

The place value of 2 in the thousands place is 2,000.

Adding 7 to the place value of 2, we get:

2,000 + 7 = 2,007.

Therefore, Vinay buys 2,007 fruits.

In number names, we can write 2,007 as "two thousand seven."

So, Vinay buys "two thousand seven" fruits.

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Answer:

1. 01= 234, 03= 255 (since the data is

already sorted)

2. I0R = 255 - 234= 21

3. Lower limit = 234- 1.5 * 21= 203.5

Upper limit = 255+ 1.5 * 21= 285.5

4. Outliers: 110, 120, 178 (below the

lower limit), and 273 (above the upper

limit)

Answer: Nope

Step-by-step explanation:

No, receiving a 20% discount followed by an additional 30% discount does not result in a total discount of 50%.

To understand why, let's consider an example with an item priced at $100.

If there is a 20% discount applied initially, the price of the item would be reduced by 20%, which is $100 * 0.20 = $20. So the new price after the first discount would be $100 - $20 = $80.

Now, if there is an additional 30% discount applied to the $80 price, the discount would be calculated based on the new price. The 30% discount would be $80 * 0.30 = $24. So the final price after both discounts would be $80 - $24 = $56.

Comparing the final price of $56 to the original price of $100, we can see that the total discount is $100 - $56 = $44.

Therefore, the total discount received is $44 out of the original price of $100, which is a discount of 44%, not 50%.

Hence, receiving a 20% discount followed by an additional 30% discount does not result in a total discount of 50%.

Answer:

first, arrange the numbers from least to greatest order. (Actually this is a longer step, if you also want to find the median. But since you are only asking for range and standard deviation, I won't do that here.) So, just find the lowest number and the highest number. Subtract the lowest from the highest. That is your range.

Your problem:

lowest: 98.6

highest: 131.9

subtract: 131.9 - 98.6 = 33.3  ← this is your range

Now, standard deviation.

Standard deviation is the amount of variety you have in your data sample.

Step 1: Find the mean

Add up all your numbers and divide by how many numbers you have.

You have 10 numbers in your sample.

115.6 + 109.3 + 126 + 104.9 + 131.9 + 113.7 + 119.8 + 98.6 + 131.9 + 131.9 

total = 1,183.6

now divide this by 10.  (n is the variable used here, so n = 10. This is because you have ten numbers. )

so n = 10

and 1,183.6/10 = 118.3

Now subtract each number by 118.3.

115.6 - 118.3 = -2.7

109.3 - 118.3 = -9

126 - 118.3 = 7.7

104.9 - 118.3 = -13.4

131.9 -118.3 = 13.6

113.7 - 118.3 = -4.6

119.8 - 118.3 = 1.5

98.6 - 118.3 = -19.7

131.9 - 118.3 = 13.6

131.9 - 118.3 = 13.6

now square all these numbers

7.29

81

59.29

179.56

184.96

21.16

0.75

388.09

184.96

184.96

Find the sum of these squares now. (We're almost done!)

sum = 1,292.02

remember our n?

it was n=10

now the formula for this is,

sum of squares ÷ n-1

substitute all this in.

1,292.02 ÷ 9 = 143.55

Remember. This is the VARIANCE. NOT the standard deviation.

The last step to find the standard deviation is, to find the square root of what we got. (143.55)

√143.55

= 11.9812353286 this is the number, but rounded two more decimal places is..

11.98 is the standard deviation.

Hope this helped!

The diameter of the engine cylinder is written in the form

|d - 5| ≤ 0.005

To represent the diameter of an engine cylinder with a tolerance of ±0.005 cm and a desired width of 5 cm, we can use an absolute value inequality.

The absolute value inequality representing the permissible limit of variation can be written as:

|d - 5| ≤ 0.005

where:

d represents the diameter of the engine cylinder.

In this inequality, the absolute value of the difference between the diameter (d) and the desired width (5 cm) must be less than or equal to the given tolerance of ±0.005 cm.

This means that the diameter of the engine cylinder can vary within ±0.005 cm of the desired width of 5 cm.

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Conditional probability, denoted as P(A|B), represents the probability of event A occurring given that event B has occurred. If events A and B are independent, P(A|B) = P(A) and P(B|A) = P(B).

The notation P(A|B) reads the probability of Event (A occurring given that Event B has occurred). If two events are independent, then the probability of one event occurring (does not affect the probability of the other event occurring). Events A and B are independent if (P(A|B) = P(A), P(B|A) = P(B), P(A|B) = P(B|A)).

To understand the relationship between conditional probability and independent events, let's consider two events A and B. The conditional probability P(A|B) represents the probability of event A occurring given that event B has already occurred. It measures the likelihood of event A happening under the condition that event B has already taken place.

On the other hand, if two events A and B are independent, it means that the occurrence or non-occurrence of one event has no effect on the probability of the other event happening. In other words, the probability of event A happening is not influenced by the occurrence or non-occurrence of event B, and vice versa.

Mathematically, if events A and B are independent, it implies that P(A|B) = P(A) and P(B|A) = P(B). This means that the probability of event A occurring is the same whether or not event B has occurred, and the probability of event B occurring is the same whether or not event A has occurred.

Therefore, the concepts of conditional probability and independent events are related in the sense that if two events are independent, the conditional probabilities P(A|B) and P(B|A) become equal to the unconditional probabilities P(A) and P(B) respectively.

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Answer: B. The minimum y-value of g(x) is -3.

Step-by-step explanation:

Based on the given functions:

f(x) = 3 - 3

g(x) = 7x² - 3

The y-value of f(x) is constant at -3, regardless of the value of x. Therefore, f(x) does not have a minimum y-value, and option A is incorrect.

The y-value of g(x) is determined by the quadratic term 7x². Since the coefficient of x² is positive (7), the parabola opens upwards, indicating that g(x) has a minimum y-value. To find the minimum value of g(x), we can look at the vertex of the parabola, which occurs when x = -b/2a in the quadratic equation ax² + bx + c. In this case, a = 7 and b = 0, so the vertex is at x = -0/2(7) = 0. Substituting x = 0 into g(x), we find: g(0) = 7(0)² - 3 = -3 Therefore, the minimum y-value of g(x) is -3, and option B is correct.

Option C, stating that g(x) has the smallest possible y-value, is incorrect because the y-value of g(x) can be larger than -3 depending on the value of x.

Option D, stating that f(x) has the smallest possible y-value, is incorrect because f(x) does not have a minimum y-value as it is constant at -3.

Therefore, the correct answer is B. The minimum y-value of g(x) is -3.

Answer:

m = undefined

Step-by-step explanation:

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

Points (-4,7) (-4,1)

We see the y decrease by 6 and the x stay the same, so the slope is

m = undefined

[tex]\pmb{Question}[/tex]

Find the slope of a line passing through (-4,7) (-4,1).

[tex]\pmb{Answer}[/tex]

Not Defined

[tex]\pmb{Formula}[/tex]

[tex]\sf{m=\dfrac{y_2-y_1}{x_2-x_1}}[/tex]

[tex]\pmb{Where}[/tex]

[tex]\pmb{Plug\:in\:the\:data}[/tex]

[tex]\sf{m=\dfrac{1-7}{-4-(-4)}}[/tex]

[tex]\sf{m=\dfrac{-6}{-4+4}}[/tex]

[tex]\sf{m=-\dfrac{6}{0}}[/tex]

[tex]\sf{Slope=Not\;De fined}[/tex]

∴ the slope is undefined

[tex]\rule{350}{3}[/tex]

[tex]\frak{-star-}[/tex]

Answer:  Choice A

The single squiggly symbol means "similar".

A squiggly line over top an equals sign is the congruence symbol.

Answer:

See attachment for the graph of the hyperbola.

Step-by-step explanation:

Given equation:

[tex](x-7)^2-\dfrac{(y-4)^2}{9}=1[/tex]

As the x²-term of the given equation is positive, the transverse axis is horizontal, and so the hyperbola is horizontal (opening left and right). Note, if the y²-term was positive, the hyperbola would have been vertical.

The general formula for a horizontal hyperbola (opening left and right) is:

[tex]\boxed{\begin{minipage}{7.4 cm}\underline{Standard equation of a horizontal hyperbola}\\\\$\dfrac{(x-h)^2}{a^2}-\dfrac{(y-k)^2}{b^2}=1$\\\\where:\\\phantom{ww}$\bullet$ $(h,k)$ is the center.\\ \phantom{ww}$\bullet$ $(h\pm a, k)$ are the vertices.\\\phantom{ww}$\bullet$ $(h\pm c, k)$ are the foci where $c^2=a^2+b^2.$\\\phantom{ww}$\bullet$ $y=\pm \dfrac{b}{a}(x-h)+k$ are the asymptotes.\\\end{minipage}}[/tex]

Comparing the given equation with the standard equation:

To find the value of c, use c² = a² + b²:

[tex]\begin{aligned}c^2&=a^2+b^2\\c^2&=1+9\\c^2&=10\\c&=\sqrt{10}\end{aligned}[/tex]

The center is (h, k). Therefore, the center is (7, 4).

The formula for the loci is (h±c, k). Therefore:

[tex]\begin{aligned}\textsf{Loci}&=(h \pm c, k)\\&=(7 \pm \sqrt{10}, 4)\\&=(7- \sqrt{10}, 4)\;\;\textsf{and}\;\;(7 +\sqrt{10}, 4)\end{aligned}[/tex]

The formula for the vertices is (h±a, k). Therefore:

[tex]\begin{aligned}\textsf{Vertices}&=(h \pm a, k)\\&=(7 \pm 1, 4)\\&=(6, 4)\;\;\textsf{and}\;\;(8, 4)\end{aligned}[/tex]

The asymptotes are:

[tex]\begin{aligned}y&=\pm \dfrac{b}{a}(x-h)+k\\\\y&=\pm \dfrac{3}{1}(x-7)+4\\\\y&=\pm 3(x-7)+4\\\\\implies y&=3x-17\\\implies y&=-3x+25\end{aligned}[/tex]

Therefore:

The graph of the hyperbola (x - 7)² - (y - 4)²/9 = 1 is attached below

The graph of a hyperbola is a curve that consists of two separate branches, each resembling a symmetrical curve. The general equation for a hyperbola in standard form is:

[(x - h)² / a²] - [(y - k)² / b²] = 1

The center of the hyperbola is represented by the coordinates (h, k). The parameters a and b determine the size and shape of the hyperbola.

Based on the standard form equation, there are two types of hyperbolas:

1. Horizontal Hyperbola:

When the major axis is parallel to the x-axis, the hyperbola is horizontal. The equation in this case is:

[(x - h)² / a²] - [(y - k)² / b²] = 1

The graph of a horizontal hyperbola opens left and right. The branches are symmetric about the x-axis and the center (h, k) is the midpoint between the branches.

2. Vertical Hyperbola:

When the major axis is parallel to the y-axis, the hyperbola is vertical. The equation in this case is:

[(y - k)² / b²] - [(x - h)² / a²] = 1

The graph of a vertical hyperbola opens up and down. The branches are symmetric about the y-axis and the center (h, k) is the midpoint between the branches.

The graph of the given hyperbola is attached below.

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Answer:

Step-by-step explanation:

To determine which option will yield the most money after three years, let's calculate the final amount for each option.

Option #1 - CD for 3 years:

Principal (initial investment) = $1000

Interest rate = 3% per year (compounded monthly)

No additional money can be added

To calculate the final amount, we can use the formula for compound interest:

A = P * (1 + r/n)^(n*t)

Where:

A = Final amount

P = Principal (initial investment)

r = Interest rate (as a decimal)

n = Number of times the interest is compounded per year

t = Number of years

For Option #1:

P = $1000

r = 3% = 0.03 (as a decimal)

n = 12 (compounded monthly)

t = 3 years

A = $1000 * (1 + 0.03/12)^(12*3)

Calculating the final amount for Option #1, we get:

A = $1000 * (1 + 0.0025)^(36)

A ≈ $1000 * (1.0025)^(36)

A ≈ $1000 * 1.0916768

A ≈ $1091.68

Option #2 - CD for 1 year:

Principal (initial investment) = $1000

Interest rate = 2% per year (compounded quarterly)

Money can be added at the end of each year

To calculate the final amount, we need to consider the annual additions and compounding at the end of each year.

First Year:

P = $1000

r = 2% = 0.02 (as a decimal)

n = 4 (compounded quarterly)

t = 1 year

A = $1000 * (1 + 0.02/4)^(4*1)

A ≈ $1000 * (1.005)^(4)

A ≈ $1000 * 1.0202

A ≈ $1020.20

At the end of the first year, the total amount is $1020.20.

Second Year:

Now we add an additional $60 to the previous amount:

P = $1020.20 + $60 = $1080.20

r = 2% = 0.02 (as a decimal)

n = 4 (compounded quarterly)

t = 1 year

A = $1080.20 * (1 + 0.02/4)^(4*1)

A ≈ $1080.20 * (1.005)^(4)

A ≈ $1080.20 * 1.0202

A ≈ $1101.59

At the end of the second year, the total amount is $1101.59.

Third Year:

Again, we add $60 to the previous amount:

P = $1101.59 + $60 = $1161.59

r = 2% = 0.02 (as a decimal)

n = 4 (compounded quarterly)

t = 1 year

A = $1161.59 * (1 + 0.02/4)^(4*1)

A ≈ $1161.59 * (1.005)^(4)

A ≈ $1161.59 * 1.0202

A ≈ $1185.39

At the end of the third year, the total amount is $1185.39.

Comparing the final amounts:

Option #1: $1091.68

Option #2: $1185.39

Therefore, Option #2 - CD for 1 year with an interest rate of 2% compounded quarterly and the ability to add money at the end of each year will yield the most money after three years.

The solution to the inequality[tex](x/4) - 3 > 2 is x > 20.[/tex]

To solve the inequality [tex](x/4) - 3 > 2,[/tex]we'll follow these steps:

Step 1: Eliminate the fraction by multiplying both sides of the inequality by the denominator, which is 4 in this case. This step allows us to get rid of the fraction and simplify the inequality.

[tex](x/4) - 3 > 2[/tex]

Multiply both sides by 4:

[tex]4 * [(x/4) - 3] > 4 * 2[/tex]

This simplifies to:

x - 12 > 8

Step 2: Isolate the variable on one side of the inequality by adding 12 to both sides:

x - 12 + 12 > 8 + 12

This simplifies to:

x > 20

So, the solution to the inequality is x > 20. This means that any value of x greater than 20 will satisfy the inequality.

To represent this solution graphically, we can plot the number line and shade the region to the right of 20, indicating that any value greater than 20 is a valid solution.

---------------------------------

   -∞     20      +∞

       --------------------------

               ●=================

---------------------------------

In the number line above, the shaded region represents the solution x > 20. Any value to the right of 20, including 20 itself, will satisfy the original inequality.

In summary, the solution to the inequality [tex](x/4) - 3 > 2 is x > 20.[/tex]

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The values of L and M in the triangle displayed are 55 and 60 respectively.

The value of angle L can be obtained thus :

125 + L = 180 (sum of angles in a triangle)

L = 180 - 125 = 55°

B.

The value of L can be calculated thus:

55 + (2x - 10) + 65 = 180 (sum of internal angles of a triangle)

120 + 2x - 10 = 180

110+2x = 180

2x = 180-110

x = 35

M = 2(35) -10 = 60°

Therefore, L = 55 and M = 60.

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Answer:

Tom worked approximately 8 overtime hours in the given pay period.

Step-by-step explanation:

Answer:

f(x)=5/(x-6)

Step-by-step explanation:

f(x)=(5x-5)/(x^2-7x+6)

f(x)=[5(x-1)]/[(x-1)(x-6)]

f(x)=5/(x-6)

[tex]\qquad\displaystyle \rm \dashrightarrow \: let \: \: Sydney's \: \: age \: \: be \: \: 'y'[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: Devaughn's \: \: age \: \: will \: \: be \: \: 3y[/tex]

Sum up ;

[tex]\qquad\displaystyle \tt \dashrightarrow \: 3y + y = 80[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: 4y = 80[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: y = 80 \div 4[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: y = 20[/tex]

So, Sydney's age is 20 years, n that of Devaughn is 20 × 3 = 60 years

Answer:

Sydney= 20, Devaughn= 60

Step-by-step explanation:

Let Sydney's age be 'x'

Devaughn's age = 3 times x = 3x

We Know That

The sum of their ages is 80.

So,

3x + x = 80

4x = 80

If we shift the 4 to the 80 side

x = 80/4

x = 20

So, Sydney's age is 20

Therefore, Devaughn's age =

3x = 3 times x

= 3 times 20

= 60

Answer:

Since P(male)xP(fail) = 0.0549 and and P(male and fail) = 0.0773, the two results are different, so the events are not independent.

Step-by-step explanation:Independent events:

Two events, A and B are independent, if:

Probability of male:

58 + 14 = 72 males out of 58 + 14 + 98 + 11 = 181

So

P(male) = 72/181 = 0.3978

Probability of failling:

14 + 11 = 25 students fail out of 181. So

P(fail) = 28/181 = 0.1381

Multiplitication of male and failling:

0.3978*0.1381 = 0.0549

Probability of being male and failing:

14 out of 181. So

14/181 = 0.0773

Different probabilities, so not independent.

Since P(male)xP(fail) = 0.0549 and and P(male and fail) = 0.0773, the two results are different, so the events are not independent.

To find the value of x, we can set the two angle measures equal to each other and solve for x.

Given:

mZA = (4x - 2)°

mZB = (6x - 20)°

Setting them equal to each other:

4x - 2 = 6x - 20

Now, we can solve for x:

4x - 6x = -20 + 2

-2x = -18

Dividing both sides by -2:

x = -18 / -2

x = 9

Therefore, the value of x is 9.

Answer:

The answer is 9.

Step-by-step explanation:

We need to use the fact that the sum of the angles in a triangle is 180 degrees. Let A, B, and C be the three angles in the triangle. Then we have:

mZA + mZB + mZC = 180°

Substituting the given values, we get:

(4x - 2)° + (6x - 20)° + mZC = 180°

Simplifying the left side, we get:

10x - 22 + mZC = 180°

Next, we use the fact that angles opposite congruent sides of a triangle are congruent. Since we know that segment AC and segment BC are congruent, we have:

mZA = mZB

Substituting the given values and simplifying, we get

4x - 2 = 6x - 20

Solving for x, we get:

x = 9

Therefore, the value of x is 9.

The solution to the equation [tex]3^x + 3^(4-2x) = 1 + 3^(4-x) is x = 2.[/tex]

To solve the equation [tex]3^x + 3^(4-2x) = 1 + 3^(4-x),[/tex] we can simplify the equation and then apply some algebraic techniques to isolate the variable x.

First, let's simplify the equation step by step:

1. Notice that [tex]3^(4-2x)[/tex] can be rewritten as[tex](3^4) / (3^2x)[/tex], using the property of exponentiation.

2. Now the equation becomes 3[tex]^x + (81 / 9^x) = 1 + 3^(4-x).[/tex]

3. We can simplify further by multiplying both sides of the equation by 9^x to eliminate the denominators.

  This gives us [tex]3^x * 9^x + 81 = 9^x + 3^(4-x) * 9^x.[/tex]

4. Simplifying the terms, we have [tex](3*9)^x + 81 = 9^x + (3*9)^(4-x).[/tex]

  Now we have [tex](27)^x + 81 = 9^x + (27)^(4-x).[/tex]

5. Notice that [tex](27)^x and (27)^(4-x)[/tex] have the same base, so we can set the exponents equal to each other.

  This gives us x = 4 - x.

6. Simplifying the equation, we get 2x = 4.

7. Dividing both sides of the equation by 2, we have x = 2.

Therefore, the solution to the equation [tex]3^x + 3^(4-2x) = 1 + 3^(4-x) is x = 2.[/tex]

Using simple language, we simplified the equation step by step and isolated the variable x by setting the exponents equal to each other. The final solution is x = 2.

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The figure that shows a reflectional symmetry would be figure C. That is option A.

The reflectional symmetry of shapes is defined as the type of symmetry where one-half of the object reflects the other half of the object.

This is also called a mirror symmetry. This is because the image seen in one side of the mirror is exactly the same as the one seen on the other side of the mirror.

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The inverse of the original statement is: "If the sum of the interior angles of a polygon is not more than 180°, then the polygon is a triangle."

The inverse of the conditional statement "If the sum of interior angles of a polygon is more than 180°, then the polygon is not a triangle" is: "If the sum of the interior angles of a polygon is not more than 180°, then the polygon is a triangle."
To find the inverse, we need to negate both the hypothesis and the conclusion of the original statement.
The hypothesis of the original statement is "the sum of the interior angles of a polygon is more than 180°". To negate this, we say "the sum of the interior angles of a polygon is not more than 180°".
The conclusion of the original statement is "the polygon is not a triangle". To negate this, we say "the polygon is a triangle".
In summary, the inverse of the original statement is "If the sum of the interior angles of a polygon is not more than 180°, then the polygon is a triangle."

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The linear rule for the sequence is f(n) = 7/20 + 3/20(n - 1)

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

35/100,5/10,65/100

In the above sequence, we can see that 15/100 is added to the previous term to get the new term

This means that

First term, a = 35/100

Common difference, d = 15/100

The nth term is then represented as

f(n) = a + (n - 1) * d

Substitute the known values in the above equation, so, we have the following representation

f(n) = 35/100 + 15/100(n - 1)

So, we have

f(n) = 7/20 + 3/20(n - 1)

Hence, the explicit rule is f(n) = 7/20 + 3/20(n - 1)

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