what is the reason behind Russia and ukarine war?​

Answer:

  (d)  none of the lines fit the data

Step-by-step explanation:

You want to know if any of the lines shown on the graph represents a good fit to the data.

The line of best fit for the data shown would be one that passes through the points (0, 5.3) and (9, 0.4). It would lie somewhat below line A and somewhat above line B.

None of the lines shown fit the data.

__

Additional comment

Line A seems to ignore the data that is fit by Line B, and vice versa. A line of best fit will minimize the measure of distance to all the data points, not just some of them.

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Answer:

Step-by-step explanation:

distribute the 3

you get 3x+24=15

subtract 24 from both sides

you get 3x=-9

divide 3 from both sides

you get x=-3

Answer:

The correct answer is C. $68.35.

Here's the breakdown of the total cost:

* 4 large pizzas at $13.99 each = $55.96

* 7.85% sales tax on $55.96 = $4.38

* $8.00 delivery charge

Total: $55.96 + $4.38 + $8.00 = $68.35

The number of fossils after 10 years is equal to 2333.59 fossils.

In order to determine the number of fossils on this barren land after 10 years, we would have to solve the given differential equation [tex]\frac{dN}{dt} =\frac{5000}{6\;+\;5t}[/tex] by applying an integration in order to create an equation that models N(t) with respect to the number of years or time (t) as follows.

First of all, we would rewrite the equation as follows;

[tex]dN =\frac{5000}{6\;+\;5t}dt[/tex]

By integrating both sides of the equation with respect to t, we have the following:

[tex]\int dN =\int \frac{5000}{6\;+\;5t}dt\\\\N=\frac{5000}{5} ln|6+5t|+C\\\\N=1000 ln|6+5t|+C[/tex]

Since the initial number of fossils (N) is equal to 100 when time (t) = 0, we would substitute the parameters into the equation as follows;

N = 1000ln(6 + 5t) + C

100 = 1000ln(6 + 5(0)) + C

C = 100 - 1000ln6

Next, we would rewrite the equation for the number of fossils (N) in terms of time (t) as follows;

N = 1000ln(6 + 5t) + 100 - 1000ln6

By simplifying the equation using the quotient log rule, we have:

N = 1000ln(6 + 5t) - 1000ln6 + 100

N = 1000ln[(6 + 5t)/6] + 100

Therefore, the number of fossils (N) after 10 years can be calculated as follows;

N = 1000ln[(6 + 5(10))/6] + 100

N = 1000ln[(6 + 50)/6] + 100

N = 1000ln(56/6) + 100

N = 2333.59 fossils.

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There is no specific minimum amount he needs to earn on the last day to meet the goal because it has already been achieved.

We have,

To find the minimum amount Gil must earn on the last day of the week to meet his weekly goal,

We need to calculate the total earnings for the first 6 days and subtract it from the target weekly goal of $500.

Given the daily earnings for the first 6 days:

425, 515, 475, 620, 525, 415

We can calculate the total earnings:

Total earnings = 425 + 515 + 475 + 620 + 525 + 415 = 2975

Now, we subtract the total earnings from the weekly goal:

Minimum amount needed on the last day.

= 500 - 2975

= -2475

Since the calculated value is negative (-2475), it implies that Gil has already exceeded his weekly goal based on the earnings from the first 6 days.

Therefore,

There is no specific minimum amount he needs to earn on the last day to meet the goal because it has already been achieved.

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Answer:

24

Step-by-step explanation:

the lengths are in proportion to one another, between triangles.

(x - 4)/12  =  5/(x +7)

(x - 4)(x + 7) = 5 X 12

x² + 7x - 4x - 28 = 60

x² + 3x - 28 = 60

x² + 3x - 88 = 0.

(x + 11) (x - 8) = 0

x = -11, 8.

x cannot be -11 since both x +7 and x - 4 would be negative lengths!!

so x = 8.

AB = 12, DE = x - 4 = 8 -4 = 4. 12/4 = 3.

the lengths of triangle ABC are 3 times longer than the lengths of DEF.

we can confirm this: AC = x + 7 = 8 + 7 = 15. DF = 5

AC is 15/5 = 3 times longer than DF.

so we expect BC to be 3 times longer than x. x = 8.

so BC = 3 X 8 = 24.

Check the picture below.

the slope goes by several names

• average rate of change

• rate of change

• deltaY over deltaX

• Δy over Δx

• rise over run

• gradient

• constant of proportionality

however, is the same cat wearing different costumes.

how fast is it traveling? or what's its average rate or slope?

to get the slope of any straight line, we simply need two points off of it, let's use those two in the picture below

[tex](\stackrel{x_1}{4}~,~\stackrel{y_1}{2})\qquad (\stackrel{x_2}{6}~,~\stackrel{y_2}{6}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{6}-\stackrel{y1}{2}}}{\underset{\textit{\large run}} {\underset{x_2}{6}-\underset{x_1}{4}}} \implies \cfrac{ 4 }{ 2 } \implies \cfrac{2~meters}{1~second}\qquad \textit{2 meters per second}[/tex]

Answer:

30

Step-by-step explanation:

y = ( a + b ) ( a - b )

Let a = 6.5 and b = 3.5

y = ( 6.5 + 3.5 ) ( 6.5 - 3.5 )

Simplify the terms in the parentheses.

y = (10)(3)

y= 30

Answer:

sandwich = $4. bottle = $2

Step-by-step explanation:

let's A = sandwich, B = bottle

elimination method:

10A + 6B = 52   (call this equation '1.' or just call it '1')

A + 3B = 10     (call this '2')

'2' X 2:   2A + 6B = 20     (call this '3')

'1' - '3':

8A + 0B = 32

A = 32/8 = 4

Sandwich costs $4.

go back to '2.'

3B = 10 - A = 10 - 4 = 6.

B = 6/3 = 2.

bottled water costs $2.

substitution method:

6B = 52 - 10A

3B = 26 - 5A.

'2.'

A + 3B = 10

A + (26 - 5A) = 10

-4A = -16

A = 4.

A + 3B = 10

3B = 10 - 4 = 6

B = 6/3 = 2.

Type I error is "Megan's results show significant improvements in health associated with a whole foods diet."

A Type I error occurs when the null hypothesis is rejected, even though it is true.

In this case, the null hypothesis states that a whole foods diet would not have any impact on general health.

Therefore, a Type I error would occur if Megan's results show significant improvements in health associated with a whole foods diet.

This means that Megan would reject the null hypothesis and conclude that a whole foods diet does have an impact on general health, even though it is not true.

So, the statement that would indicate a Type I error is "Megan's results show significant improvements in health associated with a whole foods diet."

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The surface area of the cylinder B is 2.5 cm square.

The surface area of the cylinder can be found as follows:

Cylinder A and cylinder B are similar. Let's find the surface area of cylinder B.

The radius of cylinder A is 4 cm and the volume of cylinder A is 24π,. Therefore,

volume of cylinder A = πr²h

24π = π2²h

24π = 4πh

4h = 24

h = 24 / 4

h = 6 cm

surface area of the cylinder A = 2πr(r + h) = 3π(1.5 + 6) = 22.5π cm²

Therefore, let's find the surface area of the cylinder B as follows:

(2 / 6) ² = x / 22.5π

4 / 36 = x / 22.5π

cross multiply

36x = 90π

x = 90π / 36

x = 10π / 4

surface area of cylinder B = 5 / 2 π = 2.5π cm²

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When 0.50 is written out in full, the first seven digit after the decimal point is

5 0 5 0 5 0 5

A recurring decimal also known as a repeating decimal  is a decimal representation of a number where one or more digits repeat infinitely after a certain point. In other words  there is a repeating pattern of digits that continues indefinitely.

Recurring decimals are typically denoted by placing a horizontal line or a dot above the repeated portion of the digits.

In this case the repeating decimal is represented by dot above. therefor we have 5 and 0 repeating.

The umber in fraction is 50/99

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Solution of the given expression is,

 x = 6/5 or x = -4

Given that,

|2x+5| + |3x+1| = 10

Add -|3x-1| both sides,

⇒ |2x+5| + |3x+1| - |3x+1| = 10 -  |3x+1|

⇒                          |2x+5| = 10 -  |3x+1|

Either 2x+5 = -  |3x+1| + 10 or 2x + 5 = -(-|3x-1| + 10)

Proceed:

2x+5 = -  |3x+1| + 10

⇒ -|3x-1| + 10 = 2x + 5

⇒ -|3x-1| + 10 - 10 = 2x + 5 - 10             [subtract 10 both sides]

⇒                -|3x-1| = 2x - 5

⇒                -|3x-1|/-1 = (2x - 5)/-1           [ divide both sides by -1]

We know either 3x - 1 = -2x + 5 or 3x - 1 = - (-2x+5)

       3x - 1 = -2x + 5

3x - 1 + 2x = -2x + 5 + 2x                        [ adding 2x both sides]

           5x = 6

              x = 6/5

Now,

      3x - 1 = -(-2x+5)

3x - 1 - 2x = 2x - 5 - 2x                           [ subtracting 2x both sides]

             x = -4

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The solution is: the numeric value of (A, B, C, D, E) from the following three formulas are: A = 4, B = 10, C = 2, D = 5, E = 6.

Here, we have,

given that,

Formula:

AC x 1E = 4CD (1)

ADE x 1EC = AB (2)

AB + E9 = 11A (3)

now, we have to find the numeric value of (A, B, C, D, E) from the following three formulas.

However, A, B, C, D, E represent numbers between 0 and 9.

given example: If F = 1, G = 2, then 3FG would be 312.

now, we have,

Let, A = 4 and E = 6

as we have,

A+C = E

B-D = D

A + E = B

so, we get, C = 6 - 4 = 2

B = 6 + 4 = 10

2D = 10 => D = 5

Hence, The solution is: the numeric value of (A, B, C, D, E) from the following three formulas are: A = 4, B = 10, C = 2, D = 5, E = 6.

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Answer:

number is 2

Step-by-step explanation:

let the number be n then 8 times the number is 8n, subtract 1 from this and

8n - 1 = 15 ( add 1 to both sides )

8n = 16 ( divide both sides by 8 )

n = 2

that is the number is 2

The area of the park is 12 square units.

To find the area of the park, we can use the Shoelace Formula (also known as Gauss's area formula). This formula allows us to calculate the area of any polygon given the coordinates of its vertices.

The Shoelace Formula states that if we have the coordinates of the vertices of a polygon in counterclockwise order (x₁, y₁), (x₂, y₂), ..., (xn, yn), then the area (A) of the polygon is given by:

A = 1/2 × |(x₁y₂ + x₂y₃ + ... + xn-1yn + xny₁) - (y₁x₂ + y₂x₃ + ... + yn-1xn + ynx₁)|

In our case, the coordinates of the park's vertices are (-9, -9), (-3, -9), and (-6, -5). Let's calculate the area using the Shoelace Formula:

A = 1/2 × |(-9 × -9 + -3 × -5 + -6 * -9) - (-9 × -3 + -9 × -6 + -5 × -9)|

Simplifying further:

A = 1/2 × |(81 + 15 + 54) - (27 + 54 + 45)|

A = 1/2 × |150 - 126|

A = 1/2 × |24|

A = 12

Therefore, the area of the park is 12 square units.

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Answer:

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

Plot the given vertices and connect.

See attached.

Then add the height CD.

The base is AB, with the length:

The height is CD, with length:

Find the area using equation:

Step-by-step explanation:

to find f(5)   in f(x) , just put in '5' where 'x' is in the equation

f(5) = 3 (5) + 2 = 17

Answer:

f(5) = 17

Step-by-step explanation:

Let's evaluate the function for f(5)

Insert 5 everywhere x appears:

Therefore f(5) = 17

The Separation Schema is a rule in set theory that allows us to construct a new set from an existing set based on a given property.

For example, if a set of all natural numbers, use the Separation Schema to construct a new set of all even natural numbers. The new set will contain all of the elements of the original set that satisfy the given property, in this case, the property of being even.

The property x ∉ x states that x is not a member of x. If we apply the Separation Schema to this property, we will construct a new set that contains all of the sets that are not members of themselves. This set is called the Russell set, and it is known to be paradoxical.

The Russell paradox shows that the unrestricted Comprehension Schema is inconsistent. This is because the Russell set is a set that can be constructed using the Comprehension Schema, but the Russell set also satisfies the property that it is not a member of itself. This is a contradiction, and it shows that the Comprehension Schema cannot be used to construct all sets.

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To determine the amount of binding Erin needs to cover the outside edges of the kite, we need to add up the lengths of all the sides of the kite.

Looking at the pattern, we can see that there are two sides with a length of 13 inches each, and two sides with a length of 15 inches each.

Therefore, the total length of the binding needed is:

13 inches + 13 inches + 15 inches + 15 inches = 56 inches.

Based on this calculation, Erin needs approximately 56 inches of binding to cover the outside edges of the kite.

Among the options given, the closest measurement to 56 inches is 52 inches.

Answer:

Point where the line crosses the y-axis is (0, -8)

Gradient: 3

Step-by-step explanation:

The slope-intercept form of the equation of a straight line is
y = mx + b

where m =slope or gradient

b is the y-intercept ie the y-value where the line crosses the y-axis. This is also the value of y when x = 0

Comparing the general equation with y = 3x - 8

we get the y-intercept as -8 and therefore the point (0, -8) is the point where the line crosses the y-axis

The gradient is 3 which is the coefficient of x

Answer:

Step-by-step explanation:

mag jakol

a) Population: All students who attended the college last year.

b) Population: All students who graduated from the college last year.

c) Data: 3.65, 2.80, 1.50, 3.90.

d) Sample: Group of students randomly selected from the population.

e) Variable: Cumulative GPA.

f) Sample: All students who graduated.

g) Sample: Students in the study who graduated from the college last year.

a) Population: All students who attended the college last year

Statistic: Not applicable (no specific data)

Parameter: Not applicable (no specific data)

Variable: Cumulative GPA

Data: Not applicable (no specific data)

Sample: Not applicable (no specific data)

b) Population: All students who graduated from the college last year

Statistic: Cumulative GPA of one student

Parameter: Not applicable (one student's GPA)

Variable: Cumulative GPA

Data: GPA of the specific student

Sample: Not applicable (one student)

c) Population: Not applicable (no specific population mentioned)

Statistic: Not applicable (no sample data)

Parameter: Not applicable (no population parameter)

Variable: Cumulative GPA

Data: 3.65, 2.80, 1.50, 3.90

Sample: Not applicable (no specific sample mentioned).

d) Population: All students who graduated from the college last year

Statistic: Not applicable (no specific data)

Parameter: Not applicable (no specific data)

Variable: Cumulative GPA

Data: Not applicable (no specific data)

Sample: Group of students randomly selected from the population

e) Population: All students who graduated from the college last year

Statistic: Average cumulative GPA of students

Parameter: Not applicable (no specific data)

Variable: Cumulative GPA

Data: Cumulative GPAs of students who graduated

Sample: Not applicable (no specific sample mentioned)

f) Population: All students who graduated from the college last year

Statistic: Not applicable (no specific data)

Parameter: Not applicable (no specific data)

Variable: Cumulative GPA

Data: Not applicable (no specific data)

Sample: All students who graduated

g) Population: Students in the study who graduated from the college last year

Statistic: Average cumulative GPA of students in the study

Parameter: Not applicable (no specific data)

Variable: Cumulative GPA

Data: Cumulative GPAs of students in the study

Sample: Students in the study who graduated

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The diameter of the circle is 24 units.

To find the diameter of the circle, we can use the formula that relates the length of an arc to the circumference of a circle.

The formula is:

Length of arc = (Measure of arc/360) × Circumference

Given that the length of arc AB is 14π and the measure of arc AB is 210 degrees, we can plug these values into the formula:

14π = (210/360) × Circumference

To solve for the circumference, we multiply both sides of the equation by (360/210):

14π × (360/210) = Circumference

Simplifying the right side of the equation:

Circumference = 24π

The circumference of a circle is equal to π times the diameter. Therefore,

we can write:

24π = π × diameter

To solve for the diameter, we divide both sides of the equation by π:

diameter = 24

Hence, the diameter of the circle is 24 units.

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The correct answer is: favorable outcomes divided by possible outcomes. The theoretical probability is a mathematical concept that represents the likelihood of an event occurring based on the ratio of favorable outcomes to the total number of possible outcomes.

The correct answer is: 1/5. Since there are five equally-sized pie shapes on the spinner, and only one of them is blue, the probability of landing on blue when you spin is 1 out of 5 or 1/5.

To predict how many out of 10,000 coffee drinkers will like the new coffee, we can use the concept of proportion. Since we know that 450 out of 1,000 coffee drinkers like the new coffee, we can set up the proportion:

450/1,000 = x/10,000

Cross-multiplying, we get:

1,000x = 450 * 10,000

Simplifying further:

1,000x = 4,500,000

Dividing both sides by 1,000:

x = 4,500

Therefore, out of 10,000 coffee drinkers, we can predict that approximately 4,500 will like the new coffee.

Hello !

2 1/4 ÷ 5/8

= 9/4 ÷ 5/8

= 9/4 x 8/5

= 9x8/4x5

= 72/20

= 18/5

= 3 3/5

In the given diagram, we have two parallel lines intersected by a transversal. To determine the relationship between the angles and solve for x, we can use the properties of angles formed by parallel lines and a transversal.

From the diagram, we can observe the following angle relationships:

Angle A is corresponding to the angle (4x - 3)°.

Therefore, we can write: A = (4x - 3)°.

Angle B is alternate interior to the angle (7x + 9)°.

Therefore, we can write: B = (7x + 9)°.

Angle C is alternate interior to the angle 2x°.

Therefore, we can write: C = 2x°.

Since the sum of angles in a straight line is 180°, we can set up the equation:

A + B + C = 180°

Substituting the known values, we get:

(4x - 3)° + (7x + 9)° + 2x° = 180°

Simplifying the equation, we can solve for x:

4x - 3 + 7x + 9 + 2x = 180

13x + 6 = 180

13x = 174

x = 13.38

Therefore, the value of x is approximately 13.38.

Please note that this solution assumes the given diagram accurately represents the angle relationships.

The gradients of the lines AB, XY and PQ are 3S, -3 and 77/3 respectively.

To find the gradient of a line, we need to use the formula: gradient = (change in y)/(change in x).

For the first question, we have two points A (C3,1) and B (C6,10S). The change in y is 10S - 1 = 9S, and the change in x is C6 - C3 = 3. Therefore, the gradient of the line AB is 9S/3 = 3S.

For the second question, we have two points X (5,-1) and Y (3,5). The change in y is 5 - (-1) = 6, and the change in x is 3 - 5 = -2. Therefore, the gradient of the line XY is 6/-2 = -3.

For the third question, we have two points P (C3,-2) and Q (6,75). The change in y is 75 - (-2) = 77, and the change in x is 6 - C3 = 3. Therefore, the gradient of the line PQ is 77/3.

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(a). Rounded to three significant figures, the length of the side of the cube is 30.0 cm.

(b). Rounded to three significant figures, the height of the trapezoidal prism is 2.75 cm.

Let's solve this problem step by step:

First, let's calculate the volume of the fish tank when it is filled with water to two-thirds of its height.

The base dimensions of the fish tank are 72 cm by 48 cm, and the height is 52 cm.

Filling it with water to two-thirds of the height means the water level is at 2/3 × 52 = 34.67 cm.

The volume of a rectangular prism (the fish tank) is given by V = length × width × height.

Plugging in the values, we have V = 72 cm × 48 cm × 34.67 cm

≈ 112,430.08 cm³.

Now, let's calculate the volume of the cube.

The cube is placed in the fish tank the water level rises to three-quarters of the height is 3/4 × 52 = 39 cm.

The volume of the water that spilled can be calculated by subtracting the volume of the fish tank before the cube was placed from the volume of the fish tank after the cube was placed.

The volume of the cube is 112,430.08 cm³ - 39 cm × 72 cm × 48 cm = 28,861.92 cm³.

Since the volume of a cube is given by V = side³, we can solve for the length of the side of the cube.

Thus, side³ = 28,861.92 cm³ and by taking the cubic root of both sides, we find side ≈ 30.042 cm.

Now, let's calculate the height of the trapezoidal prism.

We know that the volume of water spilled when the trapezoidal prism is placed is 500 cm³.

Since the volume of a trapezoidal prism is given by V = (1/2) * (a + b) * h * base a and b are the lengths of the parallel sides of the trapezoid, h is the height of the trapezoid, and base is the distance between the parallel sides, we can rearrange the formula to solve for the height:

h = (2 × V) / ((a + b) × base).

The cross-sectional area of the trapezoidal prism is 928 cm², we can find the base.

Since the area of a trapezoid is given by A = (1/2) × (a + b) × h, we can rearrange the formula to solve for the base:

base = (2 × A) / (a + b).

Plugging in the values, we have base = (2 × 928 cm²) / (72 cm + 48 cm)

≈ 9.28 cm.

Now we can calculate the height of the trapezoidal prism:

h = (2 × 500 cm³) / ((72 cm + 48 cm) × 9.28 cm)

≈ 2.754 cm.

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Answer:

0

Step-by-step explanation:

If x ≥ 4, then the equation simplifies to x+4 = x-4, which gives 8=0, which is not true. Therefore, there are no solutions in this case.

If x < -4, then the equation becomes -(x+4) = -(x-4), which simplifies to -x-4 = -x+4. Simplifying further gives -8 = 0, which is not true. Therefore, there are no solutions in this case either.

If -4 ≤ x < 4, then the equation becomes x+4 = -(x-4), which simplifies to 2x = 0. Therefore, x = 0 is a solution.

If we plug x = 0 back into the original equation, we can see that it works:

|0+4| = |0-4|,

which simplifies to 4 = 4.

Therefore, the solution to the equation |x+4|=|x-4| is x = 0.

Answer:

x = 0

Step-by-step explanation:

Because |-4| = |4| = 4, then the only way x+4 can be -4 is if x=0, and the only way x-4 can be -4 is if x=0 as well. x can't be 4 for instance because then we have |8|≠|0|.

Answer:

(i) x ≤ 1

(ii) ℝ except 0, -1

(iii) x > -1

(iv) ℝ except π/2 + nπ, n ∈ ℤ

Step-by-step explanation:

(i) The number inside a square root must be positive or zero to give the expression a real value. Therefore, to solve for the domain of the function, we can set the value inside the square root greater or equal to 0, then solve for x:

[tex]1-x \ge 0[/tex]

[tex]1 \ge x[/tex]

[tex]\boxed{x \le 1}[/tex]

(ii) The denominator of a fraction cannot be zero, or else the fraction is undefined. Therefore, we can solve for the values of x that are NOT in the domain of the function by setting the expression in the denominator to 0, then solving for x.

[tex]0 = x^2+x[/tex]

[tex]0 = x(x + 1)[/tex]

[tex]x = 0[/tex]     OR     [tex]x = -1[/tex]

So, the domain of the function is:

[tex]R \text{ except } 0, -1[/tex]

(ℝ stands for "all real numbers")

(iii) We know that the value inside a logarithmic function must be positive, or else the expression is undefined. So, we can set the value inside the log greater than 0 and solve for x:

[tex]x+ 1 > 0[/tex]

[tex]\boxed{x > -1}[/tex]

(iv) The domain of the trigonometric function tangent is all real numbers, except multiples of π/2, when the denominator of the value it outputs is zero.

[tex]\boxed{R \text{ except } \frac{\pi}2 + n\pi} \ \text{where} \ \text{n} \in Z[/tex]

(ℤ stands for "all integers")

Answer:

(i)  x ≤ 1

(ii)  All real numbers except x = 0 and x = -1.

(iii)  x > -1

(iv)  All real numbers except x = π/2 + πn, where n is an integer.

Step-by-step explanation:

The domain of a function is the set of all possible input values (x-values).

[tex]\hrulefill[/tex]

[tex]\textsf{(i)} \quad f(x)=\sqrt{1-x}[/tex]

For a square root function, the expression inside the square root must be non-negative. Therefore, for function f(x), 1 - x ≥ 0.

Solve the inequality:

[tex]\begin{aligned}1 - x &\geq 0\\\\1 - x -1 &\geq 0-1\\\\-x &\geq -1\\\\\dfrac{-x}{-1} &\geq \dfrac{-1}{-1}\\\\x &\leq 1\end{aligned}[/tex]

(Note that when we divide or multiply both sides of an inequality by a negative number, we must reverse the inequality sign).

Hence, the domain of f(x) is all real numbers less than or equal to -1.

[tex]\boxed{\begin{aligned} \textsf{Inequality notation:} \quad &x \leq 1\\\textsf{Interval notation:} \quad &(-\infty, 1]\\\textsf{Set-builder notation:} \quad &\left\{x \in \mathbb{R}\left|\: x \leq 1 \right\} \end{aligned}}[/tex]

[tex]\hrulefill[/tex]

[tex]\textsf{(ii)} \quad g(x) = \dfrac{1}{x^2 + x}[/tex]

To find the domain of g(x), we need to identify any values of x that would make the denominator equal to zero, since division by zero is undefined.

Set the denominator to zero and solve for x:

[tex]\begin{aligned}x^2 + x &= 0\\x(x + 1) &= 0\\\\\implies x &= 0\\\implies x &= -1\end{aligned}[/tex]

Therefore, the domain of g(x) is all real numbers except x = 0 and x = -1.

[tex]\boxed{\begin{aligned} \textsf{Inequality notation:} \quad &x < -1 \;\;\textsf{or}\;\; -1 < x < 0 \;\;\textsf{or}\;\; x > 0\\\textsf{Interval notation:} \quad &(-\infty, -1) \cup (-1, 0) \cup (0, \infty)\\\textsf{Set-builder notation:} \quad &\left\{x \in \mathbb{R}\left|\: x \neq 0,x \neq -1 \right\} \end{aligned}}[/tex]

[tex]\hrulefill[/tex]

[tex]\textsf{(iii)}\quad h(x) = \log_7(x + 1)[/tex]

For a logarithmic function, the argument (the expression inside the logarithm), must be greater than zero.

Therefore, for function h(x), x + 1 > 0.

Solve the inequality:

[tex]\begin{aligned}x + 1 & > 0\\x+1-1& > 0-1\\x & > -1\end{aligned}[/tex]

Therefore, the domain of h(x) is all real numbers greater than -1.

[tex]\boxed{\begin{aligned} \textsf{Inequality notation:} \quad &x > -1\\\textsf{Interval notation:} \quad &(-1, \infty)\\\textsf{Set-builder notation:} \quad &\left\{x \in \mathbb{R}\left|\: x > -1\right\} \end{aligned}}[/tex]

[tex]\hrulefill[/tex]

[tex]\textsf{(iv)} \quad k(x) = \tan x[/tex]

The tangent function can also be expressed as the ratio of the sine and cosine functions:

[tex]\tan x = \dfrac{\sin x}{\cos x}[/tex]

Therefore, the tangent function is defined for all real numbers except the values where the cosine of the function is zero, since division by zero is undefined.

From inspection of the unit circle, cos(x) = 0 when x = π/2 and x = 3π/2.

The tangent function is periodic with a period of π. This means that the graph of the tangent function repeats itself at intervals of π units along the x-axis.

Therefore, if we combine the period and the undefined points, the domain of k(x) is all real numbers except x = π/2 + πn, where n is an integer.

[tex]\boxed{\begin{aligned} \textsf{Inequality notation:} \quad &\pi n\le \:x < \dfrac{\pi }{2}+\pi n\quad \textsf{or}\quad \dfrac{\pi }{2}+\pi n < x < \pi +\pi n\\\textsf{Interval notation:} \quad &\left[\pi n ,\dfrac{\pi }{2}+\pi n\right) \cup \left(\dfrac{\pi }{2}+\pi n,\pi +\pi n\right)\\\textsf{Set-builder notation:} \quad &\left\{x \in \mathbb{R}\left|\: x \neq \dfrac{\pi}{2}+\pi n\;\; (n \in\mathbb{Z}) \right\}\\\textsf{(where $n$ is an integer)}\end{aligned}}[/tex]