Determine algebraically, the solution interval for the quadratic inequality 2x²-7x≤-3

Interval
Test Point
Substitution
True/False?
Solution:

The area to the right of the z-score 0.41 under the standard normal curve is approximately 0.3409.

To find the area to the right of the z-score 0.41 under the standard normal curve, we need to calculate the cumulative probability or area under the curve from 0.41 to positive infinity.

Since the standard normal distribution is symmetric around the mean (z = 0), we can use the property that the area to the right of a z-score is equal to 1 minus the area to the left of that z-score.

From the given z-table, we can look up the area to the left of 0.41, which is 0.6591.

The area to the right of 0.41 is then:

Area = 1 - 0.6591

Area = 0.3409

Therefore, the area to the right of the z-score 0.41 under the standard normal curve is approximately 0.3409.

This means that approximately 34.09% of the data falls to the right of the z-score 0.41 in a standard normal distribution.

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Answer:

A) No solutions

Step-by-step explanation:

First of all, we know that option B will always be incorrect. You cannot have two solutions. To illustrate this, try drawing two lines. You will find that they will either intersect once (one solution), or they will not intersect, (no solutions, parallel lines), or they are the same line and thus they will always intersect (infinitely many solutions).

With that in mind, let's solve the equation.

4x-4-5x=7-x+5

First, combine all like terms.

-x-4=12-x

Now add 4 to both sides to leave x by itself.

-x=16-x

This statement cannot be true. Therefore, this equation has no solutions (parallel lines. One line starts from 0, or the origin. That line is -x. The other line starts from 16. That line is -x+16.)

Hope this helps!

The future value of the loan, or the total amount due at the end of 6 months, is $9,450.

We can use the following formula to calculate the future value of a loan:

[tex]A = P + P * r * t[/tex]

Given: $9,000 principal (P).

10% interest rate (r) = 0.10

6 months is the time period (t).

When we enter these values into the formula, we get:

A=9,000+9,000*0.10*6/12

First, compute the interest portion:

Interest is calculated as = 9,000*0.10*6/12=450

We may now calculate the future value:

A=9,000+450=9,450

As a result, the loan's future value, or the total amount payable in 6 months, is $9,450.

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The rule that makes the machine work is *-5 + 6 * -5

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

4      -50

-8      10

-3     -15

A linear equation is represented as

y = mx + c

Using the points, we have

4m + c = -50

-8m + c = 10

Subtract the equations

So, we have

12m = -60

m = -5

Next, we have

-8 * -5 + c = 10

So, we have

c = 10 - 40

c = -30

This means that the operation is

-5x - 30

When expanded, we have

*-5 + 6 * -5

Hence, the rule that makes the machine work is *-5 + 6 * -5

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[tex] \sf \hookrightarrow \: {8}^{2} + {b}^{2} = {17}^{2} [/tex]

[tex] \sf \hookrightarrow \: 8 \times 8 + {b}^{2} = 17 \times 17[/tex]

[tex] \sf \hookrightarrow \: 8 \times 8 + {b}^{2} = 289[/tex]

[tex] \sf \hookrightarrow \: 64 + {b}^{2} = 289[/tex]

[tex] \sf \hookrightarrow \: {b}^{2} = 289 - 64[/tex]

[tex] \sf \hookrightarrow \: {b}^{2} = 225[/tex]

[tex] \sf \hookrightarrow \: b = \sqrt{225} [/tex]

[tex] \sf \hookrightarrow \: b = \sqrt{15 \times 15} [/tex]

[tex] \sf \hookrightarrow \: b = 15[/tex]

Answer:

The marginal pdf of X is fX(x) = x + 1/2

We need to integrate the joint pdf over the given region. This can be done as follows:

P(X < 1/2, Y > 1/4) = ∫∫[x + y] dx dy over the region 0 ≤ x ≤ 1/2 and 1/4 ≤ y ≤ 1

= ∫[x + y] dy from y = 1/4 to 1 ∫ dx from x = 0 to 1/2

= ∫[x + y] dy from y = 1/4 to 1 (1/2 - 0)

= ∫[x + y] dy from y = 1/4 to 1/2 + ∫[x + y] dy from y = 1/2 to 1 (1/2 - 0)

= ∫[x + y] dy from y = 1/4 to 1/2 + ∫[x + y] dy from y = 1/2 to 1/2

= ∫[x + y] dy from y = 1/4 to 1/2

= [(x + y)y] evaluated at y = 1/4 and y = 1/2

= [(x + 1/2)(1/2) - (x + 1/4)(1/4)]

= (1/2 - 1/4)(1/2) - (1/4 - 1/8)(1/4)

= (1/4)(1/2) - (1/8)(1/4)

= 1/8 - 1/32

= 3/32

Therefore, P(X < 1/2, Y > 1/4) = 3/32.

The marginal pdfs of X and Y can be done as follows:

For the marginal pdf of X:

fX(x) = ∫[x + y] dy over the range 0 ≤ y ≤ 1

= [xy + (1/2)y^2] evaluated at y = 0 and y = 1

= (x)(1) + (1/2)(1)^2 - (x)(0) - (1/2)(0)^2

= x + 1/2

Therefore, the marginal pdf of X is fX(x) = x + 1/2.

For the marginal pdf of Y:

fY(y) = ∫[x + y] dx over the range 0 ≤ x ≤ 1

= [xy + (1/2)x^2] evaluated at x = 0 and x = 1

= (y)(1) + (1/2)(1)^2 - (y)(0) - (1/2)(0)^2

= y + 1/2

Therefore, the marginal pdf of Y is fY(y) = y + 1/2.

To determine if X and Y are independent, we need to check if the joint pdf factors into the product of the marginal pdfs.

fX(x) * fY(y) = (x + 1/2)(y + 1/2)

However, this is not equal to the joint pdf f(x, y) = x + y. Therefore, X and Y are not independent.

To derive the conditional pdf of X given Y = y, we can use the formula:

f(xy) = f(x, y) / fY(y)

Here, we have f(x, y) = x + y from the joint pdf, and fY(y) = y + 1/2 from the marginal pdf of Y.

Therefore, the conditional pdf of X given Y = y is:

f(xy) = (x + y) / (y + 1/2)

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The function f(x) and the inverse function h(x) for which the function f(x) is defined by the values (0,3), (1,1), (2,-1) are f(x) = 3 -2x and h(x) = [tex]\frac{3 - x}{2}[/tex]

A function is a rule which takes each member x of a set and assigns, or maps it to the same value y known at its image.x → Function → yA letter such as f, g or h is often used to stand for a function.

The Function which squares a number and adds on a 3, can be written as f(x) = x2+ 5.

Let the linear function be f(x) = mx + cwhen x = 0, f(x) = 33 = m(0) + cTherefore, c = 3

when x = 1, f(x) = 11 = m(1) + c but c = 31 = m + 3

Therefore m = 1 - 3, which is -2

The linear equation f(x) = 3 - 2x

To solve for inverse function h(x)let y = 3 - 2xmaking x the subject of the equation2x = 3 - yx =[tex]\frac{3 - y}{2}[/tex]replacing x with h(x) and y with x, we haveh(x) = [tex]\frac{3 - x}{2}[/tex]

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Answer:

area = 8π/3

arc length = 4π/3

Step-by-step explanation:

θ = 60°

r = 4

Area of sector :

[tex]\frac{\theta}{360} \pi r^{2} \\\\=\frac{60}{360} \pi 4^{2} \\\\= \frac{1}{6} 16\pi \\\\= \frac{8}{3} \pi[/tex]

arc length:

[tex]\frac{\theta}{360} 2\pi r\\ \\= \frac{60}{360} 2(4)\pi \\\\= \frac{1}{6} 8\pi \\\\= \frac{4}{3} \pi[/tex]

Answer:

A ≈ 8.4 cm² , arc length ≈ 4.2 cm

Step-by-step explanation:

the area (A) of the sector is calculated as

A = area of circle × fraction of circle

  = πr² × [tex]\frac{60}{360}[/tex] ( r is the radius of the circle )

  = π × 4² × [tex]\frac{1}{6}[/tex]

  = [tex]\frac{16\pi }{6}[/tex]

  ≈ 8.4 cm² ( to 1 decimal place )

arc length is calculated as

arc = circumference of circle × fraction of circle

     = 2πr × [tex]\frac{60}{360}[/tex]

     = 2π × 4 × [tex]\frac{1}{6}[/tex]

     = [tex]\frac{8\pi }{6}[/tex]

     ≈ 4.2 cm ( to 1 decimal place )

The equation of a parabola with its vertex at the origin includes a negative coefficient 'a', the parabola opens downward. option B.

The equation y = ax² represents a parabola with its vertex at the origin. In this case, if the coefficient 'a' is negative, it determines the direction in which the parabola opens.

When 'a' is negative, the parabola opens downward. This means that the vertex, which is at the origin (0, 0), represents the highest point on the graph, and the parabola curves downward on both sides.

To understand this concept, let's consider the basic equation y = x², which represents a standard upward-opening parabola. As 'a' increases, the parabola becomes narrower. Conversely, when 'a' becomes negative, it flips the parabola upside down, resulting in a downward-opening parabola.

For example, if we have the equation y = -x², the negative coefficient causes the parabola to open downward. The vertex remains at the origin, but the shape of the parabola is now inverted.

In summary, when the equation of a parabola with its vertex at the origin includes a negative coefficient 'a', the parabola opens downward. This can be visually represented as a U-shape curving downward from the origin. So Optyion B is correct.

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The exact value of sec(-135°) is 1.

To find the exact value of sec(-135°), we need to use the relationship between secant and cosine functions.

The secant function is defined as the reciprocal of the cosine function:

sec(theta) = 1 / cos(theta).

We know that the cosine function has a period of 360°, which means that cos(theta) = cos(theta + 360°) for any angle theta.

In this case, we want to find sec(-135°). Since the cosine function is an even function (cos(-theta) = cos(theta)), we can rewrite sec(-135°) as sec(135°).

Now, let's focus on finding the value of cos(135°). The cosine function is negative in the second and third quadrants.

In the second quadrant, the reference angle is 180° - 135° = 45°. The cosine of 45° is equal to √2/2.

Therefore, cos(135°) = -√2/2.

Now, we can find sec(135°) using the reciprocal property:

sec(135°) = 1 / cos(135°).

Substituting the value of cos(135°), we have:

sec(135°) = 1 / (-√2/2).

To simplify further, we multiply the numerator and denominator by -2/√2:

sec(135°) = -2 / (√2 * -2/√2).

Simplifying the expression:

sec(135°) = -2 / -2,

sec(135°) = 1.

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The value of "c" such that the random variable cY has an x² distribution is 4.

To find the value of "c" such that the random variable cY has a chi-squared (x²) distribution, we need to consider the properties of the chi-squared distribution and the given expression for Y.

The chi-squared distribution with "k" degrees of freedom is obtained by summing the squares of "k" independent standard normal random variables. Each standard normal variable contributes one degree of freedom to the chi-squared distribution.

In the given expression for Y, we have two squared terms: (X₁ + X₃ + X₅ + X₇)² and (X₂ + X₁ + X₆ + X₈)². To obtain an x² distribution, we need to rewrite the expression in terms of squared standard normal random variables.

To achieve this, we can divide each squared term by its corresponding degrees of freedom and take the square root:

Y = (X₁ + X₃ + X₅ + X₇)² + (X₂ + X₁ + X₆ + X₈)²

= (1/4)(X₁ + X₃ + X₅ + X₇)² + (1/4)(X₂ + X₁ + X₆ + X₈)²

Now, we can rewrite Y as:

Y = (1/4)χ²₁ + (1/4)χ²₁

Here, χ²₁ and χ²₂ represent chi-squared random variables with 1 degree of freedom each.

To obtain an x² distribution, we need to make the coefficients of the chi-squared random variables equal to their degrees of freedom. In this case, we want the coefficient to be 1.

So, setting the coefficient of χ²₁ to 1, we get:

(1/4) = 1/c

Solving for "c", we find:

c = 4

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Answer:

a) 50

Step-by-step explanation:

The variance will not change as all the observations are increased uniformly.

Proof:

Variance formula:

[tex]s^{2} = \frac{\sum x_i^{2} }{n} -\frac{(\sum x_i)^{2} }{n^{2} }[/tex]

When the obervations are inc by 20,

[tex]s_1^{2} = \frac{\sum (x_i + 20)^{2} }{n} -\frac{(\sum (x_i + 20))^{2} }{n^{2} }\\\\=\frac{\sum(x_i^{2} + 2*20*x_i + 20^{2} )}{n} - \frac{(\sum x_i +20n)^{2} }{n^{2} } \\\\=\frac{\sum x_i^{2} + 40\sum x_i + 20^{2}n }{n} - \frac{(\sum x_i)^{2} +2*20n\sum x_i + 20^{2} n^{2} }{n^{2} } \\\\= \frac{\sum x_i^{2}}{n} - \frac{(\sum x_i)^{2}}{n^{2} } +\frac{40\sum x_i}{n} + 20^{2} - \frac{40\sum x_i}{n} - 20^{2}\\\\s_1^{2}= \frac{\sum x_i^{2}}{n} - \frac{(\sum x_i)^{2}}{n^{2} }\\\\=s^{2}[/tex]

Therefore variance doesn't change

The total surface area of the pyramid is option c [tex]72 cm^2[/tex].

The total surface area of a pyramid is given by the formula;S= ½Pl + BWhere B is the area of the base and P is the perimeter of the base.

To find the perimeter, add the length of all the sides of the base. Here, the base of the pyramid is a square with sides measuring 6 cm each.Therefore, its perimeter = 6 + 6 + 6 + 6 = 24 cm.

Now, to find the total surface area, we need to find the area of all four triangular faces. To find the area of one of the triangular faces, we can use the formula:

A = 1/2bhWhere b is the base of the triangle and h is the height.

To find the height, we can use the Pythagorean theorem:

[tex]h = \sqrt(6^2 - 3^2) = \sqrt(27) = 3 \sqrt(3)[/tex]

Therefore, the area of one of the triangular faces is:

A = 1/2bh = [tex]1/2(6)(3\sqrt(3)) = 9\sqrt(3)[/tex]

We have four triangular faces, so the total area of the triangular faces is:

[tex]4(9\sqrt(3)) = 36\sqrt(3)[/tex]

Finally, we can find the total surface area by adding the area of the base and the area of the triangular faces:

S = ½Pl + B = [tex]1/2(24)(3\sqrt(3)) + 6^2 = 36\sqrt(3) + 36 = 36(\sqrt(3) + 1).[/tex]

Therefore, the total surface area of the pyramid is 36(sqrt(3) + 1) cm², which is approximately 72 cm². Hence, the correct option is C. [tex]72 cm^2[/tex].

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Answer:

x = 4

Step-by-step explanation:

By property, if two tangents are drawn from an external point , then they are equal

⇒ 2x + 3 = 11

⇒ 2x = 11 - 3

⇒ 2x = 8

⇒ x = 8/2

⇒ x = 4

Answer:

x = 4

Step-by-step explanation:

To find the value of x, we can use the Two-Tangent Theorem.

The Two-Tangent Theorem states that if two tangent segments are drawn to a circle from the same external point, the lengths of the two tangent segments are equal.

Therefore:

[tex]\begin{aligned}AD &= AB\\\\2x+3&=11\\\\2x+3-3&=11-3\\\\2x&=8\\\\\dfrac{2x}{2}&=\dfrac{8}{2}\\\\x&=4\end{aligned}[/tex]

Therefore, the value of x is 4.

The surface area of a cylinder is 284m² and it's volume is 366.9m³

To find the surface area and volume of a cylinder, we need to know the radius (r) and height (h) of the cylinder. The formulas for the surface area (A) and volume (V) of a cylinder are as follows:

From the given question, the data are;

a. The surface area of the cylinder is;

SA = 2π(4)² + 2π(4)(7.3)

SA = 283.999≈284m²

b. The volume of the cylinder is

v = πr²h

v = π(4)²(7.3)

v = 366.9m³

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Answer:

[tex]y = |x - 2| + 3[/tex]

The correct answer is C.

Answer:

15

Step-by-step explanation:

5x = 4x + 3

x = 3

BC = 5x = 5(3) = 15

Answer: 15

Therefore, the value of x that solves the equation is 2/7, after eliminating the fractions and solving the resulting equation.

To eliminate fractions in the equation 6 - 3x + (1/2)x = x + 5, we can multiply each term by a number that will clear the denominators. In this case, the denominator is 2 in the term (1/2)x. The least common multiple (LCM) of 2 is 2 itself, so we can multiply each term by 2 to eliminate the fraction.

By multiplying each term by 2, we get:

2 * (6 - 3x) + 2 * ((1/2)x) = 2 * (x + 5)

Simplifying this expression, we have:

12 - 6x + x = 2x + 10

Now, the equation is free of fractions, and we can proceed to solve it.

Combining like terms, we have:

12 - 5x = 2x + 10

To isolate the variable terms, we can move the 2x term to the left side by subtracting 2x from both sides:

12 - 5x - 2x = 10

Simplifying further:

12 - 7x = 10

Next, we can move the constant term to the right side by subtracting 12 from both sides:

12 - 7x - 12 = 10 - 12

Simplifying again:

-7x = -2

Finally, we solve for x by dividing both sides by -7:

x = (-2) / (-7)

Simplifying the division of -2 by -7 gives us the solution:

x = 2/7

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Given the following diagram: We need to find the coordinates of triangle ABC, translate the triangle (-2, 5) and draw the new image on the grid above, and determine the amount each coordinate will move on the x-axis and y-axis during translation.

1. Coordinates of triangle ABC:A = (2, 6)B = (5, 8)C = (6, 3)2. Translation of triangle (-2, 5)The translation of a triangle can be done by adding or subtracting a constant value from the x-coordinates and y-coordinates of each vertex of the original triangle.

For example, if we want to translate a triangle by 3 units to the right and 2 units up, we would add 3 to the x-coordinates and add 2 to the y-coordinates of each vertex of the original triangle. Using this method, we can translate the triangle (-2, 5) as follows:

New coordinates of A = (2 + (-2), 6 + 5) = (0, 11)New coordinates of B = (5 + (-2), 8 + 5) = (3, 13)New coordinates of C = (6 + (-2), 3 + 5) = (4, 8)3. New image of triangle (-2, 5)The new image of the triangle (-2, 5) is shown in the following diagram:4. Amount each coordinate moves on x-axis During translation, each coordinate moves 2 units to the right (from -2 to 0).5. Amount each coordinate moves on y-axis During translation, each coordinate moves 6 units up (from 5 to 11).

Therefore, the coordinates of the translated triangle image are (0, 11), (3, 13), and (4, 8).

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Answer:she will actually need 1 dollar because all of that would be 31 dollars.

Step-by-step explanation:

3 pounds of lemons= $6

1 box of rice= $4

7 frozen diners= $21

6+4=10

10+21=31

The row operation on the matrix [tex]\left[\begin{array}{ccc|c}2&0&0&16\\0&8&0&3\\0&0&5&6\end{array}\right][/tex] is [tex]\left[\begin{array}{ccc|c}1&0&0&8\\0&8&0&3\\0&0&5&6\end{array}\right][/tex]

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

[tex]\left[\begin{array}{ccc|c}2&0&0&16\\0&8&0&3\\0&0&5&6\end{array}\right][/tex]

The row operation is given as

1/2R₁

This means that we divide the entries on the first row by 2

Using the above as a guide, we have the following:

[tex]\left[\begin{array}{ccc|c}2&0&0&16\\0&8&0&3\\0&0&5&6\end{array}\right] = \left[\begin{array}{ccc|c}1&0&0&8\\0&8&0&3\\0&0&5&6\end{array}\right][/tex]

Hence, the row operation on the matrix is [tex]\left[\begin{array}{ccc|c}2&0&0&16\\0&8&0&3\\0&0&5&6\end{array}\right] = \left[\begin{array}{ccc|c}1&0&0&8\\0&8&0&3\\0&0&5&6\end{array}\right][/tex]

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Answer:

(x + 13)(x - 7) = 69

x² + 6x - 91 = 69

x² + 6x - 160 = 0

(x + 16)(x - 10) = 0

x = 10, so the area of the original square is 100 m².

The distance John can ride his bike in 12 minutes is approximately 1.6 miles.

To find out the distance John can ride his bike in 12 minutes, we can use the information given about his rate of riding.

We are told that John can ride his bike 4 miles in 30 minutes. This implies that his rate of riding is 4 miles per 30 minutes.

To calculate the distance John can ride in 12 minutes, we need to determine the proportion of time he is riding compared to the given rate.

We can set up a proportion to solve for the unknown distance:

(4 miles) / (30 minutes) = (x miles) / (12 minutes)

Cross-multiplying, we get:

30 minutes * x miles = 4 miles * 12 minutes

30x = 48

Now, we can solve for x by dividing both sides of the equation by 30:

x = 48 / 30

Simplifying the fraction, we have:

x = 8/5

So, John can ride his bike approximately 1.6 miles in 12 minutes, at his current rate.

Therefore, the distance John can ride his bike in 12 minutes is approximately 1.6 miles.

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The length of the unknown side x using the concept of similar triangles is: x = 4

Similar triangles are referred to as triangles that have the same shape but different sizes. All equilateral triangles and squares of any side length are examples of similar objects. In other words, if two triangles are similar, their corresponding angles are the same and their corresponding side proportions are the same.

Now, we are told that triangle PQR is similar to Triangle STU and as such their corresponding sides are similar and therefore to find the missing side x, we have:

12/x = 15/5

x = (12 * 5)/15

x = 60/15

x = 4

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The correct statement is D. Hannah's unit rate is 1 fewer word per minute than Becky's unit rate.

To compare the unit rates, we need to determine the rate at which each person types words per minute.

For Hannah, the equation y = 11x represents her typing speed, where y is the number of words and x is the number of minutes. This means that Hannah types 11 words per minute (11 words/minute).

Looking at Becky's graph, we can determine her unit rate by calculating the change in the number of words divided by the change in time.

The change in words is 48 - 2 = 46, and the change in time is 3 - 2 = 1. So, Becky's unit rate is 46 words per minute (46 words/minute).

Comparing the unit rates:

Hannah's unit rate: 11 words/minute

Becky's unit rate: 46 words/minute

Therefore, Hannah's unit rate is 35 words per minute less than Becky's unit rate.

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Hence, the arrangements to the quadratic equation  (3x-1)(x-2) = 5x + 2 are x = and x = 4.

To unravel the quadratic equation  (3x-1)(x-2) = 5x + 2, let's to begin with grow the cleared out side of the equation:

(3x - 1)(x - 2) = 5x + 2

Growing the condition:

3x^2 - 6x - x + 2 = 5x + 2

Streamlining the condition:

3x^2 - 7x + 2 = 5x + 2

Another, let's move all terms to one side of the condition:

3x^2 - 7x - 5x + 2 - 2 =

Combining like terms:

3x^2 - 12x =

Presently, we have a quadratic condition in standard shape: ax^2 + bx + c = 0, where a = 3, b = -12, and c = 0.

To fathom the quadratic equation, able to calculate out the common calculate of x:

x(3x - 12) =

From this equation, we are able see that the esteem of x can be or unravel for 3x - 12 = 0:

3x - 12 =

Including 12 to both sides:

3x = 12

Isolating both sides by 3:

x = 4

Hence, the arrangements to the condition (3x-1)(x-2) = 5x + 2 are x = and x = 4.

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Hello!

[tex]x^2 - 3x + \dfrac{9}{4} = 0\\\\4x^2 - 12x + 9 = 0\\\\\\x = \dfrac{-b\±\sqrt{b^2 - 4ac} }{2a} \\\\\\x = \dfrac{-(-12)\±\sqrt{(-12)^2 - 4 \times 4 \times 9} }{2 \times 4} \\\\\\\\boxed{x = \dfrac{3}{2} }[/tex]

The sum of the two time durations is 7 hours, 46 minutes, and 37 seconds.

To add the given time durations, we start by adding the seconds:

12 sec + 25 sec = 37 sec.

Since 60 seconds make a minute, we carry over any excess seconds to the minutes place, which gives us a total of 37 seconds. Moving on to the minutes, we add 30 min + 16 min = 46 min.

Again, we carry over any excess minutes to the hours place, resulting in a total of 46 minutes.

Finally, we add the hours: 5 hr + 2 hr = 7 hr.

Thus, the sum of the two time durations is 7 hours, 46 minutes, and 37 seconds.

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Answer:

Sure. First, we need to find the inverse function of f(x). We can do this by using the following steps:

1. Let y = f(x).

2. Solve the equation y = 4x3 + 6x - 10 for x.

3. Replace x with y in the resulting equation.

This gives us the following inverse function:

```

f^-1(y) = (-1 + sqrt(1 + 12y)) / 2

```

Now, we need to find f^-1′ (0). This is the derivative of the inverse function evaluated at y = 0. We can find this derivative using the following steps:

1. Use the chain rule to differentiate f^-1(y).

2. Evaluate the resulting expression at y = 0.

This gives us the following:

```

f^-1′ (0) = (3 * (1 + 12 * 0) ^ (-2/3)) / 2 = 1.5

```

Therefore, f^-1′ (0) = 1.5.

Step-by-step explanation:

The correct linear inequality represented by the graph is:

y < -2x + 3. Option D

To determine which linear inequality is represented by the graph of the dashed straight line with a negative slope and going through (0, 3) and (2, -1), we can start by finding the slope of the line.

The slope of a line can be calculated using the formula:

m = (y2 - y1) / (x2 - x1).

Using the coordinates (0, 3) and (2, -1), we have:

m = (-1 - 3) / (2 - 0),

m = -4 / 2,

m = -2.

So, we know that the slope of the line is -2.

Next, we need to determine the y-intercept of the line. To do this, we can use the slope-intercept form of a linear equation: y = mx + b, where m is the slope and b is the y-intercept.

Using the point (0, 3), we can substitute the coordinates into the equation and solve for b:

3 = -2(0) + b,

3 = b.

Therefore, the y-intercept is 3.

Now that we have the slope and y-intercept, we can write the equation of the line in slope-intercept form:

y = -2x + 3.

Since we are shading everything to the left of the line, we want the region where y is less than the line. Option D is correct.

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