A recent survey asked 1200 randomly selected U.S. adults if they believe that the U.S. federal government is doing enough to keep U.S. elections safe from outside interference. After analyzing the results, the researchers were able to state that they are 95% confident that between 52.5% and 59.5% of all U.S. adults believe that the U.S. federal government is not doing enough to keep U.S. elections safe. Which statement BEST describes how to interpret these results

Answer:

The null hypothesis [tex]\mathtt{H_0 : \mu = 26500}[/tex]

The alternative hypothesis [tex]\mathtt{H_1 : \mu \neq 26500}[/tex]

Step-by-step explanation:

The summary of the given statistics is:

Population Mean = 26,500

Sample Mean = 30,150

Standard deviation = 10560

sample size = 24

The objective is to state the null hypothesis and the alternate hypothesis.

An hypothesis is a claim with  insufficient information which tends to be challenged into  further testing and experimentation in order to determine if such claim is significant or not.

The null hypothesis is a default hypothesis where there is no statistical significance between the two variables in the hypothesis.

The alternative hypothesis is the research hypothesis that the  researcher is trying to prove.

The null hypothesis [tex]\mathtt{H_0 : \mu = 26500}[/tex]

The alternative hypothesis [tex]\mathtt{H_1 : \mu \neq 26500}[/tex]

The test statistic can be  computed as follows:

[tex]z = \dfrac{\overline X - \mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]

[tex]z = \dfrac{30150 - 26500}{\dfrac{10560}{\sqrt{24}}}[/tex]

[tex]z = \dfrac{3650}{\dfrac{10560}{4.8989}}[/tex]

[tex]z = \dfrac{3650 \times 4.8989 }{{10560}}[/tex]

z = 1.6933

Answer:

For the null hypothesis to be rejected , then the conclusion of the test is that the absolute values of the z-statistic and/or the t-test statistic is greater than the critical value

Step-by-step explanation:

Here, we want to explain the conclusion of the test given that the null hypothesis is rejected.

Mathematically, the null hypothesis is as expressed as below;

H0: μ = 1,200

The alternative hypothesis H1 would be;

H1: μ > 1,200

Now, before we can reject or accept the null hypothesis, we will need a sample size and thus calculate the test statistics and the z statistics

For us to reject the null hypothesis, one of two things, or two things must have occurred.

The absolute value of the z statistic |z| or the test statistic |t| must be greater than the critical value.

If this happens, then we can make a rejection of the null hypothesis

Answer:

E(x)  [tex]= \frac{n+1}{2}[/tex]

Var(x) [tex]= \frac{1}{4} [ n - 1 ][/tex]

Step-by-step explanation:

Hint x = 1 + x1 + ......... Xn-1

[tex]X_{i} = \left \{ {{1} if the ith adjacent pair of magnets repel each other \atop {0} if ith adjacent pair of magnets join} \right.[/tex]

attached below is the detailed solutioN

usually like poles of magnets repel each other and unlike poles of magnets attract each other forming a block

Now, it look like there is some information missing in the answer. The whole problem should look like this:

Alicia Keys's new album As I Am is climbing the charts, and the manager of Tip Top Tunes expects to sell a lot of copies. Because she has limited shelf space, she can't put out all her copies of the CD at once. On Monday morning, she stocked the display with 40 copies. By the end of the day, some of the copies had been sold. On Tuesday morning, she counted the number of copies left and then added that many more to the shelf. In other words, she doubled the number that was left in the display. At the end of the day, she discovered that she had sold the exact same number of copies as had been sold on Monday. On Wednesday morning, the manager decided to triple the number of copies that had been left in the case after Tuesday. Amazingly, she sold the same number of copies on Wednesday as she had on each of the first two days! But this time, at the end of the day the display case was empty. How many copies of the As I Am CD did she sell each day?

Answer:

She sold 24 copies of the cd each day.

Step-by-step explanation:

In order to solve this problem we must first set our variable up. In this case, since we need to know what the number of sold cd's per day is, that will just be our variable:

x= Number of copies sold.

So we can start setting our equation up. So we take the first part of the problem:

"On Monday morning, she stocked the display with 40 copies. By the end of the day, some of the copies had been sold."

This can be translated as:

40-x

where this expression represents the number of copies left on the shelf by the end of monday.

"On Tuesday morning, she counted the number of copies left and then added that many more to the shelf."

so we represent it like this:

(40-x)+(40-x)

"In other words, she doubled the number that was left in the display."

so the previous expression can be simplified like this:

2(40-x)

"At the end of the day, she discovered that she had sold the exact same number of copies as had been sold on Monday."

so the expression now turns to:

2(40-x)-x   this is the number of copies left by the end of tuesday.

"On Wednesday morning, the manager decided to triple the number of copies that had been left in the case after Tuesday."

this translates to:

3[2(40-x)-x]

This is the number of copies on the shelf by the begining of Wednesday.

"Amazingly, she sold the same number of copies on Wednesday as she had on each of the first two days! But this time, at the end of the day the display case was empty."

this piece of information lets us finish writting our equation:

3[2(40-x)-x] -x = 0

since there were no copies left on the shelf, then the equation is equal to zero.

So now we proceed and solve the equation for x:

3[2(40-x)-x] -x = 0

We simplify it from the inside to the outside.

3[80-2x-x]-x=0

3[80-3x]-x = 0

we now distribute the 3 so we get:

240-9x-x=0

we combine like terms so we get:

240-10x=0

we move the 240 to the other side of the equation so we get:

-10x=-240

and divide both sides into -10 so we get:

x=24

so she sold 24 copies each day.

We have

[tex]2+\dfrac54+\dfrac{11}{16}+\dfrac{23}{64}+\cdots=\displaystyle\sum_{n=0}^\infty\frac{3\cdot2^n-1}{4^n}[/tex]

(notice that each denominator is a power of 4, and each numerator is one less than some multiple of 3, in particular 3 times some power of 2)

Recall for [tex]|x|<1[/tex], we have

[tex]\displaystyle\frac1{1-x}=\sum_{n=0}^\infty x^n[/tex]

So we have

[tex]\displaystyle\sum_{n=0}^\infty\frac{3\cdot2^n-1}4=3\sum_{n=0}^\infty\left(\frac12\right)^n-\sum_{n=0}^\infty\left(\frac14\right)^n=\frac3{1-\frac12}-\frac1{1-\frac14}=\boxed{\frac{14}3}[/tex]

Complete Question

The complete question is shown on the first uploaded image

Answer:

the null hypothesis is  [tex]H_o : \mu = 122[/tex]

the alternative hypothesis is [tex]H_a : \mu \ne 122[/tex]

The test statistics is  [tex]t = - 1.761[/tex]

The p-value is  [tex]p = P(Z < t ) = 0.039119[/tex]

so

    [tex]p \ge 0.01[/tex]

Step-by-step explanation:

From the question we are told that

   The population mean is  [tex]\mu = 122[/tex]

     The sample size is  n=  38

    The sample mean is  [tex]\= x = 116 \ feet[/tex]

     The standard deviation is [tex]\sigma = 21[/tex]

Generally the null hypothesis is  [tex]H_o : \mu = 122[/tex]

                the alternative hypothesis is [tex]H_a : \mu \ne 122[/tex]

Generally the test statistics is mathematically evaluated as

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

substituting values

         [tex]t = \frac { 116 - 122 }{\frac{ 21 }{ \sqrt{ 38} } }[/tex]

         [tex]t = - 1.761[/tex]

The p-value is mathematically represented as

      [tex]p = P(Z < t )[/tex]

From the z- table  

     [tex]p = P(Z < t ) = 0.039119[/tex]

So  

     [tex]p \ge 0.01[/tex]

Answer: 1023.75 (a)

Step-by-step explanation:

The sequence is a Geometric progression with the common ratio of ¹/₂ and first term of 512.

a = 512, r = ¹/₂. To determine the ratio, just divide the second term by the first term.

Now to calculate the sum, we consider two formula here and select the one that is most appropriate,

(1)  a( rⁿ - 1 )/r - 1, when r is greater than 1

(2) a( 1 - rⁿ )/1 - rⁿ, when r is less than 1.

In this question, formula 2 shall be appropriate because r is less than 1.

so,

S₁₂      =  512( 1 - 0.5¹² )/1 - 0.5

             512( 1 - 2.44 ₓ 10⁻⁴ )/0.5

          = 512( 0,9998 )/0.5

          = 511.875/0.5

          = 1023.75

The answer is a

Answer:

45

62x

______

  90

2700+

_________

2790

Step-by-step explanation:

Answer:

-7/6

Step-by-step explanation:

-2/3 x 7/4 = -14/12 = -7/6

Answer: -7/6

Step-by-step explanation: (-2/3) ÷ (4/7) can be rewritten as (-2/3) · (7/4).

Remember that dividing by a fraction is the same thing

as multiplying by the reciprocal of the fraction.

Before multiplying however, notice that we

can cross-cancel the 2 and 4 to 1 and 2.

So multiplying across the numerators and denominator and

remembering our negative in the first fraction, we have -7/6.

Answer:

See answer and graph below

Step-by-step explanation:

∬Ry2x2+y2dA

=∫Ry.2x.2+y.2dA

=A(2y+4Ryx)+c

=∫Ry.2x.2+y.2dA

Integral of a constant ∫pdx=px

=(2x+2.2Ryx)A

=A(2y+4Ryx)

=A(2y+4Ryx)+c

The graph of y=A(2y+4Ryx)+c assuming A=1 and c=2

The evaluation of the double integral is [tex]\mathbf{ \dfrac{105}{2}\pi }[/tex]

The double integral [tex]\mathbf{\int \int _R\ \dfrac{y^2}{x^2+y^2} \ dA}[/tex], where R is the region that lies between

the circles [tex]\mathbf{x^2 +y^2 = 16 \ and \ x^2 + y^2 = 121}[/tex].

Let consider x = rcosθ and y = rsinθ because x² + y² = r²;

Now, the double integral can be written in polar coordinates as:

[tex]\mathbf{\implies \int \int _R\ \dfrac{y^2}{x^2+y^2} \ dxdy}[/tex]

[tex]\mathbf{\implies \int \int _R\ \dfrac{r^2 \ sin^2 \theta}{r^2} \ rdrd\theta}[/tex]

[tex]\mathbf{\implies \int \int _R\ \ sin^2 \theta \ r \ drd\theta}[/tex]

Thus, the integral becomes:

[tex]\mathbf{=\int^{2 \pi}_{0} sin^2 \theta d\theta \int ^{11}_{4} rdr }[/tex]

∴

[tex]\mathbf{=\int^{2 \pi}_{0} \dfrac{1-cos 2 \theta }{2} \ \theta \ d\theta\dfrac{r}{2} \Big|^{11}_{4}dr }[/tex]  

[tex]\mathbf{\implies \dfrac{1}{2} \Big[\theta - \dfrac{sin \ 2 \theta}{2}\Big]^{2 \pi}_{0} \ \times\Big[ \dfrac{11^2-4^2}{2}\Big]}[/tex]

[tex]\mathbf{\implies \dfrac{\pi}{2} \times\Big[ 121-16\Big]}[/tex]

[tex]\mathbf{\implies \dfrac{105}{2}\pi }[/tex]

Learn more about double integral here:

https://brainly.com/question/19756166

Answer:

30,455

Step-by-step explanation:

Exponential decay

y = a(1 - b)^x

y = final amount

a = initial amount

b = rate of decay

x = time

We are looking for the rate of decay, b.

900 = 450000(1 - b)^30

1 = 500(1 - b)^30

(1 - b)^30 = 0.002

1 - b = 0.002^(1/30)

1 - b = 0.81289

b = 0.1871

The equation for our case is

y = 450000(1 - 0.1871)^x

We are looking for the amount in 13 hours, so x = 13.

y = 450000(1 - 0.1871)^13

y = 30,455

Step-by-step explanation:

x=rcosθandy=rsinθ,. 7.7. r2=x2+y2andtanθ=yx. 7.8. These formulas can be used to convert from rectangular to polar or from polar to rectangular coordinates.

Answer:

9 hours

Step-by-step explanation:

Since the group of men remains the same, number of hours is proportional to number of radios.

1300/26 = 450/h

h = 26 * 450 / 1300 = 9 hours

Answer:

See below.

Step-by-step explanation:

1 gal = 4 qt

With a full gallon oil tank, you can fill 4 1-qt bottles.

The problem does not mention the number of quarts that go in a case, so there is not enough information to answer the question.

Also, is the full tank really only 1 gallon, or is there a number missing there too?

4) 2x-2y+3 > 0

although it is spelt "26" on the choices

Answer: Transitive property.

Step-by-step explanation:

First, for the equality we have:

Reflexive:

  For all real numbers x, x = x.

Symmetric:  

 For all real numbers x, y

 if x= y, then y = x.

Transitive:

 For reals x, y and z.

 if x = y, and y = z, then x = z.

Now, let's talk about inequalities.

first, the reflexive property will say that:

x > x.

This has no sense, so this property does not work for inequalities.

Now, the reflexive.

If x > y, then y > x.

Again, this has no sense, if x is larger than y, then we can never have that y is larger than x. This property does not work for inequalities.

Not, the transitive property.

if x > y, and y > z, then x > z.

This is true.

x is bigger than y, and y is bigger than z, then x should also be bigger than z.

x > y > z.

And this also works for the inverse case:

x < y and y < z, then x < z.

So the correct option is transitive property.

Answer:

28

Step-by-step explanation:

We need to find the value of [tex]\Sigma_{x=0}^3\ 2x^2[/tex]

We know that,

[tex]\Sigma n^2=\dfrac{n(n+1)(2n+1)}{6}[/tex]

Here, n = 3

So,

[tex]\Sigma n^2=\dfrac{3(3+1)(2(3)+1)}{6}\\\\\Sigma n^2=14[/tex]

So,

[tex]\Sigma_{x=0}^3\ 2x^2=2\times 14\\\\=28[/tex]

So, the value of [tex]\Sigma_{x=0}^3\ 2x^2[/tex] is 28. Hence, the correct option is (d).

Answer:

[tex](f/g)(x) = \frac{x + 5}{3x + 5} [/tex]

Step-by-step explanation:

f(x) = 3x² + 10x - 25

g(x) = 9x² - 25

To find (f/g)(x) divide f(x) by g(x)

That's

[tex](f/g)(x) = \frac{3 {x}^{2} + 10x - 25 }{9 {x}^{2} - 25 } [/tex]

Factorize both the numerator and the denominator

For the numerator

3x² + 10x - 25

3x² + 15x - 5x - 25

3x ( x + 5) - 5( x + 5)

(3x - 5 ) ( x + 5)

For the denominator

9x² - 25

(3x)² - 5²

Using the formula

a² - b² = ( a + b)(a - b)

(3x)² - 5² = (3x + 5)(3x - 5)

So we have

[tex](f/g)(x) = \frac{(3x - 5)(x + 5)}{(3x + 5)(3x - 5)} [/tex]

Simplify

We have the final answer as

[tex](f/g)(x) = \frac{x + 5}{3x + 5} [/tex]

Hope this helps you

Answer:

The answer is below

Step-by-step explanation:

The box contains 5 red and 4 white balls.

A) The probability that at least 1 ball was​ red = P(both are red) + P(first is red and second is white) + P(first is white second is red)

Given that the first ball was (Upper A )Replaced before the second draw:

P(both are red) = P(red) × P(red) = 5/9 × 5/9 = 25/81

P(first is red and second is white) = P(red) × P(white) = 5/9 × 4/9 = 20/81

P(first is white and second is red) = P(white) × P(red) = 4/9 × 5/9 = 20/81

The probability that at least 1 ball was​ red = 25/81 + 20/81 + 20/81 = 65/81

B) The probability that at least 1 ball was​ red = P(both are red) + P(first is red and second is white) + P(first is white second is red)

Given that the first ball was not Replaced before the second draw:

P(both are red) = P(red) × P(red) = 5/9 × 4/8 = 20/72 (since it was not replaced after the first draw the number of red ball remaining would be 4 and the total ball remaining would be 8)

P(first is red second is white) = P(red) × P(white) = 5/9 × 4/8 = 20/72

P(first is white and second is red) = P(white) × P(red) = 4/9 × 5/8 = 20/72

The probability that at least 1 ball was​ red = 20/72 + 20/72 + 20/72 = 60/72

Answer:

Step-by-step explanation:

From the given information;

let the hypotenuse be a , the opposite which is the north direction be b and the west direction which is the adjacent be c

SO, using the Pythagoras theorem

a² = c² + 177²

By taking the differentiation of both sides with respect to time t , we have

[tex]2a \dfrac{da}{dt} = 2c \dfrac{dc}{dt} + 0[/tex]

[tex]\dfrac{da}{dt} = \dfrac{c}{a} \dfrac{dc}{dt}[/tex]

At c = 71 miles,[tex]a = \sqrt{ (71)^2 +(177)^2}[/tex]

[tex]a = \sqrt{ 5041+31329}[/tex]

[tex]a = \sqrt{ 36370}[/tex]

a = 190.71

SImilarly, [tex]\dfrac{dc}{dt} = \ 31 miles \ / hr[/tex]

Thus, the rate of change of the distance between Morristown and the van when the van has been travelling for 71 miles can be calculate as:

[tex]\dfrac{da}{dt} = \dfrac{c}{a} \dfrac{dc}{dt}[/tex]

[tex]\dfrac{da}{dt} = \dfrac{71}{190.71} \times 31[/tex]

[tex]\dfrac{da}{dt} = 0.37229 \times 31[/tex]

[tex]\mathbf{\dfrac{da}{dt} = 11.54}[/tex]  to the nearest hundredth.

Step-by-step explanation:

C = Amount (A) - Principal (P)

Where

C is the compound interest

To find the amount we use the formula

[tex]A = P ({1 + \frac{r}{100} })^{n} [/tex]

where

P is the principal

r is the rate

n is the period / time

From the question

P = Rs 12, 000

r = 5%

n = 3 years

Substitute the values into the above formula

That's

[tex]A = 12000 ({1 + \frac{5}{100} })^{3} \\ A = 12000(1 + 0.05)^{3} \\ A = 12000 ({1.05})^{3} [/tex]

We have the answer as

Compound interest = 13891.50 - 12000

Hope this helps you

Answer:

The median would be 7.0.

Step-by-step explanation:

The median of a set of numbers means it is the middle number. since this set has 7 numbers you would need to find the number that is in the middle of the set. This would be the 4th number since it is in the middle. 7.0 is your answer.

Step-by-step explanation:

I(S) = aS / (S + c)

As S approaches infinity, S becomes much larger than c.  So S + c is approximately equal to just S.

lim(S→∞) I(S)

= lim(S→∞) aS / (S + c)

= lim(S→∞) aS / S

= lim(S→∞) a

= a

As S approaches infinity, I(S) approaches a.

Answer: B. 25

Step-by-step explanation:

Given: Total books = 625

Number of books can fit in one box = 25

Now, the number of boxes she need to move all of the books = (Total books) ÷ (Number of books can fit in one box )

= 625÷25

= 25

hence, she requires 25 boxes in order to move all of the books.

So, correct option is B. 25.

Answer:

The width is  [tex]w = 282.8[/tex]

Step-by-step explanation:

From the question we are told that

  The sample size is n =  50

  The  population standard deviation is  [tex]\sigma = \$ 1000[/tex]

   The sample size is  [tex]\= x = \$ 15,000[/tex]

Given that the confidence level is  90%  then the level of significance can be mathematically represented as

             [tex]\alpha = 100 - 90[/tex]  

             [tex]\alpha = 10 \%[/tex]  

              [tex]\alpha = 0.10[/tex]

Next we obtain the critical value of  [tex]\frac{\alpha }{2}[/tex] from the  normal distribution table, the value is  

             [tex]Z_{\frac{0.10 }{2} } = 1.645[/tex]

Generally the margin of error is mathematically represented as

               [tex]E = Z_{\frac{0.10}{2} } * \frac{\sigma }{\sqrt{n} }[/tex]

substituting values

                 [tex]E = 1.645 * \frac{1000 }{\sqrt{50 }}[/tex]

=>                [tex]E = 141.42[/tex]

  The width of the 90%  confidence level is mathematically represented as

                      [tex]w = 2 * E[/tex]

substituting values

                       [tex]w = 2 * 141.42[/tex]

                       [tex]w = 282.8[/tex]

Answer:

3x^4+(x^3)y-8x^2y^2+9xy^3-13y^4

Step-by-step explanation:

3x^4+(nothing)=3x^4

x^3y+(nothing)=x^3y

-10x^2y^2=2x^2y^2=-8x^2y^2

9xy^3+(nothing)=0

-4y^4-9y^4=-13y^4

Add it all up and write the terms by descending order of exponent value, and u get my answer.

y = a(x  + 1.5)^2 - 12.5

y intercept is (0,-8) so:-

-8 = a(0+1.5)^2 - 12.5

-8 = 2.25a - 12.5

a =  4.5/ 2.25 = 2

so we have  

y  = 2 ( x +1.5)^2 - 12.5

solving for x  when y = 0:-

(x + 1.5)^2 = 12.5/2 = 6.25

taking sqrt's  x + 1.5  = +/- 2.5

x =  -4,   1

so the x intercepts are (-4,0) and (1,0)

Answer:

y = –1∕4(x – 8)^2 – 1

Step-by-step explanation:

I took the exam and  got  it  right.

Answer:

1) a) [tex]x = \frac{3}{2}\cdot a[/tex], b) [tex]x = 5-3\cdot a[/tex], c) [tex]x = -a[/tex], d) [tex]x = \frac{5}{2}\cdot a[/tex]

2) a) [tex]x = -\frac{3}{4}[/tex], b) [tex]x = -5[/tex], c) [tex]x = 3[/tex]

Step-by-step explanation:

1) a) [tex]5\cdot x - a = x + 5\cdot a[/tex]

[tex]5\cdot x - x = 5\cdot a + a[/tex]

[tex]4\cdot x = 6\cdot a[/tex]

[tex]x = \frac{3}{2}\cdot a[/tex]

b) [tex]4\cdot x + 3\cdot a = 3\cdot x + 5[/tex]

[tex]4\cdot x - 3\cdot x = 5 - 3\cdot a[/tex]

[tex]x = 5-3\cdot a[/tex]

c) [tex]2\cdot (3\cdot x - a) - 4\cdot (x-a) = 3\cdot (x+a)[/tex]

[tex]6\cdot x -2\cdot a -4\cdot x +4\cdot a = 3\cdot x +3\cdot a[/tex]

[tex]6\cdot x -4\cdot x -3\cdot x = 3\cdot a -4\cdot a +2\cdot a[/tex]

[tex]-x = a[/tex]

[tex]x = -a[/tex]

d) [tex]\frac{2\cdot x}{5} - \frac{x-2\cdot a}{3} = \frac{a}{2}[/tex]

[tex]\frac{6\cdot x-5\cdot (x-2\cdot a)}{15} = \frac{a}{2}[/tex]

[tex]\frac{6\cdot x - 5\cdot x+10\cdot a}{15} = \frac{a}{2}[/tex]

[tex]2\cdot (x+10\cdot a) = 15 \cdot a[/tex]

[tex]2\cdot x = 5\cdot a[/tex]

[tex]x = \frac{5}{2}\cdot a[/tex]

2) a) [tex]\frac{3}{x} + \frac{5}{x+2} = 0[/tex]

[tex]\frac{3\cdot (x+2)+5\cdot x}{x\cdot (x+2)} = 0[/tex]

[tex]3\cdot (x+2) + 5\cdot x = 0[/tex]

[tex]3\cdot x +6 +5\cdot x = 0[/tex]

[tex]8\cdot x = - 6[/tex]

[tex]x = -\frac{3}{4}[/tex]

b) [tex]\frac{7}{x-2} = \frac{5}{x}[/tex]

[tex]7\cdot x = 5\cdot (x-2)[/tex]

[tex]7\cdot x = 5\cdot x -10[/tex]

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

[tex]x = -5[/tex]

c) [tex]\frac{2}{x-3}-\frac{4\cdot x}{x^{2}-9} = \frac{7}{x+3}[/tex]

[tex]\frac{2}{x-3} - \frac{4\cdot x}{(x+3)\cdot (x-3)} = \frac{7}{x+3}[/tex]

[tex]\frac{1}{x-3}\cdot \left(2-\frac{4\cdot x}{x+3} \right) = \frac{7}{x+3}[/tex]

[tex]\frac{x+3}{x-3}\cdot \left[\frac{2\cdot (x+3)-4\cdot x}{x+3} \right] = 7[/tex]

[tex]\frac{2\cdot (x+3)-4\cdot x}{x-3} = 7[/tex]

[tex]2\cdot (x+3) -4\cdot x = 7\cdot (x-3)[/tex]

[tex]2\cdot x + 6 - 4\cdot x = 7\cdot x -21[/tex]

[tex]2\cdot x - 4\cdot x -7\cdot x = -21-6[/tex]

[tex]-9\cdot x = -27[/tex]

[tex]x = 3[/tex]

Step-by-step explanation:

For a left-hand limit, we start at the left side and move right, and see where the function goes as we get close to the x value.

For a right-hand limit, we start at the right side and move left, and see where the function goes as we get close to the x value.

If the two limits are equal, then the limit exists.  Otherwise, it doesn't.

1.  As we approach x = 2 from the left, f(x) approaches -2.

lim(x→2⁻) f(x) = -2

As we approach x = 2 from the right, f(x) approaches 1.

lim(x→2⁺) f(x) = 1

The limits are not the same, so the limit does not exist.

lim(x→2) f(x) = DNE

2. As we approach x = 2 from the left, f(x) approaches 4.

lim(x→2⁻) f(x) = 4

As we approach x = 2 from the right, f(x) approaches 2.

lim(x→2⁺) f(x) = 2

The limits are not the same, so the limit does not exist.

lim(x→2) f(x) = DNE

3. As we approach x = 2 from the left, f(x) approaches 2.

lim(x→2⁻) f(x) = 2

As we approach x = 2 from the right, f(x) approaches 2.

lim(x→2⁺) f(x) = 2

The limits are equal, so the limit exists.

lim(x→2) f(x) = 2

4. As we approach x = 2 from the left, f(x) approaches 2.

lim(x→2⁻) f(x) = 2

As we approach x = 2 from the right, f(x) approaches infinity.

lim(x→2⁺) f(x) = ∞

The limits are not the same, so the limit does not exist.

lim(x→2) f(x) = DNE