neralize
Nathan has two 8-foot boards. He cuts one board into
-foot pieces. He cuts the other board into-foot pieces.

The equation of the solution passing through (-1,0) is y = x + eˣ.

A slope field is a graphical representation of the solutions to a differential equation at different points in the xy-plane.

In this problem, we are given the equation y = -x + y and asked to sketch the solutions that pass through three different points: (0,0), (-3,1), and (-1,0). To do this, we first plot each of these points on the slope field and draw a short line segment with the same slope as the slope at that point. By repeating this process for several points, we can obtain a rough sketch of the solution curve passing through each point.

Once we have sketched the solution curves, we can try to find the equation of the solution passing through (-1,0). To do this, we need to integrate the differential equation y = -x + y with respect to x, which involves finding the antiderivative of -x + y. In this case, we can use the method of separation of variables to write the differential equation as:

dy/dx = y - x

Dividing both sides by y - x, we get:

dy / (y - x) = dx

Integrating both sides, we obtain:

ln|y - x| = x + C

where C is the constant of integration. Exponentiating both sides and simplifying, we get:

y - x = Ceˣ

where C is the constant of integration. To find the value of C, we can use the initial condition that the solution passes through (-1,0), which means that:

0 - (-1) = Ce⁻¹

Simplifying, we get:

C = e

Substituting this value of C into the equation for the solution, we get:

y - x = eˣ

which is the equation of the solution passing through (-1,0). We can verify that this solution is correct by checking that it satisfies the original differential equation:

dy/dx = y - x

Substituting y = x + eˣ, we get:

d/dx(x + eˣ) = (x + eˣ) - x

Simplifying, we get:

eˣ = eˣ

which is true.

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Complete Question:

The slope field for the equation y = -x + y is shown below EN On a print out of this slope field, sketch the solutions that pass through the points (1) (0,0); (ii) (-3,1); and (iii) (-1,0). From your sketch, what is the equation of the solution to the differential equation that passes through (-1,0)? (Verify that your solution is correct by substituting it into the differential equation.)

From the given information it cannot be determined whether it is a relation or function.

In mathematics, a function is a relationship between two sets of elements, where each element in the first set (the domain) corresponds to a unique element in the second set (the range). In other words, a function takes an input from the domain and produces a unique output from the range.

In mathematics, a relation is a set of ordered pairs that describes a relationship between two sets of elements. The ordered pairs typically consist of one element from each set, and the relationship between the two elements is defined by some rule or condition.

To determine whether the relation is a function, we need to check whether each input (type of washing machine) has a unique output (amount of water used per load), or whether there are multiple outputs for some inputs.

If each type of washing machine has a unique amount of water used per load, then the relation is a function. If there are multiple amounts of water used per load for some types of washing machines, then the relation is not a function.

Without knowing the data that Keon recorded, we cannot determine whether the relation is a function. We would need to see the data and check whether there are any types of washing machines that have multiple amounts of water used per load recorded, or whether each type of washing machine has a unique amount of water used per load recorded.

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Answer:

x = 12

Step-by-step explanation:

To find the value of x, use logarithmic rules to solve the given equation.

Given logarithmic equation:

[tex]\log_2 (x^2-100) - \log_2 (x + 10) = 1[/tex]

[tex]\textsf{Apply the log quotient law:} \quad \log_ax - \log_ay=\log_a \left(\dfrac{x}{y}\right)[/tex]

[tex]\implies \log_2 \left(\dfrac{x^2-100}{x+10}\right) = 1[/tex]

Factor the numerator x² - 100:

[tex]\implies \log_2 \left(\dfrac{(x-10)(x+10)}{x+10}\right) = 1[/tex]

Cancel the common factor (x + 10):

[tex]\implies \log_2 \left(x-10\right) = 1[/tex]

[tex]\textsf{Apply the log law:} \quad \log_ab=c \iff a^c=b[/tex]

[tex]\implies 2^1=x-10[/tex]

Simplify and solve for x:

[tex]\implies 2=x-10[/tex]

[tex]\implies 2+10=x-10+10[/tex]

[tex]\implies 12=x[/tex]

Answer:

  3

Step-by-step explanation:

You want side QR in parallelogram PQRS having side PS marked as 3.

Opposite sides of a parallelogram are the same length, so ...

  QR = PS = 3

The length of QR is 3 units.

__

Additional comment

In a parallelogram, opposite sides are parallel and are the same length. Opposite angles have the same measure, and adjacent angles are supplementary. The diagonals bisect each other, but are usually different lengths and usually do not cross at right angles.

(If diagonals are the same length, it is a rectangle; if they cross at right angles, it is a rhombus. A rectangular rhombus is a square.)

A home priced at $200,000 is the same distance from the mean price in Florida and Ohio.

The z-score of a measure X of a variable that has mean symbolized by [tex]\mu[/tex] and standard deviation symbolized by [tex]\sigma[/tex] is obtained by the rule presented as follows:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The z-score represents how many standard deviations the measure X is above or below the mean of the distribution, depending if the obtained z-score is positive or negative.

For Florida, the z-score of a home priced at $240,000 is given as follows:

Z = (200000 - 240000)/16000

Z = -2.5.

For Ohio, the z-score of a home priced at $240,000 is given as follows:

Z = (200000 - 170000)/12000

Z = 2.5.

Same absolute value of z-score, hence the houses are the same distance from the mean for each state.

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Answer:

8 Units

Step-by-step explanation:

The volume of a cone can be calculated using the formula:

V = (1/3) * π * r^2 * h

where V is the volume, π is pi, r is the radius, and h is the height of the cone. We are given the volume and the diameter, so we can use the diameter to find the radius:

d = 2r

14 = 2r

r = 7

Now we can substitute the values for the volume and radius into the formula and solve for the height:

V = (1/3) * π * r^2 * h

1230.88 = (1/3) * 3.14 * 7^2 * h

1230.88 = (1/3) * 3.14 * 49 * h

1230.88 = (49/3) * 3.14 * h

1230.88 = 153.86 * h

h = 1230.88 / 153.86

h = 8

So, the height of the cone is approximately 8 units

Area of the triangle which is the land that the university acquired is 3 miles².

Area of a two dimensional shape is the total region which is bounded by the object's shape.

The land acquired by the university is in the shape of triangle.

Half of the base has length of 1.5 miles.

Total length of base = 2 × 1.5 = 3 miles

Height of the base = 2 miles

Area of the triangle = [tex]\frac{1}{2}[/tex] × base × height

                                 =  [tex]\frac{1}{2}[/tex] × 3 × 2

                                 = 3 miles²

Hence the area of the land is 3 miles².

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The value of the function j(7) will be 1 / 10.

An inverse function is defined as the function in which the two variables are changed by each other and the function is solved for the value of an independent variable.

F(x) = 2x

y = 2x

The inverse function is calculated as:-

x = 2y

y = x / 2

F'(x) = x / 2

Given that Functions hand j are inverses. at the value of x = -10, the value of h(x) is 7, or h (-10) = 7.

The value of the function j(7) will be equal to 1 / 10 for the given data.

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All the statements are false and the reasons are given in the explanation.

The hypothesis and the conclusion are the two components of this form of statement. This is usually used in logic

Using "if" and "then," a conditional statement links its hypothesis and conclusion. Consequently, the phrase is also known as a "if-then statement."

Now in the given question.

P =  you work in a hospital

Q  = you are a doctor

a. Converse

Q →  P     False.....not all doctors work in a hospiial

b. Inverse

not P  →  not Q    False.....same as before

c. Contrapositive

not Q  →  not P        False.....other people besides doctors work at a hospital.

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The China’s expected GDP is 17lac crore.

A proportion is a fraction of a total amount, and the measures are related using a rule of three.

The relations between variables, either direct or inverse proportional, can be built to find the desired measures in the problem.

Given that China’s GDP was proportional to that of the U.S

Now,

Economic growth is defined as a rise in economic activity and product availability. Real GDP has grown throughout this time period as well proportional to that of the U.S. In light of the current circumstances, the level of prices in the United States would increase, and the GDP would increase, as there is economic growth in China and Europe.

Therefore, by the given proportion the answer will be 17lac crore.

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(d) 95% confidence; n=50 (e) n=100 at any confidence level

What is interval estimate?

Interval estimation in statistics is the use of sample data to estimate an interval of feasible values for a parameter of interest. In contrast to point estimate, which produces a single value.

The margin of error for a confidence interval estimate of the population mean decreases with an increase in the sample size and with an increase in the confidence level.

Therefore, options (b) and (d) are better than (a) and (c) because they have a larger sample size. Between options (b) and (d), a larger sample size would result in a smaller margin of error.

Therefore, option (d), which has a sample size of 50 and a higher confidence level of 95%, would result in the smallest margin of error in estimating the mean salt content μ. Option (e) with a sample size of 100 would also result in a smaller margin of error, but the confidence level is not specified, so it cannot be compared directly to the other options.

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The price that corresponds to an elasticity of 1.2 for this product is $59.85.

Elasticity is a fundamental concept in economics that measures the responsiveness of demand to changes in price. In this problem, we will use the calculus-based definition of elasticity to find the price that corresponds to a specific elasticity value.

The elasticity of demand measures the percentage change in quantity demanded that results from a 1% change in price. Mathematically, the elasticity of demand is defined as:

E = (%ΔQ / %ΔP) * (P / Q)

where E is the elasticity of demand, %ΔQ is the percentage change in quantity demanded, %ΔP is the percentage change in price, P is the price of the product, and Q is the quantity demanded.

To find the price that corresponds to an elasticity of 1.2, we need to solve for P in the elasticity equation above. Since we know the demand function D(p) = -5p² + 270,000, we can find Q by taking the derivative of the demand function with respect to price:

Q = D'(p) = -10p

Then, we can plug in Q and E = 1.2 into the elasticity equation and solve for P:

1.2 = (%ΔQ / %ΔP) * (P / Q)

1.2 = (%ΔQ / -1%) * (P / (-10p))

1.2 = (%ΔQ / -0.01) * (1 / -10)

%ΔQ = -0.0144

This means that a 1% increase in price will result in a 0.0144% decrease in quantity demanded, given an elasticity of 1.2. Now, we can plug in %ΔQ and Q into the demand function to solve for P:

-0.0144 = (D(p + 0.01P) - D(p)) / D(p)

-0.0144 = (-5(p + 0.01P)² + 270,000 - (-5p² + 270,000)) / (-5p² + 270,000)

-0.0144 = (-0.1P - 0.0005p²) / (-5p² + 270,000)

P = $59.85

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Please report this so a moderator can delete this answer. I was not finished with my answer and accidentally posted it. I will not have enough time to add it before the editing timer is up.

The inverse of the function f(x) = 64x³ - 1 is [tex]f^{-1}(x) = (\frac{3\sqrt{(x + 1)}}{4})[/tex].

First to be an inverse function that function needs to one to one function, meaning every different preimage must correspond to a different image.

We can obtain the inverse of a function by switching the variables x and y with their respective positions and solving for y in terms of x.

Given, A function f(x) = 64x³ - 1.

Let, y = 64x³ - 1.

64x³ = y + 1.

x³ = (y + 1)/64.

[tex]x = (\frac{(y + 1)}{64})^{\frac{1}{3}[/tex].

[tex]x = (\frac{3\sqrt{(y + 1)}}{4})[/tex].

[tex]y = (\frac{3\sqrt{(x + 1)}}{4})[/tex].

[tex]f^{-1}(x) = (\frac{3\sqrt{(x + 1)}}{4})[/tex].

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The range of the costs of the surfboard rentals is $ 140.

The interquartile range of the costs is $ 56.

The cost of 6 days of rentals is an outlier.

To find the range, first find the costs of renting the surfboards on the different days.

1 day :

= 28 x 1

= $ 28

2 days :

= 28 x 2

= $ 56

3 days :

= 28 x 3

= $ 84

6 days :

= 28 x 6

= $ 168

The cost can therefore be ordered to be :

28, 28, 28, 56, 56, 56, 84, 84, 84, 168

The range is :

= Largest - Least

= 168 - 28

= $ 140

The interquartile range is :

= Q 3 - Q1

Q 3 :

= 75 / 100 x 10 costs

= 7.5

= $ 84 in the cost arrangement

Q1 :

= 25 / 100 x 10

= 2. 5 position

= $ 28 in the cost arrangement

The IQR is :

= 84 - 28

= $ 56

The cost of 6 days of rentals is an outlier as it is much larger than the rest of the costs.

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The value of the equation 7.4×10⁻⁸=6.7×10⁻⁹ is 6.73×10⁻⁸

A number system is defined as a system of writing to express numbers.

The given expression is 7, point, 4, dot, 10, start superscript, minus, 8, end superscript, minus, 6, point, 7, dot, 10, start superscript, minus, 9, end superscript, equals

7.4×10⁻⁸=6.7×10⁻⁹

Now let us take 6.7×10⁻⁹ to left hand side

7.4×10⁻⁸-6.7×10⁻⁹

Factor 10⁻⁸ from both terms

10⁻⁸(7.4-0.67)

6.73×10⁻⁸

Hence, 6.73×10⁻⁸ is the value of the equation 7.4×10⁻⁸=6.7×10⁻⁹

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The complete question will be as below

Find the simplest form of the equation 7.4×10⁻⁸=6.7×10⁻⁹?

If the slope of the line is 0, that means the y axis do not change. So the equation for the line is y = -9.

The line is said to have a slope 0 and passes through point(3,-9).

Equation for the slope is

m = y-y₁/ x-x₁

Rearranging the equation, we get

y-y₁ = m (x-x₁)

Here m = 0

So, y- y₁ = 0 (x-x₁)

    y-y₁ = 0

Here a point is given as (3,-9)

3 is the x-coordinate and -9 is the y-coordinate. So taking y₁ as -9

y - ⁻9 = 0

y + 9 = 0

y = -9

So the equation for the line is y= -9

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The required height of the kite to the nearest tenth of a meter is 25.2 m.

In a right-angled triangle, the ratio of the perpendicular (with respect to an angle) to the hypotenuse gives the sine value of that angle. This relationship is used to solve problems regarding height and distance chapter.

Draw the figure as per the given instructions.

The position of the kite is at A and the string is AC which is 37 m long. The angle of elevation is ∠ACB, which is 43°. The height of the kite is AB.

So, apply the formula of sine ratio (perpendicular/hypotenuse) = sinα, where α be the opposite angle of that perpendicular.

AB/AC = sin∠ACB

AB/37 = sin43°

AB = 37sin43°

≈37×0.68

= 25.16

≈ 25.2

Therefore, the obtained answer is 25.2 m.

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Answer:

Step-by-step explanation:

48 x 10 = 480 so you can go 480 miles before you run out of gas.

In a case whereby you have 1,000,000 number cubes, each measuring one inch on a side. If you arranged the cubes on the floor to make a square, then the square would fit in your classroom.

You possess one inch on each side, one million number cubes. How tall would the cubes be if they were placed one on top of the other to form a massive tower?

The fact that the high of one cube= 1 inch

The high of 1,000,000 cubes can be calculated as :

=1*1,000,000 = 1,000,000= 1 * 10^6 inch.

The area of a square A= b^2

b= length side of the square

1,000,000 = b^2

b = 1000 inch

Therefore, the dimensions of the square  can be expressed as ( 1,000 in x 1,000 in)

But 1 in = 2.54 cm

1,000 inch = (1,000*2.54)

=2,540 cm-

=25.40 meters

Considering this in metres, the dimensions of the square can be expressed as: ( 25.40 m x 25.40 m)

We can conclude that it possible that the square will fit in the classroom.

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The area covered by the airport in 2000000 m² and 200 Ha.

A scale factor is a numerical value that can be used to alter the size of any geometric figure or object in relation to its original size. It is used to find the missing length, area, or volume of an enlarged or reduced figure as well as to draw the enlarged or reduced shape of any given figure. It should be remembered that the scale factor only affects how big a figure is, not how it looks.

Given:

A map of kano is drawn to a scale of 1:50000.

The area that airport covers = 8 cm²

Also, 1 ha = 10000 m²

if it's 1 cm, then real life would be 50,000 cm.

Then, 50,000 cm x 50,000 cm = 2,500,000,000 cm² in real life.

Now, the Area of Airport in real

= 8 x 2,500,000,000

= 20,000,000,000 cm²

= 2000000 m²

= 2000000 / 10000

= 200 Ha

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The perimeter of the triangle FGH is 7.

Perimeter is defined as space around the any two dimensional figure. It is given by the sum of all the sides of the figure.

Given the triangle FGH which is formed by connecting the midpoints of the side of the triangle CDE.

The length of the sides of the triangle are given as, CD=6, DE=4, EC=4.

To find the perimeter of the triangle FGH,

Since FGH are the midpoints of CD,DE and EC respectively.

Then, FH=1/2 DE, FG=1/2EC, GH=1/2DC.

Perimeter of a triangle = sum of the length of sides.

                                   = 1/2(DE+EC+DC)

                                   = 1/2(4+6+4)

Perimeter of a triangle = 7

Hence, the perimeter of the triangle FGH is 7.

Question:

Triangle FGH is formed by connecting the midpoint of the sides of the triangle CDE. The length of the sides is CD=6, EC=4, DE=5. find the perimeter of the triangle FGH.

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The given statement can be expressed in terms of c(x), d(x), f(x), quantifiers, and logical connectives as follows:

a) ∃x(C(x) ∧ D(x) ∧ F(x))

b) ∀x(C(x) ∨ D(x) ∨ F(x))

c) ∃x(C(x) ∧ F(x) ∧ ¬D(x))

d) ¬∃x(C(x) ∧ D(x) ∧ F(x))

e) ∀y∃x(P(x, y))

All of the pupils in your class should be a set. The domain is the set X. Denote

[tex]C(x)-'x has a cat'\\D(x)- 'x has a dog'\\F(x)- 'x has a ferret'[/tex]

a) A kid in your class owns a cat, a dog, and a ferret, so take that into consideration. This implies that ∃x ∈ X so that C(x), D(x), and F(x) are all true assertions. Using quantifiers, logical connectives, and C(x), D(x), and F(x) we can express that as follows.

∃x(C(x) ∧ D(x) ∧ F(x))

b) Take into account the claim that "Every student in your class has a cat, a dog, or a ferret." This proves ∀x ∈ X at least one of the claims made by C(x), D(x), and F(x) to be true. Using quantifiers, logical connectives, and C(x), D(x), and F(x) we can express that as follows.

∀x(C(x) ∨ D(x) ∨ F(x))

c) Take into account the claim that "Some student in your class has a cat and a ferret, but not a dog." Accordingly, ∃x ∈ X, statements C(x), F(x), and the negation of statement D(x) are true. Using quantifiers, logical connectives, and C(x), D(x), and F(x) we can express that as follows.

∃x(C(x) ∧ F(x) ∧ ¬D(x))

d) Take into account the claim that "No student in your class has a cat, a dog, and a ferret." This indicates ∀x ∈ X that neither C(x), D(x), nor F(x) is true. As a denial of the claim in section a), we may state that in terms of C(x), D(x), and F(x) by using quantifiers and logical connectives as follows.

¬∃x(C(x) ∧ D(x) ∧ F(x))

a) Take into account the adage, "There is a student in your class who has this animal as a pet. The three animals are cats, dogs, and ferrets."

This indicates that an element X from the domain exists for each of the propositions C, F, and D such that each statement is true.

Using quantifiers, logical connectives, and C(x), D(x), and F(x) we can express that as follows.

∀y∃x(P(x, y))

The complete question is:-

Let C(x) be the statement "x has a cat," let D(x) be the statement "x has a dog," and let F(x) be the statement "x has a ferret." Express each of these statements in terms of C(x), D(x), F(x), quantifiers, and logical connectives. Let the domain consist of all students in your class. a) A student in your class has a cat, a dog, and a ferret. b) All students in your class have a cat, a dog, or a ferret. c) Some student in your class has a cat and a ferret, but not a dog. d) No student in your class has a cat, a dog, and a ferret. e) For each of the three animals, cats, dogs, and ferrets, there is a student in your class who has this animal as a pet.

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The length of PQ is (1 + √(3))/2, which is answer choice (C) equilateral triangle, angle BAC measures 60 degrees.

Since ABC is an equilateral triangle, angle BAC measures 60 degrees. Also, since A' is the midpoint of BC and A is folded onto A', angle BAA' and CAA' are both 30 degrees.

Now, let's use the Law of Cosines to find the length of PQ. Let x be the length of PQ. Then, applying the Law of Cosines to triangles ABP and AQC, we have:

BP² = x² + 1 - 2x cos30

CQ² = x² + 4 - 4x cos30

Since BP = CQ, we can set the two expressions equal to each other and simplify:

x² - 2x cos30 + 3 = 0

Solving for x using the quadratic formula, we get:

x = cos30 +/- √(cos²30 - 3)

Since cos30 = √(3)/2, we have:

x = 1/2 +/- √(3)/2

Since x must be positive, we take the positive root and get:

x = 1/2 + √(3)/2 = (1 + √(3))/2

Therefore, the length of PQ is (1 + √(3))/2, which is answer choice (C).

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The question is -

Equilateral triangle ABC has P on AB and Q on AC. The triangle is folded along PQ so that vertex A now rests at A' on side BC. If BA'=1 and A'C=2 then the length of the crease PQ is

[tex]\text{(A) } \frac{8}{5} \text{(B) } \frac{7}{20}\sqrt{21} \text{(C) } \frac{1+\sqrt{5}}{2} \text{(D) } \frac{13}{8} \text{(E) } \sqrt{3}[/tex]

The new cereal box requires 11 square inches less cardboard to construct than the original cereal box.

The amount of cardboard required to build a cereal box is determined by its surface area, which includes all six sides of the box.

The original cereal box's surface area can be calculated as follows:

2 * 2.5 * 12 = 60 square inches for the front and back faces

2 * 7 * 12 = 168 square inches on the side faces

2 * 2.5 * 7 = 35 square inches for the top and bottom faces

Surface area total: 60 + 168 + 35 = 263 square inches

Using the new base dimensions, the surface area of the new cereal box can be calculated in the same way:

2 * 3 * 12 = 72 square inches for the front and back faces

2 * 6 * 12 = 144 square inches on the side faces

2 * 3 * 6 = 36 square inches for the top and bottom faces

Surface area total: 72 + 144 + 36 = 252 square inches

To calculate the difference in cardboard required to construct the two cereal boxes, subtract the new box's surface area from the original box's surface area:

263 - 252 = 11

As a result, the new cereal box uses 11 square inches less cardboard than the original cereal box.

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Answer: To find the volume of the tank, we need to determine the length and width of the base. We can use the area formula for rectangles:

Area = length * width

Since the area of the base is given as 485 square inches, we can rearrange this equation to find the length or width:

length * width = 485

Let's call the length "l". Then, the width is 485/l. The volume of the tank is the product of the area of the base and the height:

Volume = Area * height = (l * 485/l) * 16 = 485 * 16 = 7760 cubic inches.

So the volume of Tiny's tank is 7760 cubic inches.

Step-by-step explanation:

The value of x in 3v(1/8-x)=-1/2 is 10 2/7

A linear equation only has one or two variables. No variable in a linear equation is raised to a power greater than 1 or used as the denominator of a fraction.

The highest power of variable in a linear equation is 1

3v(1/8-x)=-1/2

square both sides to eliminate the root

3²v(1/8-x)² =(-1/2)²

= 9(1/8-x) = 1/4

= 9/8 - 9x = 1/4

multiply through by 8

9 - 72x = 2

72x = 9-2

72x = 7

x = 72/7

x = 10 2/7

therefore the value of x is 10 2/7

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