8. Given AABC~AEDC
What is the value of x?
C. 30
D. 20
A. 15
B. 12
E
60
X
C
D
10
40
B

Given the information in the diagram, the theorem that best justifies why lines j and k must be parallel include the following: D. converse alternate exterior angles theorem.

In Mathematics and Geometry, parallel lines are two (2) lines that are always the same (equal) distance apart and never meet or intersect.

In Mathematics and Geometry, the alternate exterior angle theorem states that when two (2) parallel lines are cut through by a transversal, the alternate exterior angles that are formed lie outside the two (2) parallel lines, are located on opposite sides of the transversal, and are congruent angles.

Since the alternate exterior angles are congruent, we can logically deduce the following based on the converse alternate exterior angles theorem;

93° ≅ 93° (lines j and k are parallel lines).

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Complete Question:

Given the information in the diagram, which theorem best justifies why lines j and k must be parallel?

alternate interior angles theorem

alternate exterior angles theorem

converse alternate interior angles theorem

converse alternate exterior angles theorem

The sides of the rectangle that have the same length as the circumference of the circles are the sides parallel to the bases of the cylinder. The sides of the rectangle that have the same length as the height of the cylinder are the sides perpendicular to the bases. While the model provides a simplified representation, practical calculations might have slight differences due to real-world factors.

In a cylinder, the two circular bases are connected by a curved surface, forming a three-dimensional shape. The rectangular shape that wraps around the curved surface of the cylinder is called the lateral surface or the lateral area.

To determine which sides of the rectangle have the same length as the circumference of the circles, we need to understand the geometry of a cylinder. The circumference of a circle is calculated using the formula:

Circumference = 2πr,

where r is the radius of the circle. In a cylinder, the bases are identical circles, so the circumference of each base is equal. Therefore, the sides of the rectangle that are parallel to the bases have the same length as the circumference of the circles.

Now, let's consider the height of the cylinder. The height is the distance between the two bases and is perpendicular to the bases. In the rectangular representation of the cylinder, the sides that are perpendicular to the bases represent the height. Hence, the sides of the rectangle that are perpendicular to the bases have the same length as the height of the cylinder.

When comparing the model (rectangular representation) with practical calculations, we may notice some differences. The model provides a simplified representation of the cylinder, assuming that the lateral surface is perfectly wrapped around the curved surface. However, in practical calculations, there might be slight variations due to factors like material thickness, manufacturing processes, or measuring precision. These variations can result in minor deviations between the model and the practical calculations.

It's important to consider that the model is an approximation and serves as a visual aid to understand the basic properties of the cylinder. In real-life applications or engineering calculations, precise measurements and considerations of tolerances are crucial.

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Answer:

Step-by-step explanation:

The area of the obtuse triangle is 34 square feet with a base of 10 ft and height of 6.8 ft.

To find the area of the obtuse triangle, we can use the formula A = (1/2) * base * height. Let's denote the unknown part of the base as x.

In the given triangle, we have the width of the obtuse angle triangle as 10 ft, the height (perpendicular) as 6.8 ft, and the unknown part of the base as x.

Using the formula, we can calculate the area as:

A = (1/2) * (10 + x) * 6.8

Simplifying this expression, we get:

A = 3.4(10 + x)

Now, we need to determine the value of x. From the given information, we know that the width of the obtuse angle triangle is 10 ft. This means the sum of the two parts of the base is 10 ft. Therefore, we can write the equation:

x + 10 = 10

Solving for x, we find:

x = 0

Since x = 0, it means that one part of the base has a length of 0 ft. Therefore, the entire base is formed by the width of the obtuse angle triangle, which is 10 ft.

Now, substituting this value of x back into the area formula, we have:

A = 3.4(10 + 0)

A = 3.4 * 10

A = 34 square feet

Hence, the area of the obtuse triangle is 34 square feet.

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1a.) The amount that the eldest son received would be =GHç 1,360

b .) The amount received by the daughter would be =G Hç 2176

c.) The difference between the amount the two sons received would be =GHç1,904

For 1a.)

The amount that the land is worth= $8,600

The amount received for various purposes= $1,800

The remaining amount shared to the sons= 8,600-1800= $6,800

The percentage amount received by the eldest son= 20% of 6800

That is;

= 20/100×6800/1

= 136000/100

= $1,360

The remaining amount= 6800-1360= $5,440

For 1b.)

The ratio that the remaining amount was shared between the other son and the daughter = 3 : 2 respectively.

The total ratio= 3+2=5

For daughter= 2/5× 5440

= 10880/5 = 2176

The other son= 5440-2176 = 3264

For 1c.)

The difference between the amount the two sons received would be =3264-1,360 = GHç1,904.

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1. Using a calculator, we find that cos 10 ≈ 0.98.

2. Using a calculator, we find that sin 30 ≈ 0.50.

3. Using a calculator, we find that sin 20 ≈ 0.34.

4. Using a calculator, we find that tan 25 ≈ 0.47.

5. Using a calculator, we find that tan 48.5 ≈ 1.14.

Using a calculator to evaluate the given trigonometric functions, rounded to the nearest hundredth, we have:

Cos 10:

Using a calculator, we find that cos 10 ≈ 0.98.

Sin 30:

Using a calculator, we find that sin 30 ≈ 0.50.

Sin 20:

Using a calculator, we find that sin 20 ≈ 0.34.

Tan 25:

Using a calculator, we find that tan 25 ≈ 0.47.

Tan 48.5:

Using a calculator, we find that tan 48.5 ≈ 1.14.

These values represent the approximate decimal values of the trigonometric functions at the given angles, rounded to the nearest hundredth.

Just a reminder, when using a calculator, make sure it is set to the correct angle mode (degrees or radians) as per the given problem.

It's important to note that these values are approximate since they are rounded to the nearest hundredth. If you need more precise values, you can use a calculator that allows for a greater number of decimal places or use trigonometric tables.

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Answer:

1. x = 18

2. x = 31

Step-by-step explanation:

1. Angles in a quadrilateral sum to 360°.

360 -(60 +130 +70) = 360 -260 = 100

So, x +82 = 100. x = 100 -82 = 18

2. Angles in a triangle sum to 180°. 180 -(29 +75) = 180 -104 = 76.

So, -17 +3x = 76. 3x = 93. x = 93/3 = 31

Answer:

y = π/3

Step-by-step explanation:

To find the value of y, we can use the trigonometric identity for the sum of angles:

sin(x + y) = sin x * cos y + cos x * sin y

Comparing this with the given equation:

sin(x + y) = 1/2(sin x) + √3/2(cos x)

We can equate the corresponding terms:

sin x * cos y = 1/2(sin x) ----(1)

cos x * sin y = √3/2(cos x) ----(2)

From equation (1), we can see that cos y = 1/2.

From equation (2), we can see that sin y = √3/2.

To determine the values of y, we can use the trigonometric values of cosine and sine in the first quadrant of the unit circle.

In the first quadrant, cos y is positive, so cos y = 1/2 corresponds to y = π/3 (60 degrees).

Similarly, sin y is positive, so sin y = √3/2 corresponds to y = π/3 (60 degrees).

Therefore, the value of y is y = π/3 (or 60 degrees).

The slope of a line that is perpendicular to the given line x + 6y = 3 is 6.

To determine the slope of a line that is perpendicular to the given line, we need to find the negative reciprocal of the slope of the given line.

The equation of the given line is x + 6y = 3.

To find the slope of the given line, we can rearrange the equation into slope-intercept form (y = mx + b), where m represents the slope:

x + 6y = 3

6y = -x + 3

y = (-1/6)x + 1/2

From the equation y = (-1/6)x + 1/2, we can see that the slope of the given line is -1/6.

To find the slope of a line that is perpendicular, we take the negative reciprocal of -1/6.

The negative reciprocal of -1/6 can be found by flipping the fraction and changing its sign:

Negative reciprocal of -1/6 = -1 / (-1/6) = -1 * (-6/1) = 6

Therefore, the slope of a line that is perpendicular to the given line x + 6y = 3 is 6.

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Answer:

Third option

Step-by-step explanation:

-3(2x - 5) < 5(2 - x)

-6x + 15 < 10 - 5x <-- Third option

15 < 10 + x

5 < x

x > 5

There only appears to be one option. The solution to the inequality is x>5, not x<5.

Answer:

9

Step-by-step explanation:

a) The distribution of X, the student loan debt of a randomly selected college graduate, is normal with a mean of $25,200 and a standard deviation of $11,200. b) The probability is approximately 7.28%.

c) The middle lies between approximately $22,164 and $28,536.

a. The distribution of X, the student loan debt of a randomly selected college graduate, is a normal distribution (bell-shaped curve) with a mean (μ) of $25,200 and a standard deviation (σ) of $11,200. We can represent this as X ~ N(25200, 11200).

b. To find the probability that the college graduate has between $27,250 and $43,650 in student loan debt, we need to calculate the z-scores for these two values and then find the area under the normal curve between those z-scores.

First, we calculate the z-score for $27,250:

z1 = (X1 - μ) / σ = (27250 - 25200) / 11200 ≈ 1.8304

Next, we calculate the z-score for $43,650:

z2 = (X2 - μ) / σ = (43650 - 25200) / 11200 ≈ 1.6518

Now, we need to find the area under the normal curve between these two z-scores. We can use a standard normal distribution table or a calculator to find this area.

Using a standard normal distribution table or a calculator, the probability is approximately P(1.6518 ≤ Z ≤ 1.8304) ≈ 0.0728.

c. To find the middle 20% of college graduates' loan debt, we need to find the range of values that contain the central 20% of the distribution. This range corresponds to the values between the lower and upper percentiles.

The lower percentile is the 40th percentile (50% - 20%/2 = 40%) and the upper percentile is the 60th percentile (50% + 20%/2 = 60%).

Using a standard normal distribution table or a calculator, we can find the z-scores corresponding to these percentiles:

For the lower percentile (40th percentile):

z_lower = invNorm(0.40) ≈ -0.2533

For the upper percentile (60th percentile):

z_upper = invNorm(0.60) ≈ 0.2533

Now, we can convert these z-scores back to the corresponding loan debt values:

Lower debt value:

X_lower = μ + z_lower * σ = 25200 + (-0.2533) * 11200 ≈ $22,164

Upper debt value:

X_upper = μ + z_upper * σ = 25200 + 0.2533 * 11200 ≈ $28,536

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The solutions to the quadratic equation 2x² + 6x - 10 = x² + 6 are -8 and 2.

Given the quadratic equation in the question:

2x² + 6x - 10 = x² + 6

First, reorder the quadratic equation in standard form:

2x² + 6x - 10 = x² + 6

2x² - x² + 6x - 10 - 6 = x² - x² + 6 - 6

2x² - x² + 6x - 10 - 6 = 0

x² + 6x - 10 - 6 = 0

x² + 6x - 16 = 0

Next, factor the equation using the AC method:

( x - 2 )( x + 8 ) = 0

Equate each factor to 0 and solve for x:

( x - 2 ) = 0

x - 2 = 0

x = 2

( x + 8 ) = 0

x + 8 = 0

x = -8

Therefore, the solutions are -8 and 2.

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Answer:

Step-by-step explanation:

x² + 3x - 2

(5)² + 3 × 5- 2

25 + 15 - 2

40 - 2

38...

118 be the median of a positively skewed distribution with a mean of 122. Option D.

To determine which of the given values could be the median of a positively skewed distribution with a mean of 122, we need to consider the relationship between the mean, median, and skewness of a distribution.

In a positively skewed distribution, the tail of the distribution is stretched towards higher values, meaning that there are more extreme values on the right side. Consequently, the median, which represents the value that divides the distribution into two equal halves, will typically be less than the mean in a positively skewed distribution.

Let's examine the given values in relation to the mean:

A. 122: This value could be the median if the distribution is perfectly symmetrical, but since the distribution is positively skewed, the median is expected to be less than the mean. Thus, 122 is less likely to be the median.

B. 126: This value is higher than the mean, and since the distribution is positively skewed, it is unlikely to be the median. The median is expected to be lower than the mean.

C. 130: Similar to option B, this value is higher than the mean and is unlikely to be the median. The median is expected to be lower than the mean.

D. 118: This value is lower than the mean, which is consistent with a positively skewed distribution. In such a distribution, the median is expected to be less than the mean, so 118 is a plausible value for the median.

In summary, among the given options, (118) is the most likely value to be the median of a positively skewed distribution with a mean of 122. So Option D is correct.

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Note the complete question is

If the mean of a positively skewed distribution is 122, which of these values could be the median of the distribution?

A. 122

B. 126

C. 130

D. 118

Answer:

B.     [tex] 9^{\frac{1}{8}x} [/tex]

Step-by-step explanation:

[tex] \sqrt[4]{9}^{\frac{1}{2}x} = [/tex]

[tex] = ({9}^{\frac{1}{4}})^{\frac{1}{2}x} [/tex]

[tex] = 9^{\frac{1}{4} \times \frac{1}{2}x} [/tex]

[tex] = 9^{\frac{1}{8}x} [/tex]

The resulting matrix after the rows are interchanged is given as follows:

[tex]\left[\begin{array}{cccc}2&9&4&5\\8&-2&1&7\\1&4&-4&9\end{array}\right][/tex]

The matrix for this problem is defined as follows:

[tex]\left[\begin{array}{cccc}8&-2&1&7\\2&9&4&5\\1&4&-4&9\end{array}\right][/tex]

The row 1 is given as follows:

[8 -2 1 7].

The row 2 is given as follows:

[2 9 4 5].

Interchanging the rows means that the elements of the row 1 in the matrix is exchanged with the elements of row 2, hence the resulting matrix is given as follows:

[tex]\left[\begin{array}{cccc}2&9&4&5\\8&-2&1&7\\1&4&-4&9\end{array}\right][/tex]

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a) The summation (Ex) of the given data set, we add up all the values

Ex = 77

b) n = 5

c) x = 15.4

a) To compute the summation (Ex) of the given data set, we add up all the values:

19 + 10 + 15 + 17 + 16 = 77

b) The sample size (n) is the total number of data points in the set. In this case, there are 5 data points, so:

n = 5

c) To calculate the sample mean (x), we divide the summation (Ex) by the sample size (n):

x = Ex / n

x = 77 / 5

x = 15.4

Therefore, the answers are:

a) Ex = 77

b) n = 5

c) x = 15.4

The summation is 77, the sample size is 5, and the sample mean is 15.4.

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Answer:

[tex]5a2b2/4[/tex]

Step-by-step explanation:

Step-by-step explanation:

To find the difference in total revenue projected by the two models over the six-year period ending at t = 5, we need to calculate the revenue for each model from t = 0 to t = 5 and subtract the results.

For R1:

R1 = 7.23 + 0.25t + 0.03t^2

Substituting t = 5:

R1(5) = 7.23 + 0.25(5) + 0.03(5^2)

R1(5) = 7.23 + 1.25 + 0.75

R1(5) = 9.23 + 0.75

R1(5) = 9.98 million dollars

For R2:

R2 = 7.23 + 0.1t + 0.01t^2

Substituting t = 5:

R2(5) = 7.23 + 0.1(5) + 0.01(5^2)

R2(5) = 7.23 + 0.5 + 0.25

R2(5) = 7.73 + 0.25

R2(5) = 7.98 million dollars

To find the difference, we subtract R2(5) from R1(5):

Difference = R1(5) - R2(5)

Difference = 9.98 - 7.98

Difference = 2 million dollars

Therefore, the model R1 projects 2 million dollars more in total revenue than R2 over the six-year period ending at t = 5.

The probability that the dial stops on yellow is 2/5 or 0.4 (or 40%).

Therefore, there is a 40% chance that the spinner will stop on yellow.

To find the probability that the spinner stops on yellow, we need to determine the number of favorable outcomes (yellow) and the total number of possible outcomes.

The spinner has 10 equally sized slices, with 4 yellow, 4 red, and 2 blue.

The number of favorable outcomes (yellow) is 4 because there are 4 yellow slices.

The total number of possible outcomes is 10 because there are 10 slices in total.

Therefore, the probability of the spinner stopping on yellow can be calculated as:

Probability = Number of favorable outcomes / Total number of possible outcomes

Probability = 4 / 10

Simplifying this fraction, we get:

Probability = 2 / 5

So, the probability that the dial stops on yellow is 2/5 or 0.4 (or 40%).

Therefore, there is a 40% chance that the spinner will stop on yellow.

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Step-by-step explanation:

In a randomized block design blocked by gender, treatments should be assigned randomly within each gender block. The correct assignment maintains a distribution of one treatment for each gender. Looking at the given options, only one meets this criterion:

OA: (1f, 2f), B: (1m, 2m). C: (3f, 3m). D: (4f, 4m)

Each treatment group A, B, C, and D contains one male and one female, making the distribution of treatments blocked by gender.

Answer:

p =1 0

Step-by-step explanation:

x³(x - y)⁹ - y³(x - y)⁹ = (x - y)⁹[x³ - y³]

= (x - y)⁹(x - y)(x² + y² + xy)

= (x - y)¹⁰(x² + y² + xy)

where p = 10 and q = 1

Answer:

p = 10

Step-by-step explanation:

Given expression:

[tex]x^3(x-y)^9 - y^3(x-y)^9[/tex]

Factor out the common term (x - y)⁹:

[tex](x-y)^9(x^3- y^3)[/tex]

[tex]\boxed{\begin{minipage}{5cm}\underline{Difference of two cubes}\\\\$x^3-y^3=(x-y)(x^2+y^2+xy)$\\\end{minipage}}[/tex]

Rewrite the second parentheses as the difference of two cubes.

[tex](x-y)^9(x-y)(x^2+y^2+xy)[/tex]

[tex]\textsf{Apply\:the\:exponent\:rule:} \quad a^b\cdot \:a^c=a^{b+c}[/tex]

[tex](x-y)^{9+1}(x^2+y^2+xy)[/tex]

[tex](x-y)^{10}(x^2+y^2+xy)[/tex]

Comparing the rewritten original expression with the given expression:

[tex](x-y)^{10}(x^2+y^2+xy)=(x-y)^p(x^2+y^2+qxy)[/tex]

We can see that [tex](x-y)^p[/tex] corresponds to [tex](x-y)^{10}[/tex] in the given expression.

Therefore, we can conclude that p = 10.

The vertex of h(x) is (-3, 3), and the range is y ≥ 3.

To find the vertex of the quadratic function h(x), we can use the formula x = -b/2a, where the quadratic function is in the form [tex]ax^2 + bx + c[/tex].

From the given table, we can observe that the x-values of the vertex correspond to the minimum points of the function.

The minimum point occurs between -4 and -3, which suggests that the x-coordinate of the vertex is -3. Therefore, x = -3.

To find the corresponding y-coordinate of the vertex, we look at the corresponding h(x) value in the table, which is 3. Hence, the vertex of the function h(x) is (-3, 3).

To determine the range of h(x), we need to consider the y-values attained by the function.

From the table, we see that the lowest y-value is 3 (the y-coordinate of the vertex), and there are no other y-values lower than 3. Therefore, the range of h(x) is all real numbers greater than or equal to 3.

The vertex of h(x) is (-3, 3), and the range is y ≥ 3.

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The vertex of the quadratic function is (-4, 12).

The range of h(x) is [3, ∞).

To find the vertex and range of the quadratic function h(x) based on the given table, we can use the properties of quadratic functions.

The vertex of a quadratic function in the form of f(x) = ax² + bx + c can be determined using the formula:

x = -b / (2a)

The domain of h(x) is all real numbers, we can assume that the quadratic function is of the form h(x) = ax² + bx + c.

Looking at the table, we can see that the x-values are increasing from left to right.

Additionally, the y-values (h(x)) are increasing from -6 to -4, then decreasing from -4 to -1.

This indicates that the vertex of the quadratic function lies between x = -4 and x = -3.

To find the exact x-coordinate of the vertex, we can use the formula mentioned earlier:

x = -b / (2a)

Based on the table, we can choose two points (-4, 4) and (-3, 3).

The difference in x-coordinates is 1, so we can assume that a = 1.

Plugging in the values of (-4, 4) and a = 1 into the formula, we can solve for b:

-4 = -b / (2 × 1)

-4 = -b / 2

-8 = -b

b = 8

The equation of the quadratic function h(x) can be written as h(x) = x² + 8x + c.

Now, let's find the y-coordinate of the vertex.

We can substitute the x-coordinate of the vertex, which we found as -4, into the equation:

h(-4) = (-4)² + 8(-4) + c

12 = 16 - 32 + c

12 = -16 + c

c = 28

The equation of the quadratic function h(x) is h(x) = x² + 8x + 28.

The range of the quadratic function can be determined by observing the y-values in the table.

From the table, we can see that the minimum y-value is 3.

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The percent increase in recreational boating participation over the sixteen-year period is approximately 8.72%. This means that the number of participants increased by around 8.72% from 52.7 million to 57.3 million.

To determine the percent increase in recreational boating participation over the sixteen-year period, we can use the following formula:

Percent Increase = ((New Value - Old Value) / Old Value) * 100

Using the given information, we have an old value of 52.7 million and a new value of 57.3 million.

Percent Increase = ((57.3 million - 52.7 million) / 52.7 million) * 100

= (4.6 million / 52.7 million) * 100

= 0.0872 * 100

= 8.72%

This increase indicates a positive trend in recreational boating, reflecting a growing interest in this activity over time. Factors such as improved accessibility, marketing efforts, and increasing disposable income may have contributed to this upward trend.

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Answer:

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

Angle G is central angle and angle E is inscribed angle, both with same endpoints.

According to the inscribed angle theorem the inscribed angle is half of the central angle.

Hence the central angle G measures:

Answer: 10 and -4

Step-by-step explanation: 10 + - 4 = 6 and 10 x -4 = -40

El supplemento y el complemento de cada ángulo son, respectivamente:

Caso A: m ∠ A' = 43°

Caso B: m ∠ A' = 31°

De acuerdo con la geometría, la suma de un ángulo y su complemento es igual a 90° and la suma de un ángulo y su suplemento es igual a 180°. Matemáticamente hablando, cada situación es descrita por las siguientes formulas:

Ángulo y su complemento

m ∠ A + m ∠ A' = 90°

Ángulo y su suplemento

m ∠ A + m ∠ A' = 90°

Donde:

Ahora procedemos a determinar cada ángulo faltante:

Caso A: Complemento

47° + m ∠ A' = 90°

m ∠ A' = 43°

Caso B: Suplemento

149° + m ∠ A' = 180°

m ∠ A' = 31°

El enunciado se encuentra escrito en español y la respuesta está escrita en el mismo idioma.

The statement is written in Spanish and its answer is written in the same language.

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The steps used in the solution of the equation -6 = -2/3(x + 12) + 1/3x are based on the principles of the Distributive Property, combining like terms, addition property of equality, symmetric property, subtraction property of equality, and multiplication property of equality.

Let's analyze the steps of the solution for the given equation -6 = -2/3(x + 12) + 1/3x:

Step 1: Distributive Property

The equation begins with the Distributive Property, which states that you can distribute a factor to each term inside parentheses. In this case, we distribute -2/3 to (x + 12), resulting in -2/3 * x and -2/3 * 12.

Step 2: Simplification

We simplify the expression -2/3 * 12 to -8, as multiplying -2/3 by 12 gives us -24, and simplifying the fraction -24/3 yields -8.

Step 3: Combine Like Terms

We combine the like terms -2/3x and -8. The equation becomes -2/3x - 8 + 1/3x.

Step 4: Combine Like Terms

We combine the like terms -2/3x and 1/3x by adding their coefficients. The sum of -2/3x and 1/3x is -1/3x.

Step 5: Addition Property of Equality

We add -1/3x to both sides of the equation to isolate the constant term. The equation becomes -6 - 1/3x = -1/3x.

Step 6: Symmetric Property

Since the equation has a form of -1/3x = -6 - 1/3x, we can rearrange the terms using the Symmetric Property.

Step 7: Addition Property of Equality

We add 1/3x to both sides of the equation to isolate the constant term. The equation becomes -6 = 0.

Step 8: Subtraction Property of Equality

We subtract 0 from both sides of the equation to simplify it further. The equation remains -6 = 0.

Step 9: Multiplication Property of Equality

We multiply both sides of the equation by any non-zero number to check for consistency. In this case, there is no need for multiplication as the equation is already in its simplified form.

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