Find the domain of the equation.
ln[tex]\frac{x^3+5x^2+3x-9}{x^2-x-2}[/tex]

The second root is in the form [tex]a+\sqrt{b}[/tex].

Polynomial roots are the solutions to any given polynomial for which the value of the unknown variable must be determined. We can evaluate the value of polynomial to zero if we know the roots. A polynomial of degree 'n' in variable x is an expression of the form [tex]a^nx^n + a^{n-1}x^{n-1} +....+ a^1x^1 + a^0[/tex], where each variable has a constant accompanying it as its coefficient. The term refers to each variable in the expression that is separated by an addition or subtraction symbol. The degree of a polynomial is defined as the maximum power of a polynomial variable.

Given,

one root of polynomial is [tex]a-\sqrt{b}[/tex]

If a given root is complex, its conjugate is the other root. As a result, if a+ib is a root, a-ib is another a root.

If the root given is a surd of the form [tex]a+\sqrt{b}[/tex], then [tex]a-\sqrt{b}[/tex] is another root as well.

If the given root is rational, we can apply the product of the roots rule.

Thus, The polynomial's second root will be [tex]a+\sqrt{b}[/tex].

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The point that must be on the graph of y=f(2x+1)+3 is (-6, 0)

An ordered pair is a composition of the x coordinate (abscissa) and the y coordinate (ordinate), having two values written in a fixed order within parentheses.

Given that, (-2,3) is on the graph of y=f(x), we need to find a point that must be graphed of y=f(2x+1)+3

3 = f(2X(-3)+1) + 3

0 = f(-6)

Hence, the point on the graph is (-6, 0)

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x>2 might the original value of both goods have been if the initial price had been a natural number and the pen had been more expensive than the notebook.

Given that,

Pens and notebooks used to cost the same, however recently the price of the pen went up by $4 while the price of the notebook went down by $6 from its previous value of $5.

We have to find what might the original value of both goods have been if the initial price had been a natural number and the pen had been more expensive than the notebook.

We know that,

We get the equation as

x+4>5x-6

By solving the x

The inequality with x<2.5

So, natural number, n=2

x>2

Therefore, x>2 might the original value of both goods have been if the initial price had been a natural number and the pen had been more expensive than the notebook.

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The test statistic is  3.10

Hypothesis :

In statistics, hypothesis testing is used to identify the variance in the group of data that results from genuine variation. Based on the presumptions, the sample data are taken from the population parameter. The hypothesis can be divided into many categories. A hypothesis is described in statistics as a formal statement that explains the relationship between two or more variables belonging to the specified population. It aids the researcher in converting the stated issue into an understandable justification for the study's findings. It provides examples of various experimental designs and guides the investigation of the research procedure.

Complete question:

In this scenario what is the test statistic? • The digital marketing specialist would like to test the claim that the percent of customers who use online coupons when making an online purchase is different than 75% Sample size = 80 online customers • Sample proportion = 0.90 Calculate the test statistic using the formula: P-Po where • p = sample proportion, n sample size, and po - population proportion under the null hypothesis Round your answer to 2 decimal places. Provide your answer below:

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Answer:

The slope is y=-2x.

If solving the equation for the line it is y=-2x-1

the alternative hypothesis will state that the proportion is not significantly different than 0.75.

This should be researched with a hypothesis test on the proportion of tourists visiting Rock City.

The claim that want to be tested is if the proportion has increased from the past proportion (π=0.75).

Then, the alternative hypothesis will state that the proportion is significantly higher than 0.75.

On the contrary, the alternative hypothesis will state that the proportion is not significantly different than 0.75.

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Answer:

All the best

The answer is fully correct

The probability that the marksman hits the target at most 13 times is 11.57.

Define binomial probability.

The likelihood of precisely x successes on n further trials in an experiment with two possible outcomes is known as a binomial probability (commonly called a binomial experiment). A random variable's binomial probability is indicated by the notation X B(n, p) (where the prefix stands for "has distribution..."). The distribution's parameters are known as n and p.

The binomial probability is nCx⋅px⋅(1−p)n−x when the probability of success on a given trial is p. In this case, nCx denotes the number of various combinations of x objects chosen from a set of n objects.

Solution Explained:

Given,
P(HT) = 0.89

so if we multiply the chances of hitting he target with 13, we get

P(H13) = 0.89 X 13 = 11.57

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Answer: 32

Step-by-step explanation:

50% = x/2

50% of 64 = 64/2

50%=32

Answer:  The correct answer is 32 tickets were early admission

Step-by-step explanation:

T = Total tickets sold

E = Early admission tickets

If 50% of the total tickets are for early admission, then our equation would be:

E = T * 50% (or .5)

Substitute 64 for the total tickets

E = 64 * .5

E = 32

The solution to the system equation  is (x, y, z, w) = (23, 12, 3, 1).

What is equation?

An equation could be a formula that expresses the equality of 2 expressions, by connecting them with the sign =

Main body:

Here is a system of linear equations that represents the situation.

x +5y +10z +20w = 133 . . . total amount earned

x +y +z +w = 39 . . . . . . . . . total number of bills

y = 4z . . . . . . . . . . . . . . . . . . the number of 5s is 4 times the number of 10s

x = 2y -1 . . . . . . . . . . . . . . . . the number of 1s is 1 less than twice the number of 5s

_____

We can substitute for x and z in the first two equations:

... (2y-1) +5y +10(y/4) +20w = 133

... (2y-1) +y +(y/4) +w = 39

These simplify to

... 9.5y +20w = 134

... 3.25y +w = 40

Solving by your favorite method, you get

... y = 12

... w = 1

So the other values can be found to be

... x = 2·12 -1 = 23

... z = 12/4 = 3

hence ,The solution to the system is (x, y, z, w) = (23, 12, 3, 1).

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It will take 51.9 seconds to return to sea level.

The rate of change of position of an object in any direction. Speed is measured as the ratio of distance to the time in which the distance was covered.

Using the speed - distance relationship, the time taken for the scientist to travel back to sea level would be 51.9 seconds

Distance = Speed × time

First travel :

Distance covered = 90 × 3.5 = 315 feet

Second travel :

Distance covered = 30 × 2.2 = 66 feet

Net change in position from sea level :

(315 - 66) feet = 249 feet

Maximum speed = 4.8 feet per second

Time taken = Distance / speed

Time taken = 249 ÷ 4.8 = 51.875 seconds

Hence, it will take 51.9 seconds to return to sea level.

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There are 201376 ways can a group of 5 lucky students be selected to sit in the front of the classroom.

What are combinations?

Combinations are mathematical operations that count the variety of configurations that can be made from a set of objects, where the order of the selection is irrelevant. You can choose any combination of the things in any order.

Permutations and combinations are often mistaken. The chosen components' order is crucial in permutations, though. For instance, whereas permutations treat the arrangements differently, combinations treat the arrangements ab and ba equally (as one arrangement).

Use the combinations formula:

n = 32, r=5

[tex]^{n}C_{r}=\frac{n!}{(n-r)! r!}[/tex]

[tex]^{32}C_{5}=\frac{32!}{(32-5)! 5!}[/tex]

       = (32*31*30*29*28*27!)/(5!*27!)

       = (32*31*30*29*28)/ (5*4*3*2*1)

       = 201376

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The probability that the couple will have more than 2 girls is 1/2.

Here, we are given that a couple is planning to have 3 children.

Let us list down all the possible combination of outcomes-

GGG, GGB, GBB, BBB

Here G stands for a girl and B for a boy.

Thus, there are a total of 4 outcomes. We need to find the probability that the number of girls is greater than or equal to 2.

Out of the listed outcomes, 2 combinations- GGG and GGB have number of girls ≥ 2.

We know that probability = Number of favorable outcomes/ total number of outcomes

P(x ≥ 2) = 2/4

= 1/2

Thus, the probability that the couple will have more than 2 girls is 1/2.

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The greatest possible error for her actual measurement 0.5 pounds

What is a mistake, exactly?

A mistake is an inaccurate or incorrect action (the Latin word error, which means "wandering"). Sometimes the terms "mistake" and "error" are used synonymously. The Latin word "errare," which means "to stray," is the source of the etymology.

Given:

Beatriz calculated the error range to be 23.5 to 24.5 pounds.

How much room did she have for error in her actual measurement?

Solution:

We initially locate the median of the supplied numbers before calculating the margin of error.

23.5 and 24.5 as the median

[tex]M=\frac{23.5+34.5}{2}= \frac{48}{2}=24[/tex]

Therefore, the largest inaccuracy is 24.5-24 = 0.5 pounds.

As a result, 0.5 pounds is accurate.

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As per the concept of z - score, the percentile would this client's score is 0.4772

Z - score:

In statistics, z - score is also termed the standard score, is used to determine how much each data point position is away from its mean. Where as in other words, it measures the deviation of x (data point) in terms of the standard deviations. Here the percentage of the population above or below the score can be obtained using z tables.

Given,

A test has a mean of 100 and a standard deviation of 15. a client scores 130 on the test.

Here we need to find at what percentile (rounded off) would this client's score place her.

As per the formula of z score, here we have the values,

mean = 100

standard deviation = 15

Score = 130

Therefore, the z score is calculated as,

=> z score = (130 - 100) / 15

=> z score = 30 / 15

=> z score = 2

According to the z score table the resulting value is 0.4772.

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The shape of the molecule having formula ab2 has been linear or trigonal planar. Hence, the correct option will be e) linear or trigonal planar

The hybridization is been given as the combining of the valence shells with the formation of the given hybridized shell which have equal energy distribution.

The molecule that has A as a central atom, and B bonded to the fixed central atom. The molecular hybridization of the given molecule will be sp hybridization and that accounts for the linear shape of the molecule according to VSEPR theory.

Hybridization of A having the lone pair, and the shape of the molecule will be trigonal planar.

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Answer:

B is in the 1st quadrant

J is slightly in the 2nd quadrant

T is in the 3rd quadrant

P is in the 4th quadrant

Positions:

B.) 4, 3

J.) 0, 7

T.) -4, -4

P.) 4, -6

When you are looking at an axis graph, picture a big "C" over it, the tip of the "C" is the 1st quadrant, as you go down the "C" it symbolizes each quadrant. When the point is only on one line or axis (depending on which axis), then it is, for example, (0, 3) or (3, 0).

The best buy will be;

''28 pounds for $49.56''

What is Division method?

Division method is used to distributing a group of things into equal parts. Division is just opposite of multiplications. For example, dividing 20 by 2 means splitting 20 into 2 equal groups of 10.

Given that;

The two options are,

16 pounds for $31.68

28 pounds for $49.56

Now,

Since, The two options are,

16 pounds for $31.68

28 pounds for $49.56

Hence, The unit rate for each are;

Since, The cost of 16 pounds = $31.68

Hence, The cost of 1 pounds = $31.68/16

                                            = $1.98

And, The cost of 28 pounds = $49.56

Hence, The cost of 1 pounds = $49.56/28

                                            = $1.77

Thus, The best buy will be;

''28 pounds for $49.56''

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The volume of the solid S in the given question is 5.48unit³.

A three-dimensional space's occupied volume is measured.

It is frequently expressed numerically in a variety of imperial or US-standard units as well as SI-derived units.

The definition of length and volume are connected.

So, the volume of the solid S:
An equilateral triangle's sides are shown as a cross-section.

An equilateral triangle's height is determined by:

[tex]h = sSin60 = \frac{\sqrt{3} }{2} s[/tex]

Consequently, one triangle's area is:

[tex]A=\frac{1}{2} s h=\frac{1}{2} s \cdot \frac{\sqrt{3}}{2} s=\frac{\sqrt{3}}{4} s^2[/tex]

The line equation that depicts the diagonal is:

[tex]\begin{aligned}& x+y=1 \\& y=-x+1 \\& x=-y+1\end{aligned}[/tex]

This will indicate the s value integrate from 0 to 2 if we integrate along the y-axis.

[tex]\begin{aligned}& V=\int_0^2 \frac{\sqrt{3}}{4} s^2 d x \\& =\frac{\sqrt{3}}{4} \int_0^2(-y+1)^2 d x \\& =\frac{\sqrt{3}}{4} \int_0^2\left(y^2-2 y+1\right) d x \\& =\frac{\sqrt{3}}{4}\left[\frac{1}{3} y^3-y^2+y\right] \\& \left.=\frac{\sqrt{3}}{4}\left[\frac{1}{3}(2)^3-(2)^2+2\right)\right] \\& =5.48\end{aligned}[/tex]

Therefore, the volume of the solid S in the given question is 5.48unit³.

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Correct question:
Find the volume V of the described solid S. The base of S is the triangular region with vertices (0, 0), (2, 0), and (0, 2). Cross-sections perpendicular to the y-axis are equilateral triangles.

The required value of V for the given equation is 115.34

What is a simple equation?

A simple equation is a mathematical formula that, on both sides of the "equal to" sign, expresses the relationship between two expressions. It primarily consists of a variable, sometimes with a numerical constant in addition. There cannot be a simple equation without a variable. A known, fixed quantity that is used in a straightforward equation is referred to as a constant.

Given an expression, V/7.9 equals 14.6

or, V/7.9 = 14.6

or, V = 14.6*7.9

or, V = 115.34

Hence, the required value of V is 115.34

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The bottom of the ladder is at 8 feet from the wall.

The position of the ladder will form a right angled triangle with the wall. The perpendicular is 15 feet. Let the distance between ladder and wall be x. Thus, the length of the ladder will be 2x + 1. Forming a equation according to Pythagoras theorem.

(2x + 1)² = x² + 15²

Expanding the equation

4x² + 1 + 4x = x² + 225

Rewriting the equation

4x² - x² + 4x = 225 - 1

Performing subtraction

3x² + 4x = 224

3x² + 4x - 224 = 0

Factorizing the quadratic equation, we get -

x = -9.3 and 8

The distance can not be negative. Hence, the value of x is 8.

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a) The student's work is False and the strategy is incorrect as the fraction should be 1/16

b) The fraction 1/16  as decimal is 0.0625

What is an Equation?

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given data ,

Let the number be = A

Now the value of A is A = 6 1/4 %

The value of A = 6.25 %

Now , the value of 6.25 % = 6.25 / 100

The fractional form can be substituted by dividing the numerator and denominator by 25 , we get

A = 6.25 / 100

A = ( ( 6.25 ) / 25 ) / ( 100 / 25 )

A = ( 0.25 ) / 4

The value of A = 0.25 / 4

The value of A = 0.0625

And it can be represented in the fractional form as

A = 0.25 / 4

The value of 0.25 = 1/4

Substituting the value of 0.25 in the equation , we get

A = ( 1/4 ) / 4

A = 1/16

Therefore , the value of A = 1/16

b)

The decimal from of the number 1/16 is 0.0625

The mistake the student did was while dividing the number by 100 to convert the percentage , the student evaluated the number 625 instead of 6.25 , so after avoiding the error , we get the fraction as 1/16

The decimal from of the number 1/16 is 0.0625

Hence ,

a) The student's work is False and the strategy is incorrect as the fraction should be 1/16

b) The fraction 1/16  as decimal is 0.0625

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The airplane with 17 gallons will weight 746.3 pounds.

With the unitary method, you first determine the value of a single unit before determining the value of the necessary quantity of units. What can values and units be?

Consider going to the store to buy 6 apples. You are informed by the merchant that 10 apples are available for Rs 100. In this scenario, the units are apples and the value is their cost. It's crucial to understand the units and values while using the unitary technique to solve a problem.

Airplane weight with 50 gallons = 2195 pounds

So, airplane weight with 1 gallons = 2195 / 50

                                                        = 43.9 pounds

Hence,

Airplane with 17 gallons = 17 * 43.9 pounds

                                        = 746.3 pounds

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The estimate value of the standard deviation of the population ( manufacturing process) is 0.125..

The standard deviations is estimated to be one fourth of the sample range (as most of data values are within two standard deviations of the mean).

We have given that,

A sample of manufacturing process.

Sample size, n = 5

Mean of sample ranges = 0.50

we have to calculate the estimate of standard deviations of population.

thus , we estimate the standard deviations as fourth of the mean of the sample ranges is

S = Mean of sample ranges/4

=> S = 0.50/4

=> S = 0.125

Hence, the standard deviation of the population is estimated as 0.125..

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Answer:

Solution = (5, 2)

Step-by-step explanation:

Step 1: Solve for y in 9x + 2y = 49

    1. 9x + 2y = 49 → 2y = -9x + 49 → y = -4.5x + 24.5

Step 2: Substitute -4.5x + 24.5 for y in -3x + 5y = -5

    2. -3x + 5(-4.5x + 24.5) = -5

Step 3: Solve for x

    3. -3x - 22.5x + 122.5 = -5 → -25.5x + 122.5 = -5 → -25.5x = -127.5 → x = 5

Step 4: Substitute 5 for x in -3x + 5y = -5

    4. -3(5) + 5y = -5 → -15 + 5y = -5 → 5y = 10 → y = 2

Hope this helps :)

Answer:

4x + 15

Step-by-step explanation:

1. Add the numbers

4x + 2(2 + 5) + 1

4x + 2(7) + 1

2. Multiply the numbers

4x + 2(7) + 1

4x + 14 + 1

3. Add the numbers

4x + 14 + 1

4x + 15

Solution:

4x + 15

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Answer: 195/22 or 8[tex]\frac{19}{22}[/tex]

Step-by-step explanation:

The probabilities are given as follows:

The probabilities are called identifying the desired outcomes from the distribution of the number of children per parent.

Hence the probability of fewer than 4 siblings is of:

P(X < 4) = P(X = 1) + P(X = 2) + P(X = 3) = 0.266 + 0.322 + 0.149 = 0.737.

The probability of at least 5 siblings is of:

P(X >= 5) = P(X = 5) + P(X = 6) + P(X > 6) = 0.059 + 0.032 + 0.02 = 0.111.

The probability that the student is not an only child is given as follows:

P(X > 1) = 1 - P(X = 1) = 1 - 0.266 = 0.734.

The distribution is given as follows:

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The Wason selection test measures a person's ability to recognize information that challenges a certain hypothesis, in this case, a type of conditional hypothesis. if P, then Q.

Given,

Wason's Card;

A popular tool for studying problem resolution that was developed in 1966 by English psychologist Peter C(athcart) Wason (1924–2003). The uppermost faces of the four cards, which are arranged on a table, display the letters and numerals E, K, 4, and 7.

What is demonstrated by the Wason selection task?

As a result, the Wason selection test gauges how well people can spot evidence that refutes a certain hypothesis, in this case, a conditional hypothesis of the type. P, then Q if.

For example;-

The majority of people have no trouble choosing the proper cards ("16" and "drinking beer") if the rule is "If you are drinking alcohol, then you must be over 18" and the cards contain an age and beverage on one side, respectively.

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The equivalent expressions of each number are respectively;

1) 2(x - 2) = -7 + 6x - 4x + 3

2) (x + 14) - (8 - 2x) = 9x - 2(3x - 3)

3) 3(x + 5) = -2x + 9 + 5x + 6

4) -4(x + 1) + 5x =  (7 - 2x) + (3x - 11)

5) (7 + 5x) + (4x - 1) = -3x + 6 + 4x

The properties of algebra include associative property, distributive property, Identity property, Inverse property, e.t.c.

Now, let us simplify the terms on the right;

a) (7 - 2x) + (3x - 11)

Expanding the brackets gives us;

7 - 2x + 3x - 11

= x - 4

This can also be expressed as;

-4(x + 1) + 5x

b) -7 + 6x - 4x + 3

Simplifying gives;

2x - 4

= 2(x - 2)

c) 9x - 2(3x - 3)

Simplifying gives;

9x - 6x + 6

3x + 6

It can also be written as;

(x + 14) - (8 - 2x)

d) -3x + 6 + 4x

Simplifying gives;

x + 6

This can also be expressed as;

(7 + 5x) + (4x - 1)

e) -2x + 9 + 5x + 6

This can also be expressed as;

3x + 15

3(x + 5)

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Answer:

[tex]\boxed{4.} \quad (7-2x)+(3x-11)[/tex]

[tex]\boxed{1.} \quad -7+6x-4x+3[/tex]

[tex]\boxed{2.} \quad 9x-2(3x-3)[/tex]

[tex]\boxed{5.} \quad -3x+6+4x[/tex]

[tex]\boxed{3.} \quad -2x+9+5x+6[/tex]

Step-by-step explanation:

Simplify the given expressions numbered 1 through 5:

[tex]\begin{aligned}\textsf{1.} \quad 2(x-2)&=2 \cdot x + 2 \cdot (-2)\\&=2x-4\end{aligned}[/tex]

[tex]\begin{aligned}\textsf{2.} \quad (x+14)-(8-2x)&=x+14-8+2x\\&=x+2x+14-8\\&=3x+6\end{aligned}[/tex]

[tex]\begin{aligned}\textsf{3.} \quad 3(x+5)&=3 \cdot x + 3 \cdot 5\\&=3x+15\end{aligned}[/tex]

[tex]\begin{aligned}\textsf{4.} \quad -4(x+1)+5x&=-4 \cdot x -4 \cdot 1+5x\\&=-4x-4+5x\\&=5x-4x-4\\&=x-4\end{aligned}[/tex]

[tex]\begin{aligned}\textsf{5.} \quad (7+5x)+(-4x-1)&=7+5x-4x-1\\&=5x-4x+7-1\\&=x+6\end{aligned}[/tex]

Simplify the given answer expressions:

[tex]\begin{aligned}(7-2x)+(3x-11)&=7-2x+3x-11\\&=3x-2x+11-11\\&=x-4\end{aligned}[/tex]

[tex]\begin{aligned}-7+6x-4x+3&=6x-4x+3-7\\&=2x-4\end{aligned}[/tex]

[tex]\begin{aligned}9x-2(3x-3)&=9x-2 \cdot 3x-2 \cdot (-3)\\&=9x-6x+6\\&=3x+6\end{aligned}[/tex]

[tex]\begin{aligned}-3x+6+4x&=4x-3x+6\\&=x+6\end{aligned}[/tex]

[tex]\begin{aligned}-2x+9+5x+6&=5x-2x+9+6\\&=3x+15\end{aligned}[/tex]

Therefore, the matching pairs of equivalent expressions are:

[tex]\boxed{4.} \quad (7-2x)+(3x-11)[/tex]

[tex]\boxed{1.} \quad -7+6x-4x+3[/tex]

[tex]\boxed{2.} \quad 9x-2(3x-3)[/tex]

[tex]\boxed{5.} \quad -3x+6+4x[/tex]

[tex]\boxed{3.} \quad -2x+9+5x+6[/tex]