Apyrotechnician plans for two fireworks to explode together at the same
height in the air. They travel at speeds shown below. Firework B is launched
0.25 s before Firework A. How many seconds after Firework B launches will
both fireworks explode?
Firework A Firework B
320 fts
240 fuis
Both fireworks will explode _?seconds after Firework B launches.

Based on the calculations, the depth of tent is equal to 12 feet.

Based on the diagram (see attachment) and information provided, we can logically deduce the following parameters (points):

Next, we would determine the height of the right-angled triangle (ADC) by applying Pythagorean theorem:

AC² = AD² + DC²

AD² = AC² - DC²

AD² = 5² - 3²

AD² = 25 - 9

AD² = 16

AD = √16

AD = 4 feet.

Also, we would determine the area of the triangle (ABC):

Area = 1/2 × b × h

Where:

Substituting the given parameters into the formula, we have;

Area = 1/2 × 6 × 4

Area = 12 feet².

Depth of tent = 3 × height of ADC

Depth of tent = 3 × 4

Depth of tent = 12 feet.

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The answer is y = -5.5x - 52.

First, let's find the slope of the line.

m = Δy/Δx

Now, substitute the slope and one of the given points in the point slope equation.

y - y₁ = m (x - x₁)

Answer: y=-5,5x-52.

Step-by-step explanation:

                        Equation of a straight line

                               [tex]\displaystyle\\\boxed {\frac{x-x_1}{x_2-x_1}=\frac{y-y_1}{y_2-y_1} }[/tex]

[tex](-10;3)\ \ \ \ \ (-8;-8).\\\displaystyle\frac{x-(-10)}{-8-(-10)} =\frac{y-3}{-8-3}\\ \frac{x+10}{-8+10}=\frac{y-3}{-11} \\ \frac{x+10}{2}=\frac{y-3}{-11} \\-11*(x+10)=2*(y-3)\\-11x-110=2y-6\\2y=-11x-104\ |:2\\y=-5,5x-52.[/tex]

Answer:

36

Step-by-step explanation:

Honestly, I'd work this out visually (see image) by drawing a rectangle and dividing it up. But if you need to show the math, here's how to do it:

Let x = the number of students in the homeroom

1/3 x have drama

1/4 (x - 1/3x) have band, and we are told 6 students have band, so

1/4 (x - 1/3 x) = 6

1/4 (2/3 x) = 6    

2/12 x = 6          

1/6 x = 6            

6 × 1/6 x = 6 × 6

1 x = 6 × 6            

x = 36

Answer:

36

Step-by-step explanation:

Let n be the total number of students in Mr. Short's homeroom. Then, working backward, we see that 

6 = 1/4(1 - 1/3)n = 1/4(2/3)n = 1/6n,$

so n = 6 x 6 = 36 students.

The surface integral of the given surface [tex]z=\frac{2}{3}(x^\frac{3}{2}+y^\frac{3}{2})[/tex] in the intervals 0 ≤ x ≤ 4, 0 ≤ y ≤ 1 is 7.36615.

To evaluate the surface integral, the formula applied is

[tex]S=\int\limits^b_a\int\limits^d_c {f(x,y)} \ dy\ dx[/tex]

⇒ S = [tex]\int\limits^b_a \int\limits^d_c {\sqrt{1+(\frac{dz}{dx})^2+(\frac{dz}{dy})^2 } } \, dy \, dx[/tex]

Where the differentiation is partial differentiation and f(x, y) = z.

The given function is

f(x, y) = [tex]z=\frac{2}{3}(x^\frac{3}{2}+y^\frac{3}{2})[/tex]

Then, applying the partial differentiation w.r.t x and y respectively,

dz/dx = [tex]\frac{2}{3}(\frac{3}{2})x^{1/2}[/tex] = [tex]x^{1/2}[/tex]

dz/dy = [tex]\frac{2}{3}(\frac{3}{2})y^{1/2}[/tex] = [tex]y^{1/2}[/tex]

Then,

we have surface integral  formula as

S = [tex]\int\limits^b_a \int\limits^d_c {\sqrt{1+(\frac{dz}{dx})^2+(\frac{dz}{dy})^2 } } \, dy \, dx[/tex]

On substituting,

⇒ S = [tex]\int\limits^4_0 {\int\limits^1_0 {\sqrt{1+(x^{1/2})^2+(y^{1/2})^2} \, dy } \, dx[/tex]

⇒ S = [tex]\int\limits^4_0 {\int\limits^1_0 {\sqrt{1+x+y} \, dy } \, dx[/tex]

On integrating w.r.t y

⇒ S = [tex]\int\limits^4_0 {\frac{2}{3}(1+x+y)^{3/2}} \,|_0^1 dx[/tex]

Applying limits,

⇒ S = [tex]\int\limits^4_0 {\frac{2}{3}[(1+x+1)^{3/2}-(1+x+0)^{3/2} } \, dx[/tex]

⇒ S = [tex]\frac{2}{3} \int\limits^4_0 {[(x+2)^{3/2}-(x+1)^{3/2}}] \, dx[/tex]

Again applying the integration w.r.t x

⇒ S = [tex]\frac{2}{3}[\frac{2}{5}(x+2)^{5/2}-\frac{2}{5}(x+1)^{5/2}]|_0^4[/tex]

Applying the limits and simplifying,

⇒ S = [tex]\frac{4}{15}[(4+2)^{5/2}-(4+1)^{5/2}-(0+2)^{5/2}+(0+1)^{5/2}][/tex]

⇒ S = [tex]\frac{4}{15}[1+36\sqrt{6}-25\sqrt{5}-4\sqrt{2}][/tex]

∴ S = 7.36615

Therefore, the surface integral of the given function is 7.36615.

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The area of the given parallelogram is 8 square units

A parallelogram is a 2 dimensional figure with 4 sides and angles. From the given diagram, the opposite sides are equal and the sum of the adjacent angles are supplementary

The formula for calculating the area of a parallelogram is expressed as:

A = base * height

From the given diagram

Base = (4, 0) = 4 units

Height. = (2, 0) = 2 units

Substitute the given values into the formula to have:

A = 4 units * 2 units

A = 8 square units

Hence the area of the given parallelogram is 8 square units

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The discount to be increased so that he can gain only 15% is 8%.

How much is the discount to be increased so that he can gain only 15%:

The cost price of the goods - y.

The selling price of the goods - x.

The selling price is

1.2y (100% + 20% = 120%).

The refrigerator was sold at 13% VAT at Rs. 10848:

=> 1.2y + 13%(1.2y) = 10848

=> 1.2y + 0.156y = 10848

=> 1.356y = 10848

=> y = 8000

Thus, the cp is Rs 8000.

The selling price is:

=> 0.96x = 1.2y

=> 0.96x = 1.2(8000)

=> x = 10000

The selling price is Rs. 10000

For a profit of 15%, that is 1.15(8000) = 9200, the discount is:

(1 - discount)10000 = 9200

1 - discount = 0.92

Discount = 1 - 0.92 = 0.08 or 8%

How much is the discount to be increased so that he can gain only 15%: Discount=8%

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The mean of the data set A is 4.58 while the mean of data set B is 5.

An equation is an expression that shows the relationship between two or more numbers and variables.

The mean of a set of numbers is the average of that group of number. It is the ratio of the sum of the data to the total number of data.

Mean of data set A = [(3 * 1) + (4 * 3) + (5 * 2) + (6 * 2) + (7 * 3) + (8 * 1)] / (1 + 3 + 2 + 2 + 3 + 1) = 4.58

Mean of data set B = [(4 * 2) + (5 * 2) + (6 * 2)] / ( 2 + 2 + 2) = 5

The mean of the data set A is 4.58 while the mean of data set B is 5.

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Translate the triangle by [tex](^{-2}_{-4})[/tex] means translating 2 units left and 4 units down.

Transformation is the movement of a point from its initial location to a new location. Types of transformations are reflection, translation, rotation and dilation.

Rigid transformation preserves the shape and size of the figure. Reflection, translation, rotation are rigid transformations.

Translate the triangle by [tex](^{-2}_{-4})[/tex] means translating 2 units left and 4 units down.

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Each element in the range of g(x) is 3 less than the corresponding element in the range of f(x).

Answer:

See below.

In short, by having vertical shift will affect range from -1 ≤ y ≤ 1 to -1 + a ≤ y ≤ 1 + a

Step-by-step explanation:

Generally, a cosine function without vertical shift will always have range equal to -1 ≤ y ≤ 1

If there is given vertical shift, our range will change to -1 + a ≤ y ≤ 1 + a

An example is if we are given the function of cos(x), this always has range of -1 ≤ y ≤ 1 because there is no vertical shift.

But if we have cos(x) + 1, we have vertical shift which is 1. Then the range will be -1+1 ≤ y ≤ 1+1 which equals to 0 ≤ y ≤ 2.

Hence, the function of -5cos(2x) has only range of -1 ≤ y ≤ 1 but the function of -5cos(2x) - 3 will have range of -1-3 ≤ y ≤ 1-3 which equals to -4 ≤ y ≤ -2

Answer:

8

Step-by-step explanation:

I belive what they mean is how many dots thier are i am not complety sure though but just by counting them up i came up with 8.

hope this helped

The solutions to the questions are given below

a)

sample(n) word length sample mean

1                    5,4,4,2    3.75

2                    3,2,3,6     3.5

3                         5,6,3,3    4.25

b)R =0.75

c)

The table below shows sample means in the table.

sample(n) word length sample mean

1                    5,4,4,2    3.75

2                    3,2,3,6     3.5

3                         5,6,3,3    4.25

b)

Generally, the equation for is  mathematically given as

variation in the sample means

R =maximum -minimum

R=4.25-3.5

R =0.75

c)

In conclusion, In most cases, the mean of many samples will provide a more accurate estimate than the mean of a single sample.

They may have a higher level of confidence in their estimate if the range of the sample means is closer to 0 than it is now.

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Answer:

C.  They most likely used an observational study asking students to keep track of their TV habits and weight.

A.  This is an experiment because a treatment was applied to a group.

Step-by-step explanation:

According to our question:

1. The study concluded that there is a relation between watching TV and the weight gain in college freshmen.

Now, this is possible when a group of college students were asked to participate to watch TV for more than 1 hour for a specific amount of time and have a record of their weights.

So, the study is an observational study where the habits and weights of college students were observed.

Hence, option C is correct.

2. The students were divided onto two groups where one group used the computer program and the other did not to check the improvement in the reading levels.

This means that the students were given a procedure to follow during the study in order to see the results..

So, the study is an experiment where a treatment is applied to the group of students.

Hence, option A is correct.

Answer: C

For the people who want only one answer and not 2 for the question if you can only submit one answer :)

Step-by-step explanation:

let's name the variables for the age of the people as the people themselves :

orlando = omar + 12

carl = 3×orlando

carl + orlando + omar = 68

using the first equation in the second equation, and then that result and the first equation in the third equation :

carl = 3×(omar + 12)

3×(omar + 12) + (omar + 12) + omar = 68

3×omar + 36 + omar + 12 + omar = 68

5×omar + 48 = 68

5×omar = 20

omar = 4

orlando = omar + 12 = 4 + 12 = 16

carl = 3×orlando = 3×16 = 48

the ratio omar : carl : orlando

4 : 48 : 16 = 1 : 12 : 4

Using the Fundamental Counting Theorem, it is found that there are 1024 ways of delivering the letters.

It is a theorem that states that if there are n things, each with [tex]n_1, n_2, \cdots, n_n[/tex] ways to be done, each thing independent of the other, the number of ways they can be done is:

[tex]N = n_1 \times n_2 \times \cdots \times n_n[/tex]

In this problem, each house has a correct letter, however the letter cannot be used for the house, hence the parameters are given as follows:

n1 = n2 = n3 = n4 = n5 = 5 - 1 = 4.

Thus the number of ways is:

N = 4 x 4 x 4 x 4 x 4 = 4^5 = 1024.

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April's work in checking for the function, [tex]f(x) = x^3 - 1[/tex], to be even, odd, or neither, is incorrect. She did a mistake in step 2, where she took [tex]f(-x) = - f(x)[/tex], which was not true, since [tex]f(-x) = -x^3 - 1[/tex], whereas [tex]- f(x) = -x^3 + 1[/tex].

A function f(x) is said to be odd or even when it completes the condition:

In the question, we are given that April was asked to determine whether  [tex]f(x) = x^3 - 1[/tex], is even, odd, or neither, and are shown her work.

We are asked whether April's work is correct, and if not, what is the first step where April made a mistake.

To check for a function is even or odd, we first calculate f(-x).

April's first step was also finding f(-x), which she found accurately as  [tex]f(-x) = -x^3 - 1[/tex].

Next, we need to check whether f(-x) is equal to f(x), -f(x), or neither, to check if the function is even, odd, or neither, respectively.

April, also choose the right step of checking, but did a mistake in checking, since f(-x) ≠ f(x), and also, f(-x) ≠ - f(x), since, [tex]- f(x) = -(x^3 - 1)[/tex] = [tex]-x^3 + 1[/tex], which is not equal to f(-x).

Thus, April's work in checking for the function, [tex]f(x) = x^3 - 1[/tex], to be even, odd, or neither, is incorrect. She did a mistake in step 2, where she took f(-x) = - f(x), which was not true, since [tex]f(-x) = -x^3 - 1[/tex], whereas [tex]- f(x) = -x^3 + 1[/tex].

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Answer:

See image

Step-by-step explanation:

Using a graphing calculator, first press the Y= button at the top left.

Then enter the equation given in the question.

Next, press 2nd then graph, and you will get the table. Put in whatever values for x to get y.

So your answer is 48, 18, 0, -6, 0, -18, 0, 48

Answer:

6 children passes

Step-by-step explanation:

a = adult and c = children

25a + 15c = 440           and          a + c = 20

                                                     c = 20 - a

25a + 15(20-a) = 440

25a + 300 - 15a = 440

25a - 15a = 440 - 300

10a = 140

a = 140/10

a = 14 and this is 14 adult tickets

Now we plug the a value into the equation to find the c value

14 + c = 20

c = 20 - 14

c = 6

Verify answer:

25(14) = 15(6) = 440

350 + 90 = 440

440 = 440

Answer: [tex]\Large\boxed{a_n=6n-4}[/tex]

Step-by-step explanation:

Given sequence

2, 8, 14, 20, 26

Determine the pattern

2 + 6 = 8

8 + 6 = 14

14 + 6 =20

20 + 6 = 26

For each term, add 6 to get the next term

Determine the type of sequence

Since it continuously adds 6, the common difference is 6, which means it is an arithmetic sequence

Given the arithmetic sequence formula

aₙ = a₁ + d (n - 1)

Substitute values into the formula

aₙ = (2) + (6) (n - 1)

aₙ = 2 + 6 (n - 1)

aₙ = 2 + 6n - 6

aₙ = 2 - 6 + 6n

[tex]\Large\boxed{a_n=6n-4}[/tex]

Hope this helps!! :)

Please let me know if you have any questions

Given sequence :-

Solution :-

We need to calculate the nth term of the given sequence.

We know that,

Applying the values here.

>> tn = 2 + (n - 1) 6

>> tn = 2 + (n - 1) × 6

>> tn = 2 + 6n - 6

>> tn = 6n - 4

Therefore, nth term of the sequence is (6n - 4).

The value of dividing x^2-3x+9 by x-2 is x + 3

There are several ways to divide polynomial functions; some of these ways include

The expression for the polynomial division is given as:

x^2-3x+9/x-2

To divide polynomial functions, we make use of the division by factorization method

Start by expanding the numerator of the polynomial division

x^2-3x+9/x-2 = x^2 + 3x - 6x + 9/x-2

Factorize the equation

x^2-3x+9/x-2 = x(x + 3) - 2(x + 3)/x - 2

Factor out x + 3 from the numerator

x^2-3x+9/x-2 = (x- 2)(x + 3)/x - 2

Cancel out the common factors

x^2-3x+9/x-2 = x + 3

Hence, the value of dividing x^2-3x+9 by x-2 is x + 3

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Using a quadratic regression equation, it is found that the prediction of volunteers in year 10 is of:

A. 47.

To find the equation, we need to insert the points (x,y) in the calculator.

In this problem, we have that the function initially increases, then it decreases, which means that it is a quadratic function. The points (x,y) to  be inserted into the calculator are given as follows:

(1, 20), (2,17), (3, 16), (4,16), (5,18), (6,21), (7,25), (8,31)

Inserting these points into the calculator, the prediction of y for a value of x is given as follows:

y = 0.702x^2 - 4.726x + 23.857

Hence, when x = 10, the prediction is:

y = 0.702 x 10² - 4.726 x 10 + 23.857

y = 47.

Hence option A is correct.

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There were 204 participants from the three schools

The given parameters are:

X = 12 + Y

X = Z - 18

Girls = Boys + 50

The ratios of the number of boys to girls in the schools are:

1 : 2

1 : 5

2 : 1

This means that:

So, we have:

Boys = 1/3X + 1/6Y + 2/3Z

Girls = 2/3X + 5/6Y + 1/3Z

Substitute the above equations in Girls = Boys + 50

2/3X + 5/6Y + 1/3Z = 1/3X + 1/6Y + 2/3Z + 50

Evaluate the like terms

1/3X + 2/3Y - 1/3Z = 50

Multiply through by 3

X + 2Y - Z = 150

So, we have:

X = 12 + Y

X = Z - 18

X + 2Y - Z = 150

Substitute X = 12 + Y in X = Z - 18 and X + 2Y - Z = 150

12 + Y = Z - 18 ⇒ Y - Z = 30

12 + Y + 2Y - Z = 150 ⇒ 3Y - Z = 138

Subtract Y - Z = 30 from the equation 3Y - Z = 138 to eliminate Z

2Y = 108

Divide by 2

Y = 54

Substitute Y = 54 in X = 12 + Y

X = 12 + 54

X = 66

Substitute X = 66 in X = Z - 18

66 = Z - 18

Solve for Z

Z = 66 + 18

Z = 84

So, the total number of participants from the 3 schools is

Total = X + Y + Z

This gives

Total = 66 + 54 + 84

Evaluate the sum

Total = 204

Hence, there were 204 participants from the three schools

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The length of the rectangle is 77cm.

According to the statement

we have given that the area of a painting is 4081 c m squared. and width of the painting is 53 cm.

And we have to find the Length of the painting.

So, For this purpose

Area of rectangle is the region occupied by a rectangle within its sides. It depends on its sides.

So, the formula to find the area of rectangle is

Area = Length*Breadth

4081 = L*53

L = 4081/53

L = 77 cm.

So, The length of the rectangle is 77cm.

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Using the z-distribution, it is found that since the p-value is less than 0.05, there is evidence to support the claim that the average cost of private universities are decreasing.

At the null hypothesis, it is tested if the average cost is still of $50,000, that is:

[tex]H_0: \mu = 50000[/tex]

At the alternative hypothesis, it is tested if the average cost is decreasing, that is:

[tex]H_1: \mu < 50000[/tex]

The test statistic is:

[tex]z = \frac{\overline{x} - \mu}{\frac{\sigma}{\sqrt{n}}}[/tex]

In which:

The parameters for this problem are:

[tex]\overline{x} = 49450, \mu = 50000, \sigma = 2500, n = 100[/tex]

Hence:

[tex]z = \frac{\overline{x} - \mu}{\frac{\sigma}{\sqrt{n}}}[/tex]

[tex]z = \frac{49450 - 50000}{\frac{2500}{\sqrt{100}}}[/tex]

z = -2.2.

Using a z-distribution calculator, for a left-tailed test, as we are testing if the mean is less than a value, with z = -2.2, the p-value is of 0.0139.

Since the p-value is less than 0.05, there is evidence to support the claim that the average cost of private universities are decreasing.

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Answer:

204

Step-by-step explanation:

6*10=60

6*6=36

6*8=48

2((10*6)/2)=60

60+60+36+48=204

After 16 days, the total homework questions that have been assigned would be 80 questions.

Ratio expresses the equivalent relationship between two numbers. It shows the frequency of the number of times that one value is contained within other value(s).

A ratio table can be described as table that shows equivalent values between two numbers.

The first step is to determine the ratio between the number of days and the total number of questions.

Ratio = money saved / money earned

8/10 =  total number of questions / number of days

55 / 11 = 5

Now, multiply the ratio gotten in the previous step by the number of days

Total number of homework questions = number of x ratio

5 x 16 = 80 questions

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Answer:

[tex]4\ln \left| \dfrac{1}{3}\sqrt{9+(\ln x)^2} + \dfrac{1}{3}\ln x \right|+\text{C}[/tex]

Step-by-step explanation:

Fundamental Theorem of Calculus

[tex]\displaystyle \int \text{f}(x)\:\text{d}x=\text{F}(x)+\text{C} \iff \text{f}(x)=\dfrac{\text{d}}{\text{d}x}(\text{F}(x))[/tex]

If differentiating takes you from one function to another, then integrating the second function will take you back to the first with a constant of integration.

Given indefinite integral:

[tex]\displaystyle \int \dfrac{4}{x\sqrt{9+(\ln(x))^2}}\:\:\text{d}x[/tex]

Rewrite 9 as 3²:

[tex]\implies \displaystyle \int \dfrac{4}{x\sqrt{3^2+(\ln(x))^2}}\:\:\text{d}x[/tex]

Integration by substitution

[tex]\boxed{\textsf{For }\sqrt{a^2+x^2} \textsf{ use the substitution }x=a \tan\theta}[/tex]

[tex]\textsf{Let } \ln x=3 \tan \theta[/tex]

[tex]\begin{aligned}\implies \sqrt{3^2+(\ln x)^2} & =\sqrt{3^2+(3 \tan\theta)^2}\\ & = \sqrt{9+9\tan^2 \theta}\\ & = \sqrt{9(1+\tan^2 \theta)}\\ & = \sqrt{9\sec^2 \theta}\\ & = 3 \sec\theta\end{aligned}[/tex]

Find the derivative of ln x and rewrite it so that dx is on its own:

[tex]\implies \ln x=3 \tan \theta[/tex]

[tex]\implies \dfrac{1}{x}\dfrac{\text{d}x}{\text{d}\theta}=3 \sec^2\theta[/tex]

[tex]\implies \text{d}x=3x \sec^2\theta\:\:\text{d}\theta[/tex]

Substitute everything into the original integral:

[tex]\begin{aligned} \implies \displaystyle \int \dfrac{4}{x\sqrt{9+(\ln(x))^2}}\:\:\text{d}x & = \int \dfrac{4}{3x \sec \theta} \cdot 3x \sec^2\theta\:\:\text{d}\theta\\\\ & = \int 4 \sec \theta \:\: \text{d}\theta\end{aligned}[/tex]

Take out the constant:

[tex]\implies \displaystyle 4 \int \sec \theta\:\:\text{d}\theta[/tex]

[tex]\boxed{\begin{minipage}{7 cm}\underline{Integrating $\sec kx$}\\\\$\displaystyle \int \sec kx\:\text{d}x=\dfrac{1}{k} \ln \left| \sec kx + \tan kx \right|\:\:(+\text{C})$\end{minipage}}[/tex]

[tex]\implies 4\ln \left| \sec \theta + \tan \theta \right|+\text{C}[/tex]

[tex]\textsf{Substitute back in } \tan\theta=\dfrac{1}{3}\ln x :[/tex]

[tex]\implies 4\ln \left| \sec \theta + \dfrac{1}{3}\ln x \right|+\text{C}[/tex]

[tex]\textsf{Substitute back in } \sec\theta=\dfrac{1}{3}\sqrt{9+(\ln x)^2}:[/tex]

[tex]\implies 4\ln \left| \dfrac{1}{3}\sqrt{9+(\ln x)^2} + \dfrac{1}{3}\ln x \right|+\text{C}[/tex]

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The solution to the inequalities 3x + 5 < 6x + 8 and 4x + 3 < 7x + 5 is x > -1 and x > -2/3 respectively

Inequality signs are;

3x + 5 < 6x + 8

3x - 6x < 8 - 5

-3x < 3

x > 3/-3

The inequality sign is flipped from < to greater than because of the division with negative value.

x > -1

2. 4x + 3 < 7x + 5

4x - 7x < 5 - 3

-3x < 2

The inequality sign is flipped from < to greater than because of the division with negative value.

x > -2/3

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