Reduce to first order and solve. (1 - x2 )y" - 2xy' + 2y = 0, y1 = x. Use: -Sp dx = In [1/(1-x2)] find U with its integral. to get y2 = uyi. O a. Y2 = yıu = Y SU dx = 2 + 5 x In [(x+1)/(x-1)] O b. y2 = yıu = yısu dx = -1 + 0.5 x In [(x+1)/(x-1)]. O c. y2 = yıu = yılU dx = 10 + 20 x In [(x+1)/(x-1)] O d. y2 = yıu = YISU dx = 1 + 2 x In [(x+1)/(x-1)].

The average speed of the car is 51 kph. The result is obtained by dividing the total distance by the total time taken.

Average speed is the overall speed of a moving object during a given amount of time. It can be expressed as

Avg speed = Total distance covered / Total time taken

The speed itself can be found by the following formula.

s = d/t

Where

A car can travel at the following speeds.

Find the average speed!

We should convert the unit of time from minutes to hour.

t₃ = 45 minutes = ¾ h

t₄ = 15 minutes = ¼ h

We count the time and distance in each travels.

t₁ = d₁/s₁ = 20/40 = ½ h

t₂ = d₂/s₂ = 25/50 = ½ h

d₃ = s₃×t₃ = 60 × ¾ = 45 km

d₄ = s₄×t₄ = 48 × ¼ = 12 km

Avg speed = (s₁ + s₂ + s₃ + s₄)/(t₁ + t₂ + t₃ + t₄)

Avg speed = (40 + 50 + 60 + 48)/(½ + ½ + ¾ + ¼)

Avg speed = 102/2

Avg speed = 51 kph

Hence, the car has the average speed of 51 kph.

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[tex]\frac{1}{4}[/tex] must be added to the expression [tex]x^{2}[/tex] + x to make it a perfect-square trinomial

An integer that can be expressed as the square of another integer is called a perfect square. In other words, it is defined as the product of some integer with itself.Perfect square numbers are not only limited to the numerals but also exist in algebraic identities and polynomials. These can be identified with the help of a factorisation technique.

For this case we have the following expression: [tex]x^{2}[/tex]+ x

We want to find a constant to create a perfect-square trinomial.

The value of the constant is given by: k = [tex]\frac{b}{2} ^{2}[/tex]Where, b belongs to the coefficient that accompanies the term of exponent 1 in the quadratic equation.

b=1,Substituting values: k =  [tex]\frac{1}{2} ^{2}[/tex]

k=[tex]\frac{1} ^{4}[/tex]

Rewriting the expression we have: [tex]x^{2}[/tex]+x+    [tex]\frac{1}{4}[/tex] = (x + [tex]\frac{1}{2}[/tex])(x + [tex]\frac{1}{2}[/tex])

[tex]x^{2}[/tex]+x+ [tex]\frac{1} ^{4}[/tex]: = [tex]x+1^{2}[/tex]

So,k =[tex]\frac{1}{4}[/tex] must be added to the expression [tex]x^{2}[/tex] +x to make it a perfect-square trinomial.

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The value that must be added to the expression x² + x to make it a perfect-square trinomial is 1/4.

Therefore the answer is 1/4.

To make the expression x² + x a perfect-square trinomial, we need to add a value that makes the expression equal to (x + a)², where a is a constant.

The expression (x + a)² can be expanded as follows:

(x + a)² = x² + 2ax + a²

Comparing this to the expression x² + x, we can see that we need to add the value a² to the expression x² + x to make it a perfect-square trinomial. By observing the coefficient of x-term we can obtain value of a.

2a = 1

a = 1/2

a² = 1/4

So the expression will be

x² + x + 1/4 = (x + 1/2)²

So x² + x + 1/4 is a perfect-square trinomial.

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The donuts are cheaper in bakery B which implies that Kevin is right.

In order to solve a problem for two different values of a quantity, its unit value is first derived. This method is known as unitary method. The ratio of two numbers are taken in this method. There can be two relations between the ratio a direct or n indirect. For the direct relation the quantities simultaneously increase or decrease while in the indirect it happens reverse.

The price of donuts for both bakeries are given as,

For Bakery A,

The price of 10 donuts is $15.

Unit rate is given as 15/10 = $1.50 per donut.

For Bakery B,

The price of 12 donuts is $17.

Unit rate is given as 17/12 = $1.41 per donut.

Thus, price of donuts from Bakery B is less than Bakery A.

It implies that Kevin is right.

Hence, the price of donuts is cheaper in case of bakery B which implies that Kevin is right.

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Kevin is correct. And he used the correct unit rate formula to find the inequality.

In mathematics, "inequality" refers to a relationship between two expressions or values that are not equal to each other. To solve the inequality, you may multiply or divide each side by the same positive number, add the same amount to each side, take the same amount away from each side, and more. You must flip the inequality sign if you multiply or divide either side by a negative number.

Given:

Bakery A.

She used the equation 10 donuts $15 to show that their donuts cost about $0.67 per donut.

This is less than Bakery B's price, 12 donuts $17= about $0.71 per donut.

Kevin thinks they should order donuts from Bakery B.

He used the equation

$17 ÷ 12 donuts to show that their donuts cost about $1.42 per donut.

This is less than Bakery A's price,

$15÷ 10 donuts = about $1.50 per donut.

The inequality:

1.42 < 1.50

To find the unit rate:

Unit rate = the total price of donut / the number of total donuts

Kevin used the above formula to find the unit rate.

So, Kevin is right.

Therefore, Kevin is right.

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The reason for the first and second steps in the equation are Addition Property of Equality and Division Property of Equality respectively

An equation is a mathematical statement that shows that two expressions are equal. It consists of two expressions separated by an equal sign (=) e.g. 2x + 3 = 7

Given: the equation 3x-2=4

Match the reason with the correct step in the equation as follows:

3x-2= 4

Add to 2 to both sides of the equation:

3x-2+2 = 4+2    (Addition Property of Equality)

3x = 6

Divide both sides by 3:

3x/3 = 6/3        (Division Property of Equality)

x = 2

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The area of the region enclosed by the graphs of y=x^3+8 and y=x+8 is 24. This is found by calculating the area under each graph and subtracting the area of the overlap and the area outside the region.

The area of the region can be calculated by finding the area under the two graphs, subtracting the area of the overlap and then subtracting the area outside the region bounded by the two graphs.

First, find the area under the graph of y=x^3+8 and above the x-axis. This can be calculated using the definite integral

∫ y dx from 0 to 8, which is equal to 8^4/4+8 = 288.

Next, find the area under the graph of y=x+8 and above the x-axis. This can be calculated using the definite integral

∫ y dx from 0 to 8, which is equal to 8^2/2+8 = 72.

The area of the overlap between the two graphs is the area of the region below both graphs and above the x-axis. This can be calculated using the definite integral

∫ y dx from 0 to 8, which is equal to 8^3/3+8 = 168.

Finally, subtract the area outside the region bounded by the two graphs, which is the area below the line y=x+8 and above the x-axis. This can be calculated using the definite integral

∫ y dx from 0 to 8, which is equal to 8^2/2 = 32.

Therefore, the area of the region in the first quadrant that is enclosed by the graphs of y=x^3+8 and y=x+8 is 288 - 72 - 168 + 32 = 24.

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Answer:

Step-by-step explanation:

thats it

Answer:

1

Step-by-step explanation:

The step by step explanation is in the diagram above

I hope you understand

Answer: $119.00

Step-by-step explanation:

To calculate the pay for the day, we need to determine the number of hours worked in the morning and the number of hours worked in the afternoon.

The morning shift starts at 8:00 AM and ends at 12:15 PM, which is a total of 12:15 - 8:00 = 4 hours and 15 minutes. The afternoon shift starts at 1:00 PM and ends at 5:15 PM, which is a total of 5:15 - 1:00 = 4 hours and 15 minutes.

Since there are 60 minutes in an hour, the total number of hours worked in the morning is 4 hours + 15 minutes / 60 minutes = 4.25 hours. The total number of hours worked in the afternoon is also 4.25 hours.

The total number of hours worked on the day is 4.25 + 4.25 = 8.5 hours.

To find the pay for the day, we need to multiply the number of hours worked by the wage rate:

Pay = 8.5 hours * $14/hour = $119

Rounding this number to the nearest hundredth gives us $119.00.

The monthly paycheck for state income tax is $1,356.67

The unitary technique involves first determining the value of a single unit, followed by the value of the necessary number of units.

Given:

Annual Gross pay = $48,840

He is married with 3 dependents.

So, every month he earns:

monthly earnings = $48,840/12

                               = $4,070

Also, he has 3 dependents

= $4,070/3

= $1,356.67

Thus, his monthly paycheck for state income tax is $1,356.67

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I) and II) are true because the limit of the function at x=2 is 5, which implies that f is continuous and differentiable at x=2. III) is false because the derivative of f does not need to be continuous at x=2.

Let f be a function such that lim h→0 (f(2+h) - f(2)) / h = 5.

We can calculate this limit as follows:

lim h→0 (f(2+h) - f(2)) / h = lim h→0 (f(2+h) / h) - (f(2) / h)

= lim h→0 f(2+h) / h - lim h→0 f(2) / h

= lim h→0 f(2+h) / h - 0

= lim h→0 f(2+h) / h

= 5

Therefore, I) and II) are true because the limit of the function at x=2 is 5, which implies that f is continuous and differentiable at x=2. III) is false because the derivative of f does not need to be continuous at x=2.

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The graph of f(x) = x^3 -6x^2 + 4x +1 hits 3 times to the x-axis

Hence, option (D) is the correct choice.

We have the function:

f(x) = x^3-6x^2+4x+1

The values of the variable in a function that cause the function to equal zero are known as the zeros of the function. The places on the x-axis where the graph crosses the x-axis are known as a function's zeros graphically. In other words, we may say that a function's zeros are the graph's x-intercepts.

The zeros of the function will be:

x^3-6x^2+4x+1 = 0

Since the last term is positive, so (x-1) will be the factor of the above equation.

Factorising, we will get,

(x-1)(x^2-5x-1)=x^3-6x^2+4x+1

So, one factor will be x = 1

Now solving the quadratic equation,

The roots will be: [tex]x^2 = \frac{( 5 \pm \sqrt 29)}{2}[/tex]

So, the value of both roots will be: (x-5.19)(x+0.192)

Therefore, the graph will hit x-axis at these three points.

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The number of days for which Jaya can rent the car as calculated from the given data is 4 days.

Jaya is planning to drive 125 miles, replacing y's value and solving for x we get,

53.75x+0.12(125)≤230

53.75x+15≤230

53.75x≤230-15

53.75x≤215

x≤215/53.75

x≤4

Therefore , as calculated Jaya can rent the car for 4 days.

Inequality is a mathematical statement that uses the symbol of inequality to indicate the relationship between any two given expressions. Both sides of an inequality sign have different expressions.

It signifies that the expression on the left should be more or smaller than the expression on the right and the reverse is also true. Literal inequalities occur when the relationship between two algebraic expressions is defined using inequality symbols.

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A statement is the best interpretation of one of the values in this function include the following: C. The number of bacteria increases at a rate of 12% each day.

In Mathematics, an exponential function can be defined as a type of mathematical function whose values are generated by a constant that is raised to the power of the argument.

Mathematically, an exponential function can be modeled by using this equation:

f(x) = ab^x

Where:

Based on the information provided by this exponential function f(x) = 245(1.12)^x, we can logically deduce that the rate of change is equal to 12 percent:

Rate of change = 112 - 100

Rate of change = 12%

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Complete Question:

Scientists are studying a bacteria sample. The following function f(x) = 245(1.12)^x gives the number of bacteria in the sample at the end of x days.

Which statement is the best interpretation of one of the values in this function?

answer choices

The initial number of bacteria is 12.

The initial number of bacteria decreases at a rate of 88% each day.

The number of bacteria increases at a rate of 12% each day.

The number of bacteria at the end of one day is 245.

Answer:

x = 10

y = 5

Step-by-step explanation:

Interior angles of a triangle sum to 180°.  Therefore, from inspection of the given diagram, the interior angles of the given right triangle are:

This means the triangle is a special right triangle and that the 30-60-90 theorem can be applied to quickly find the measures of the missing sides.

30-60-90 Triangle Theorem

A 30-60-90 triangle is a special right triangle where the measures of its sides are in the ratio  1 : √3 : 2.

Therefore, the formula for the ratio of the sides is b : b√3 : 2b where:

From inspection of the given triangle, the side opposite the 60° angle is 5√3.  Therefore:

⇒ b√3 = 5√3

⇒ b = 5

Substitute the found value of b into the expressions for the other two sides to find the values of x and y:

⇒ x = 2b = 2 · 5 = 10

⇒ y = b = 5

μ is a population parameter, it represents the mean of a population.

In statistics, μ (also denoted as "mu") is a symbol commonly used to represent the population mean. The population mean is a measure of central tendency that describes the average value of a given set of data, calculated by summing all the values in a population and dividing by the total number of observations in that population. Therefore, μ is a population parameter, as it describes a characteristic of the entire population, not just a sample of it. It is important to note that when working with samples, we use mean of samples which is denoted as (x-bar) instead of μ.

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Using the points A(3,1) B(3,4) C(-4,1) D(-4,5) the following deductions are made

slope of (line AB): undefined

slope of (line CD) undefined

type of line parallel lines

The slope, m of the linear function is calculated using the points A(3,1) and B(3,4)

m = (y₀ - y₁) / (x₀ - x₁)

m = (1 - 4) / (3 - 3)

m = (-3) / (0)

m = undefined

The slope, m of the linear function is calculated using the points C(-4,1) and D(-4,5)

m = (y₀ - y₁) / (x₀ - x₁)

m = (1 - 5) / (-4 - (-4))

m = (-4) / (0)

m = undefined

The slope ae equal and undefined hence they are vertical parallel lines

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The two more recipes that have an equivalent ratio of sugar and water would be: 6 cups of sugar: 16 cups of water and 9 cups of sugar : 24 cups of water.

Ratio equivalent is defined as the ratios that appear the same when compared after simplification.

For example, the ratio equivalent of 4:6 would be 2:3 after being simplified by division. Also 2:3 is equivalent to 4:6 after being multiplied by a number.

Therefore to make a larger recipe for the nectar, 3:8 ratio of the sugar to water should be multiplied by 2 and 3. Thus;

3 cups sugar : 8 cups of water × 2 = 6 cups of sugar: 16 cups of water

3 cups sugar : 8 cups of water × 3 = 9 cups of sugar : 24 cups of water.

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The triangle that is similar to triangle PQR using the Pieces of Right Triangles Similarity Theorem is the (a) triangle RQS

How to determine the similar triangle?

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

The triangle set

From the triangles, we have

Original triangle = Triangle PQR

The theorem to use in this question is the Right Triangles Similarity Theorem

This theorem implies that:

We make use of the similar angles (i.e. the right angle) to determine the similar triangles

In this case, we have

Triangle PQR is right-angled at point Q

Also, we have

Triangle RQS is right-angled at point Q

This implies that both triangles are similar

Hence, the required triangle is (a) triangle RQS

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Answer:

x=4

Step-by-step explanation:

To find the value of x in equation 3x + 1 = x + 9, we need to solve for x.

First, we can subtract x from both sides of the equation: 3x + 1 - x = x + 9 - x

This simplifies to: 2x + 1 = 9

Next, we can subtract 1 from both sides of the equation: 2x + 1 - 1 = 9 - 1

This simplifies to 2x = 8

Finally, we can divide both sides of the equation by 2: 2x / 2 = 8 / 2

This simplifies to: x = 4

So the value of x in equation 3x + 1 = x + 9 is 4.

The circumcenter is the point of intersection of three perpendicular bisectors of the sides of a triangle. All three of the perpendicular bisectors are congruent, and the circumcenter itself is congruent to itself.

The circumcenter of a triangle is the point of intersection of all three perpendicular bisectors of the sides of the triangle. It is the center of the circumcircle which is the circle that passes through all three vertices of the triangle. The perpendicular bisectors of the sides of the triangle are all congruent, meaning that they are all the same length and have the same angle of inclination. Additionally, the circumcenter itself is congruent to itself, meaning that it has the same location and coordinates. This is because it is the point of intersection of all three perpendicular bisectors, which all have the same coordinates. The circumcenter is an important concept in geometry and trigonometry as it is the center of the circumcircle and provides insight into the properties of the triangle.

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The parts of circumcenter that are congruent are all the three perpendicular bisectors and the circumcenter itself .

What is Circumcenter ?

The Circumcenter of a triangle is defined as the point of intersection of all the three perpendicular bisectors of the side . It is the center of  circumcircle which is the circle that passes through the vertices of the triangle.

the parts of circumcenter that are congruent are :

(i) The Perpendicular bisectors of sides of triangle are all congruent, which means  they are of the same length and have the same angle of inclination .

(ii) The Circumcenter of the circle is congruent to itself, because it has the same location and coordinates because  it is point of intersection of all three perpendicular bisectors, all of which have the same coordinates .

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Answer: Quadrant III

Step-by-step explanation:

     If both numbers are less than 0, both are negative. If both numbers are negative, this point will be in the bottom left-hand quadrant, known as Quadrant III.

I have attached an image from Cuemath that gives a good visual of the four quadrants.

423

To solve the above equation follow the BODMAS rule

That is,

Brackets of Division Multiplication Addition Subtraction

BODMAS rule states that mathematical expressions with multiple operations need to be solved from left to right in the order of BODMAS.

so,

first, calculate the value in the brackets

(144-4)*(12/4)+3 = (140)*(12/4)+3

=140*(3)+3

According to the rule, a division operation is to be performed but there is no division operation in the equation we skip this step and proceed to the next operation

so, multiply 140*3

=420+3

the next operation is the addition

=423

the next operation according to the rule is subtraction which is not required to be performed in the equation.

therefore,

(144-4)*(12/4)+3 = 423

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74%

Find how many times 50 goes into 100. Then, take that number and multiply it by 37. Then you should get your answer.

The factored polynomials are given as follows:

1. x² + 11x + 30 = (x + 6)(x + 5).

2. x² + 8x + 7 = (x + 7)(x + 1).

3. a² + 20a + 36 = (a + 18)(a + 2).

4. x² + 16x + 60 = (x + 10)(x + 6).

5. n² + 14n + 45 = (n + 5)(n + 9).

6. x² + 12x + 32 = (x + 8)(x + 4).

7. x² - 6x + 8 = (x - 4)(x - 2).

8. c² - 10c + 16 = (c - 8)(c - 2).

9. x² - 20x + 36 = (x - 18)(x - 2).

10. x² - 7x + 10 = (x - 5)(x - 2).

11. x² - 9x + 8 = (x - 1)(x - 8).

12. x² - 13x + 40 = (x - 8)(x - 5).

13. x² + 37x + 36 = (x + 1)(x + 36).

14. x² + 12x + 36 = (x + 6)².

15. x² + 13x + 36 = (x + 9)(x + 4).

16. x² - 25x + 100 = (x - 20)(x - 5).

17. x² + 30x + 200 = (x + 20)(x + 10).

18. m² + 12m + 20 = (m + 10)(m + 2).

All the 18 quadratic polynomials are factored using the same procedure, obtaining it's roots, and then writing the polynomial as the product of the linear factors.

For item 1, the function is given as follows:

x² + 11x + 30.

Then the coefficients of the quadratic function are:

a = 1, b = 11, c = 30.

The discriminant is of:

D = 11² - 4 x 1 x 30 = 1.

The roots are given as follows:

Hence the factored polynomial, considering the product of the linear factors, is given as follows:

(x - 5)(x - 6).

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The coordinate point (0, -3) is located on y-axis. Therefore, option A is the correct answer.

A coordinate plane is a two-dimensional surface formed by two number lines. It is formed when a horizontal line (the X-axis) and a vertical line (the Y-axis) intersect at a point called the origin. The numbers on a coordinate grid are used to locate points.

The point is on the y-axis and is of the form (0, a) and since the distance is y units below the x-axis the coordinates are (0, -y).

Thus, point (0, -3) is located on y-axis

Therefore, option A is the correct answer.

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well, what's 19% of 378000?

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{\textit{\LARGE a}\% of \textit{\LARGE b}}\\ \cline{1-1} \\ \left( \cfrac{\textit{\LARGE a}}{100} \right)\cdot \textit{\LARGE b} \\\\ \cline{1-1} \end{array}~\hspace{5em}\stackrel{\textit{19\% of 378000}}{\left( \cfrac{19}{100} \right)378000}\implies 71820~\hfill \underset{current~value}{\stackrel{378000-71820}{\text{\LARGE 306,180}}}[/tex]

The statement that describes the graph is (c)  Scott pays his nephew $10 per hour, and he initially had $900 for extra labor.

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

The graph (added as an attachment)

The equation is a linear equation and a linear equation is represented as

y = mx + c

Where

From the graph, the two points are (0, 900) and (800, 10).

So, we start by calculating the slope using

slope = (y₂ - y₁)/(x₂ - x₁)

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

m = (900 - 800) / (0 - 10)

m = - 100 / 10

Evaluate

m = -10

So, we have

y = -10x + c

Substitute (0, 900) in y = -10x + c

900 = -10 x 0 + c

So, we have

c = 900

Then the equation becomes

y = - 10x + 900

This means that Scott pays his nephew $10 per hour, and he initially had $900 for extra labour.

Hence, the correct option is C.

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The best method to compare the lengths of two line segments is by using a ruler and divider.

The most precise way to measure a line segment is by penetrating the two needles of a divider at its ends, then set it next to a ruler without moving the divider's arms once they have been fastened at the ends of the line segment.

A line segment in geometry has two different points on it that define its boundaries. A line segment is sometimes referred to as a section of a line that links two places.

The difference between a line and a line segment is that a line has no endpoints and can go on forever in any direction.

Thus, The best method to compare the lengths of two line segments is by using a ruler and divider.

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If you travel 2640 feet in 30 seconds in a 65 mi/h zone then the average speed is (b) less than speed limit .

The Distance travelled in 30 seconds is = 2640 feet ,

Average Speed can be calculated by formula; Speed = Distance / Time ,

So , the speed of the person in feet per second will be = 2640/30 ft/sec .

we know the conversion rate that 1 mile = 5280 feet and 1 hr = 3600 sec ,

So , On converting the speed to mi/hr ,

it can be written as = (2640 × 3600)/(5280 × 30)

= 60 mi/hr .

The Speed Limit is given as 65 mi/hr .

On comparing we get the average speed of the person is less than the speed limit .

Therefore , The Average Speed is Less than the Limit .

The given question is incomplete , the complete question is

You travel 2640 feet in thirty seconds while in a 65 mi/h zone. (There are 5280 ft in one mi). Your average speed is:

(a) exactly the speed limit.

(b) less than the speed limit.

(c) larger than the speed limit.

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Your average speed is 98.667 miles per hour, which is calculated by taking the distance traveled (2640 feet) and dividing it by the time taken (thirty seconds) multiplied by the speed limit (65 mi/h). This is based on the fact that there are 5280 feet in one mile.

Step 1: Convert 2640 feet to miles.

1 mile = 5280 feet

2640 feet / 5280 feet = 0.5 miles

Step 2: Calculate speed.

Speed = Distance / Time

Speed = 0.5 miles / (30 seconds/60 seconds)

Speed = 0.5 miles / 0.5 minutes

Speed = 1 mile / 0.5 minutes

Speed = 2 miles/minute

Speed = 2 miles/minute * 60 minutes/hour

Speed = 120 miles/hour

Speed = 120 miles/hour * 65 mi/h

Speed = 78 miles/hour

Speed = 98.667 miles/hour

The average speed is calculated by dividing the distance traveled (2640 feet) by the time taken (thirty seconds) and then multiplying it by the speed limit (65 mi/h). This is based on the fact that there are 5280 feet in one mile. First, the distance (2640 feet) was converted to miles (0.5 miles). Then, the speed (0.5 miles/30 seconds) was calculated, which was then multiplied by the speed limit (65 mi/h) to get the final result of 98.667 miles/hour.

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To solve this inequality, we need to split it into two cases based on the sign of 2x - 6.

If 2x - 6 is nonnegative, then the inequality becomes 2x - 6 >= 4, which simplifies to x >= 5.

If 2x - 6 is negative, then the inequality becomes -(2x - 6) >= 4, which simplifies to x <= 1.

Therefore, the solution to the inequality is x <= 1 or x >= 5, which is the same as the second answer choice x ≤ 1 or x ≥ 5.

The other answer choices are not valid solutions because they do not include all values of x that satisfy the inequality.